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II. Riesz spaces

§ 1. Riesz spaces and fully lattice-ordered spaces

Definition 1 An ordered vector space is said to be a Riesz space (or lattice-ordered vector space)¹ if its order structure is a lattice-ordering (that is, if every pair of elements x, y of E ha… 08JO
Example The space Rᴬ of all real-valued functions defined on any set A is a Riesz space (for the order relation ⟨ x(t) ≤ y(t) for all t ∈ A ⟩); for, any two real-valued functions x, y defi… 08JP
Definition 2 A Riesz space E is said to be fully lattice-ordered if every nonempty subset of E that is bounded above has a supremum in E. 08JQ
Example — 1) If A is any set, the space Rᴬ of real-valued functions defined on A is fully lattice-ordered, the supremum in Rᴬ of a family that is bounded above being its upper envelope (GT… 08JR
Proposition 1 For an ordered vector space E to be fully lattice-ordered, it is necessary and sufficient that E be a Riesz space and that it satisfy one of the following two conditions: ↩ 1 08JS
Proposition 2 — Let E be a Riesz space equipped with a Hausdorff topology that is compatible with its ordered vector space structure (TVS, II, §2, No. 7). If, for every set H ⊂ E that is bounded… 08JT
Proposition 3 Let (E_ι)_ι ∈ I be a family of ordered vector spaces. For the product space E = ∏_ι ∈ I E_ι to be a Riesz space (resp. a fully lattice-ordered space), it is necessary and sufficien… ↩ 1 08JU
Definition 3 Let E be an ordered vector space, V and W two supplementary linear subspaces of E. One says that E is the ordered direct sum of V and W if the canonical mapping (x, y) ↦ x + y of t… 08JV
Proposition 4 For an ordered vector space E to be the ordered direct sum of two supplementary linear subspaces V, W, it is necessary and sufficient that the relations x ∈ V, y ∈ W, x + y ≥ 0 imp… ↩ 1 08JW
Definition 4 In a fully lattice-ordered space E, a linear subspace B of E is said to be a band if it satisfies the following conditions: 1) the relations x ∈ B, y ∈ E and |y| ≤ |x| imply y ∈ B… ↩ 1 08JX
Example In the space Rᴬ of real-valued functions defined on a set A, the set of functions that are zero at all the points of a subset M of A is a band. 08JY
Remark In the space Rᴬ, the subspace B(A) of bounded real-valued functions on A satisfies condition 1) of Def. 4; moreover, for every subset X of B(A) that is bounded above in B(A), the u… 08JZ
Proposition 5 — Let E be a fully lattice-ordered space, A a non-empty subset of E consisting of elements geqslant 0, such that: 1) A + A ⊂ A, and 2) the relations x ∈ A,\ 0 leqslant y leqslant x… ↩ 3 08K0
Theorem 1 (F. Riesz) — Let A be a subset of a fully lattice-ordered space E. The set A' of elements that are alien to every element of A is a band; the band A'' of elements that are alien to every elem… ↩ 6 08K1
Proposition 6 — Let E be a fully lattice-ordered space, M a subset of E, and B the band generated by M. Let M₁ be the set of elements geqslant 0 of E each of which is leqslant some element of th… ↩ 1 08K2
Corollary — Let a be an element of a fully lattice-ordered space E, Bₐ the band generated by a, B'ₐ the band of elements alien to a. For every element x geqslant 0 of E, the component of x i… ↩ 2 08K3

§ 2. Linear forms on a Riesz space

Definition 1 Given an ordered vector space E, a linear form L on E is said to be positive if L(x) ≥ 0 for every x ≥ 0 in E. ↩ 2 08KI
Example 1 Let A be any set, E a linear subspace of the space Rᴬ of all real-valued functions defined on A. For every element a ∈ A, the mapping x ↦ x(a) is a positive linear form on E. ↩ 1 08KJ
Example 2 Let I = [a, b] be a compact interval of R, E the Riesz space formed by the regulated real-valued functions on I (FRV, II, §1, No. 3); the mapping x ↦ ∫ₐᵇ x(t) dt is a positive line… 08KK
Example 3 Let F be any set, U an ultrafilter on F (GT, I, §6, No. 4), E the Riesz space B(F) of bounded real-valued functions on F. For every x ∈ E, lim_U x(t) exists, because x(U) is a base… 08KL
Proposition 1 Let E be an ordered vector space, L a mapping of E into R such that L(x + y) = L(x) + L(y) and such that the relation x ≥ 0 implies L(x) ≥ 0; then L(λ x) = λ L(x) for every scalar… ↩ 1 08KM
Proposition 2 Let E be a real vector space, C a convex cone with vertex 0 in E such that E = C - C, and x ↦ M(x) a mapping of C into R such that M(λ x + μ y) = λ M(x) + μ M(y) for all x ∈ C, y ∈… ↩ 4 08KN
Proposition 3 Let E be a directed ordered vector space, P the set of elements geqslant 0 of E, and x ↦ M(x) a mapping of P into R, with values geqslant 0, such that M(x + y) = M(x) + M(y) for al… ↩ 3 08KO
Definition 2 Given a Riesz space E, a linear form L on E is said to be relatively bounded if, for every x geqslant 0 in E, L is bounded on the set of y ∈ E such that |y| leqslant x. 08KP
Theorem 1 1° In order that a linear form L on a Riesz space E be relatively bounded, it is necessary and sufficient that it be the difference of two positive linear forms. ↩ 3 08KQ
Lemma — Let E be a directed ordered vector space, E^* its dual, ordered by taking as positive elements the positive linear forms. Let (u_α) be an increasing directed family of elements o… 08KR
Proposition 4 — In order that two positive linear forms L, M on a Riesz space E be alien to each other in the space Ω, it is necessary and sufficient that, for every number ε > 0 and every x ≥ 0… ↩ 2 08KS
Proposition 5 — Let L be a positive linear form on a Riesz space E. In order that a positive linear form M on E belong to the band generated by L in Ω, it is necessary and sufficient that, for e… ↩ 2 08KT
Example Let E be a Riesz space equipped with a locally convex topology compatible with its ordered vector space structure (TVS, II, §2, No. 7). Let E' be the topological dual of E, and sup… 08KU

III. Measures on locally compact spaces

§ 1. Measures on a locally compact space

Definition 1 — Let X be a topological space, let E be either overlineR or a vector space over R, and let f be a mapping of X into E. The smallest closed set S in X such that f(x) = 0 on X - S (… 08L4
Proposition 1 Let X be a locally compact space, E a Hausdorff locally convex space. ↩ 3 08L5
Proposition 2 — Let X be a locally compact space, E a Hausdorff locally convex space. ↩ 4 08L6
Corollary Let X be a paracompact, locally compact space; if E is a quasi-complete locally convex space, then the space K(X; E) is quasi-complete. 08L7
Lemma 1 Let X be a locally compact space, K a compact subset of X, and (Vₖ)_1 ≤ k ≤ n a finite covering of K by open sets of X. Then, there exist n continuous mappings fₖ of X into [0, 1]… ↩ 19 08L8
Lemma 2 Let X be a locally compact space, K a compact subset of X, E a locally convex space, q a continuous semi-norm on E, and Φ an equicontinuous set of mappings of X into E whose suppor… ↩ 2 08L9
Proposition 3 — Let X be a locally compact space, X' the compact space obtained by adjoining to X a point at infinity ω (GT, I, §9, No. 8, ↩ 4 08LA
Proposition 4 — Let X be a locally compact space, E a locally convex space; then, the space K(X;E) is dense in C(X;E) for the topology of compact convergence. ↩ 1 08LB
Proposition 5 — Let X be a locally compact space, E a real (resp. complex) locally convex space. For every compact subset K of X, the vector space K(X,K;R) ⊗_R E (resp. K(X,K;C) ⊗_C E) (identifi… ↩ 3 08LC
Definition 2 — A continuous linear form on K(X; C), X a locally compact space, is called a measure (or complex measure) on X. ↩ 3 08LD
Proposition 6 — Let X be a locally compact space, (K_α) a family of compact subsets of X whose interiors overset∘K_α form an open covering of X. For a linear form μ on K(X; C) to be a measure on… ↩ 3 08LE
Proposition 7 For every set S of bounded subsets of K(X;C) that is a covering of K(X;C), the space M_S(X;C) is Hausdorff and quasi-complete. 08LF
Remark The giving of a measure μ on a locally compact space X defines on X (along with the topology of X) a structure S. Let X₁ be a second set, φ a bijective mapping of X onto X₁; in con… 08LG
Example Lebesgue measure on R is invariant under every translation of the additive group R. Indeed, for every function f ∈ K(R; C) and every real number a, we have 08LH
Theorem 1 Every positive linear form on the Riesz space K(X; R) is a (positive) real measure on X. ↩ 7 08LI
Theorem 2 Every real measure on a locally compact space X is the difference of two positive measures. ↩ 2 08LJ
Theorem 3 The space M(X; R) of real measures on a locally compact space X is fully lattice-ordered. 08LK
Proposition 8 If μ is a measure on X then, for every function h ∈ C(X; C), 08LL
Proposition 9 Let V be a linear subspace of K(X; R) having the following property: (P) For every compact subset K of X, there exists a function f ∈ V such that f ≥ 0 and such that f(x) > 0 for a… ↩ 3 08LM
Definition 3 A measure on a locally compact space X is said to be bounded if it is continuous on K(X; C) for the topology of uniform convergence. ↩ 1 08LN
Example — 1) The measure εₐ defined by a unit mass at a point a ∈ X is bounded, and |εₐ| = 1. 08LO
Proposition 10 — For every measure μ on X, ↩ 1 08LP
Corollary 1 — For every measure μ on X, the norms of μ and |μ| are equal; μ is bounded if and only if |μ| is bounded. ↩ 2 08LQ
Corollary 2 — For every measure μ on a compact space X, 08LR
Corollary 3 — For every real measure μ on a locally compact space X, 08LS
Proposition 11 — If μ and ν are two positive measures on X, then | μ + ν | = | μ | + | ν |. ↩ 1 08LT
Corollary 1 — If μ and ν are two positive measures on X such that μ ≤ ν, then | μ | ≤ | ν |; in particular, if ν is bounded then so is μ. Indeed, | ν | = | μ | + | ν - μ |. 08LU
Corollary 2 — For every real measure μ on X, 08LV
Proposition 12 — If μ is a bounded measure on X and if g is a bounded continuous mapping of X into C, then the measure g · μ is bounded and | g · μ | ≤ | g | · | μ |. ↩ 1 08LW
Proposition 13 Let X be a locally compact space and, for every x ∈ X, let εₓ be the Dirac measure at the point x. The mapping x ↦ εₓ is a homeomorphism of X onto a subspace of the space M(X; C) o… ↩ 5 08LX
Proposition 14 In the space M(X; C) of measures on a locally compact space X, the cone M₊(X) of positive measures is complete for the uniform structure deduced from the vague topology (hence is v… ↩ 1 08LY
Corollary If A and B are two vaguely closed subsets of M₊(X), then A + B is vaguely closed in M₊(X) (hence also in M(X; C)). 08LZ
Proposition 15 — Let H be a subset of M(X; C). The following properties are equivalent: ↩ 6 08M0
Corollary 1 — Let ν be a positive measure on X; the set of measures μ such that |μ| ≤ ν is vaguely compact. 08M1
Corollary 2 — The set of measures μ such that |μ| ≤ a (a any finite number > 0) is vaguely compact. ↩ 6 08M2
Corollary 3 — If X is compact, the set of positive measures μ on X such that |μ| = 1 is vaguely compact. ↩ 1 08M3
Corollary 4 — In the space M(X; C), the mapping μ ↦ |μ| is lower semi-continuous for the vague topology. ↩ 1 08M4
Proposition 16 — Let K be a compact subset of X, H a vaguely bounded subset of M(X; C); then, the bilinear form (f, μ) ↦ ⟨ f, μ ⟩ is continuous on K(X, K; C) × H when K(X, K; C) is equipped with… 08M5
Proposition 17 On the space M(X; C), consider the following topologies: T₁: the topology of pointwise convergence in a total subset T of K(X; C); T₂: the vague topology; T₃: the topology of stric… ↩ 4 08M6
Proposition 18 On the cone M₊(X), the topologies induced by the following topologies coincide: T₁: the topology of pointwise convergence in a linear subspace V of K(X; C) that is dense in K(X; C)… ↩ 1 08M7
Corollary — If X is paracompact then the topologies induced on M₊(X) by the vague topology and the topology of compact convergence coincide. 08M8

§ 2. Support of a measure

Proposition 1 Let (Y_α)_α ∈ A be an open covering of X and suppose given, on each subspace Y_α, a measure μ_α, in such a way that for every pair (α, β), the restrictions of μ_α and μ_β to Y_α ∩… ↩ 2 08MR
Corollary (Principle of localization) — Let μ and ν be two measures on X, and let (Y_α) be a family of open sets of X such that, for every α, the restrictions to Y_α of μ and ν are equal; then the restrictions of μ and… ↩ 2 08MS
Definition 1 If μ is a measure on a locally compact space X, one defines the support of μ, denoted Supp(μ), to be the closed set complementary to the largest of the open sets in X on which the… 08MT
Example — 1) For a measure on X to be zero, it is necessary and sufficient that its support be empty. 2) The support of Lebesgue measure on R is the entire line R; for, it is nonempty and… 08MU
Proposition 2 The support of a measure μ is identical to the support of the measure |μ|; if μ is real, its support is the union of the supports of the measures μ⁺ and μ⁻. ↩ 1 08MV
Proposition 3 If μ and ν are two measures on a locally compact space X such that |μ| ≤ |ν|, then Supp(μ) ⊂ Supp(ν). 08MW
Proposition 4 The support of the sum of two measures is contained in the union of their supports. 08MX
Proposition 5 The support of the restriction of a measure μ to an open set U is the trace on U of the support of μ. ↩ 1 08MY
Proposition 6 The set of measures on a locally compact space X, whose support is contained in a closed set F, is a vaguely closed linear subspace of M(X; C). ↩ 3 08MZ
Proposition 7 — If Φ is a filter on M(X; C) such that the support of μ recedes indefinitely along Φ, then μ converges vaguely to 0 with respect to Φ. ↩ 1 08N0
Proposition 8 — Let μ be a measure on a locally compact space X. For every function f ∈ K(X; C) that is zero on Supp(μ), μ(f) = 0. ↩ 5 08N1
Corollary 1 — If two functions f, g in K(X; C) are equal on Supp(μ), then μ(f) = μ(g). ↩ 1 08N2
Corollary 2 — Let μ be a positive measure on X; if f ∈ K(X; C) is such that f(x) ≥ 0 on Supp(μ), then μ(f) ≥ 0. ↩ 4 08N3
Corollary 3 — Let μ be a bounded measure on X; if f ∈ K(X; C) is such that |f(x)| ≤ a on Supp(μ), then |μ(f)| ≤ a |μ|. ↩ 2 08N4
Proposition 9 — Let μ be a positive measure on X; if f is a function in K₊(X) such that μ(f) = 0, then f is zero on Supp(μ). ↩ 1 08N5
Proposition 10 — Let μ be a measure on a locally compact space X; for every function g ∈ C(X; C), the support of the measure g · μ is the closure T of the set of points x ∈ Supp(μ) such that g(x)… ↩ 1 08N6
Corollary — In order that the measure g · μ be zero, it is necessary and sufficient that g be zero on the support of μ. 08N7
Proposition 11 — Every measure with compact support is bounded. ↩ 2 08N8
Proposition 12 — Let aᵢ (1 ≤ i ≤ n) be distinct points in a locally compact space X. Every measure on X whose support is contained in the set of the aᵢ is a linear combination of the measures ε_a… ↩ 1 08N9
Theorem 1 — Every measure μ on a locally compact space X is in the vague closure of the vector space V of measures whose support is finite and contained in Supp(μ). ↩ 3 08NA
Corollary 1 — Every bounded measure μ on X is in the vague closure of the convex set A of measures whose support is finite and is contained in that of μ, and whose norm is ≤ | μ |. Moreover, i… ↩ 2 08NB
Corollary 2 — Every bounded measure μ on X is in the vague closure of the set of measures whose support is finite and contained in that of μ and whose norm is equal to | μ |. ↩ 1 08NC
Corollary 3 Every bounded positive measure μ on X is in the vague closure of the convex set of positive measures whose support is finite and contained in that of μ and whose norm is equal to |… ↩ 1 08ND
Corollary 4 In the space M(X; C), the set of point measures is total for the topology of strictly compact convergence (§1, No. 10). ↩ 1 08NE
Proposition 13 Let μ be a measure on a locally compact space X. For a point x₀ to belong to Supp(μ), it is necessary and sufficient that the point measure ε_x₀ be in the vague closure of the set… ↩ 2 08NF
Corollary Let μ be a measure on X. For a measure ν on X to be in the vague closure of the set of measures g · μ, where g runs over K(X; C), it is necessary and sufficient that Supp(ν) ⊂ Supp… 08NG
Proposition 14 For a measure μ on a locally compact space X to be a discrete measure (§ 1, No. 3, Example I), it is necessary and sufficient that Supp(μ) be a discrete closed subspace of X. 08NH

§ 3. Integrals of continuous vector-valued functions

Definition 1 For every function f ∈ widetildeK(X;E) we call integral of f with respect to μ, and denote by ∫ f dμ or ∫ f(x) dμ(x), or ∫ f μ, or ∫ f(x)μ(x), the element of E'^* defined by ↩ 1 08NN
Example — 1) Suppose that E is finite-dimensional over C and Hausdorff, so that if (eᵢ)_1 ≤ i ≤ n is a basis of E then the mapping ↩ 1 08NO
Remark 1 If E is a locally convex space and N is the closure of 0 in E, so that E₁ = E/N is the Hausdorff locally convex space associated with E, we know that the duals E' and E'₁ are ident… 08NP
Remark 2 Let E be a locally convex space over C, and let E₀ be the locally convex space over R underlying E; we know that the mapping z' ↦ Re z' which, to every continuous (complex) linear… 08NQ
Proposition 1 — The mapping ↩ 1 08NR
Proposition 2 — Let u be a continuous linear mapping of E into a locally convex space F; for every function f ∈ widetildeK(X;E), the function u ∘ f belongs to widetildeK(X;F) and ↩ 2 08NS
Proposition 3 — For every function g ∈ C(X;C) and every function f ∈ widetildeK(X;E), the function gf belongs to widetildeK(X;E) and ↩ 1 08NT
Proposition 4 — Let μ be a positive measure on X, S its support, and f a function in widetildeK(X;E). Suppose E is Hausdorff, and equip the space E'^* with the weak topology σ(E'^*, E'). ↩ 3 08NU
Corollary Let μ be a positive measure on X, f a mapping belonging to K(X;E), and D the closed convex envelope of f(X) in E'. There exists a number a > 0 such that ∫ f dμ ∈ a · D. ↩ 2 08NV
Proposition 5 Suppose that the space X is compact, and let f be a continuous mapping of X into a Hausdorff locally convex space E. The closed convex envelope of f(X) in E' (for σ(E', E')) is equ… ↩ 3 08NW
Proposition 6 For every continuous mapping with compact support f of X into a Hausdorff locally convex space E, every continuous semi-norm q on E and every measure μ on X such that ∫ f dμ ∈ E, ↩ 2 08NX
Proposition 7 Let E be a Hausdorff locally convex space, and f ∈ K(X; E). If f(X) is contained in a complete convex subset A of E, then ∫ f dμ ∈ E. ↩ 8 08NY
Corollary 1 — Let E be a Hausdorff locally convex space; for every function f ∈ K(X; E), ∫ f dμ belongs to the completion widehatE of E. ↩ 2 08NZ
Corollary 2 — If E is a quasi-complete Hausdorff locally convex space, then ∫ f dμ ∈ E for every function f ∈ K(X; E). ↩ 8 08O0
Proposition 8 — Suppose that E is Hausdorff; let μ be a measure on X. The mapping f ↦ ∫ f dμ of K(X;E) into widehatE (No. 3, Cor. 1 of Prop. 7) is the unique continuous linear mapping Φ such tha… ↩ 1 08O1
Proposition 9 — Suppose that E is Hausdorff; let f be a continuous mapping of X into E with compact support. When the space M(X;C) is equipped with the topology of strictly compact convergence (… ↩ 2 08O2
Corollary — With hypotheses and notations as in Prop. 9, the restriction of the mapping μ ↦ ∫ f dμ to the set M₊(X) of positive measures, or to a vaguely bounded subset B of M(X; C), is vagu… 08O3

§ 4. Products of measures

Theorem 1 — Let X and Y be two locally compact spaces, λ a measure on X, μ a measure on Y; there exists one and only one measure ν on X × Y such that, for every function g ∈ K(X; C) and ever… ↩ 19 08O8
Lemma 1— Let X and Y be two locally compact spaces, K (resp. L) a compact subset of X (resp. Y). 08O9
Lemma 2 With notations as in Lemma 1, for every function f ∈ K(X × Y, K × L; C) the function ↩ 3 08OA
Definition 1 Given two measures λ, μ defined, respectively, on two locally compact spaces X, Y, the product measure of λ by μ is defined to be the unique measure ν on X × Y satisfying the relat… 08OB
Theorem 2 Let λ, μ be two measures defined, respectively, on two locally compact spaces X, Y. For every function f in K(X × Y; C), the integral of f with respect to the product measure ν of… ↩ 3 08OC
Example 1 The product measure of the Dirac measures εₓ (x ∈ X) and ε_y (y ∈ Y) is the Dirac measure ε_(x, y). 08OD
Example 2 Let us take X = Y = R, and for λ and μ the Lebesgue measure on R (§1, No. 3); their product is called Lebesgue measure on R²; the integral of a function f ∈ K(R²; C) with respect t… 08OE
Remark Let E be a Hausdorff locally convex space, and let f be a mapping in K(X × Y; E) such that f(X × Y) is contained in a complete convex subset C of E. Then, for every y ∈ Y, the inte… 08OF
Proposition 1 Let λ be a measure on X, μ a measure on Y; if g ∈ C(X; C),\ h ∈ C(Y; C), then (g · λ) ⊗ (h · μ) = (g ⊗ h) · (λ ⊗ μ). 08OG
Proposition 2 The support of the product λ ⊗ μ is equal to the product of the support of λ and the support of μ. ↩ 1 08OH
Proposition 3 Let λ ∈ M(X; C),\ μ ∈ M(Y; C). Then ↩ 9 08OI
Corollary — Let λ ∈ M(X; R), \ μ ∈ M(Y; R). Then 08OJ
Proposition 4 — Let λ ∈ M(X; C), \ μ ∈ M(Y; C). Then ↩ 2 08OK
Proposition 5 For every measure λ₀ ∈ M(X; C), the mapping μ ↦ λ₀ ⊗ μ of M(Y; C) into M(X × Y; C) is vaguely continuous. 08OL
Proposition 6 When M(X; C) , \ M(Y; C) and M(X × Y; C) are equipped with the topology of strictly compact convergence (§ 1, No. 10), the bilinear mapping (λ, μ) ↦ λ ⊗ μ of M(X; C) × M(Y; C) into… 08OM
Proposition 7 ('associativity of the product of measures') — Let (Iₖ)_1 ≤ k ≤ r be a partition of the interval [1, n] of N; then ⨂ₖ₌₁ʳ ( ⨂_i ∈ Iₖ μᵢ ) = ⨂ᵢ₌₁ⁿ μᵢ 08ON
Definition 2 A family (μ_α)_α ∈ I, where, for every α ∈ I, μ_α is a measure on X_α, is said to be an inverse system of measures if, whenever α ≤ β, μ_α = (p_αβ)_*(μ_β). A measure μ on X = lim_←… 08OO
Proposition 8 (i) If an inverse system (μ_α) of measures on the X_α has an inverse limit, then that limit is unique. (ii) If an inverse system (μ_α) has an inverse limit, then the family of norm… ↩ 2 08OP
Lemma 3 — Let F be the set of functions f ∈ C(X; C) having the following property: there exist an α ∈ I and a function f_α ∈ C(X_α; C) such that f = f_α ∘ p_α. Then F is a dense linear sub… ↩ 1 08OQ
Example — Let (X_λ)_λ ∈ L be a family of compact spaces; set X = ∏_λ ∈ L X_λ and, for every finite subset J of L, set X_J = ∏_λ ∈ J X_λ; denote by pr_J : X → X_J and pr_J, K : X_K → X_J (f… 08OR
Proposition 9 Suppose that all of the measures μ_λ are positive and that the family (μ_λ(1))_λ ∈ L is multipliable in R₊ (with product possibly 0). Then there exists one and only one measure μ o… ↩ 2 08OS
Corollary Assume that the conditions of Prop. 9 are verified, and let (L_ρ)_ρ ∈ R be a partition of L. Then each of the families of measures (μ_λ)_λ ∈ L_ρ admits a product measure, the famil… 08OT

IV. Extension of a measure. Lp spaces

§ 1. Upper integral of a positive function

Lemma — Every function f ∈ I₊ is the upper envelope of the set (directed for the relation ≤) of all functions g ∈ K₊ such that g ≤ f. 08P2
Definition 1 Given a positive measure μ on X, one calls upper integral of a function f ∈ J₊ (with respect to μ) the positive number (finite or equal to +∞) ↩ 5 08P3
Example — Let X be a discrete space, μ a positive measure on X, and set α(x) = μ(φ_x) for all x ∈ X. Every numerical function f defined on X is then continuous; for such a function f ≥ 0… 08P4
Proposition 1 For every finite real number α > 0 and every function f ∈ J₊, μ^*(α f) = α μ^*(f). ↩ 4 08P5
Proposition 2 On the set J₊, the function μ^* is increasing. The proofs are immediate from Def. 1. ↩ 1 08P6
Theorem 1 Let H be a nonempty set of functions in J₊, directed for the relation ≤. For every positive measure μ on X, μ^*( sup_g ∈ H g ) = sup_g ∈ H μ^*(g) = lim_g ∈ H μ^*(g). ↩ 12 08P7
Theorem 2 If f₁ and f₂ are two functions in I₊ then ↩ 4 08P8
Proposition 3 — For every family (f_ι)_ι ∈ I of functions in J₊, 08P9
Proposition 4 — Let f be a function in J₊. The mapping μ ↦ μ^*(f) of the set M₊(X) of positive measures on X, into the extended real line overlineR, is lower semi-continuous for the vague topolo… ↩ 2 08PA
Definition 2 — Given a positive measure μ on X, for every open set G ⊂ X the upper integral μ^*(φ_G) is called the outer measure of G and is denoted μ^*(G). ↩ 1 08PB
Proposition 5 — The outer measure of a relatively compact open set G is finite. ↩ 1 08PC
Proposition 6 — If G₁ and G₂ are two open sets such that G₁ ⊂ G₂, then μ(G₁) ≤ μ(G₂). For, the relation G₁ ⊂ G₂ is equivalent to φG₁ ≤ φG₂. ↩ 1 08PD
Proposition 7 — Let 𝔅 be a set of open subsets of X that is directed for the relation ⊂; then ↩ 4 08PE
Proposition 8 — Let (Gι){ι \in I} be any family of open sets; then ↩ 2 08PF
Example — Let X = \mathbf{R} and let μ be Lebesgue measure on \mathbf{R} (Ch. III, §1, No. 3); we are going to determine the outer measure of an open interval G = ]a, b[ (−∞ ≤ a < b ≤ +∞)… 08PG
Proposition 9 For Lebesgue measure on R, the outer measure of an open set in R is equal to the sum of the lengths of its connected components. ↩ 2 08PH
Definition 3 Let μ be a positive measure on X; for every numerical function f ≥ 0 (finite or not) defined on X, the positive number ↩ 4 08PI
Example — If X is a discrete space, μ a positive measure on X, and if one sets α(x) = μ(φ_x), then μ^*(f) = ∑_x ∈ X α(x) f(x) for every numerical function f ≥ 0 defined on X, since such a… 08PJ
Proposition 10 — If f and g are two numerical functions geqslant 0 defined on X such that f leqslant g, then μ^*(f) leqslant μ^*(g). 08PK
Proposition 11 — For every finite real number α > 0 and every numerical function f geqslant 0 defined on X, ↩ 1 08PL
Proposition 12 — If f₁ and f₂ are two numerical functions geqslant 0 defined on X, then ↩ 1 08PM
Theorem 3 — For every increasing sequence (fₙ) of numerical functions geqslant 0 defined on X, ↩ 8 08PN
Corollary — Let 𝔉 be a set of numerical functions ≥ 0, directed for the relation ≤, such that there exists a countable cofinal subset 𝔊 of 𝔉 (S, III, §1, No. 7); then ↩ 1 08PO
Remark — 1) The relation (11) does not necessarily hold when 𝔉 is an uncountable directed set of functions ≥ 0 that are not lower semi-continuous. Take for example X = R, μ being Lebesgue… ↩ 2 08PP
Proposition 13 For every sequence (fₙ) of numerical functions geqslant 0 defined on X, ↩ 2 08PQ
Proposition 14 (Fatou’s lemma) — For every sequence (fₙ) of numerical functions geqslant 0, 08PR
Corollary Let (fₙ) be a sequence of numerical functions geqslant 0 such that, for every x ∈ X, lim_n → ∞ fₙ(x) = +∞. If μ is not the zero measure, then lim_n → ∞ μ^*(fₙ) = +∞. 08PS
Proposition 15 For every scalar α > 0 and every pair of positive measures μ, ν on X, ↩ 3 08PT
Definition 4 Let μ be a positive measure on X; for every subset A of X, the upper integral μ^*(φ_A) is called the outer measure of A (with respect to the measure μ) and is denoted μ^*(A). 08PU
Proposition 16 If A and B are two subsets of X such that A ⊂ B, then μ^*(A) ≤ μ^*(B). 08PV
Corollary Every relatively compact set in X has finite outer measure. 08PW
Proposition 17 — If (Aₙ) is an increasing sequence of subsets of X, then μ^*(⋃ₙ Aₙ) = supₙ μ^*(Aₙ). 08PX
Proposition 18 — For every sequence (Aₙ) of subsets of X, ↩ 2 08PY
Proposition 19 — For every subset A of X, μ^*(A) is the infimum of the outer measures of the open sets containing A. ↩ 3 08PZ

§ 2. Negligible functions and sets

Definition 1 — Given a measure μ on a locally compact space X, a numerical function f ≥ 0 (finite or not) defined on X is said to be negligible for the measure μ if |μ|^*(f) = 0. ↩ 1 08Q8
Proposition 1 — If f is a negligible function ≥ 0, then every numerical function g such that 0 ≤ g ≤ α f (α a scalar > 0) is negligible. For, 0 ≤ |μ|^*(g) ≤ α |μ|^*(f) = 0. 08Q9
Proposition 2 — The sum and upper envelope of a sequence (fₙ) of negligible functions ≥ 0 are negligible. ↩ 1 08QA
Proposition 3 For a lower semi-continuous function f ≥ 0 on X to be negligible, it is necessary and sufficient that f be zero on the support of μ. ↩ 1 08QB
Definition 2 Given a measure μ on a locally compact space X, a subset A of X is said to be negligible for the measure μ if |μ|^*(A) = 0. 08QC
Proposition 4 Every subset of a negligible set is negligible; every countable union of negligible sets is negligible. ↩ 1 08QD
Example — Let μ be Lebesgue measure on R. Every set x₀ reduced to a point is negligible (cf. § 1, No. 3, Remark 1). It follows that every countable subset of R is negligible for Lebesgue m… 08QE
Proposition 5 The complement of the support S of μ is the largest negligible open set in X. ↩ 4 08QF
Theorem 1 — In order that a numerical function (finite or not) f geqslant 0 defined on X be negligible, it is necessary and sufficient that f(x) = 0 almost everywhere. ↩ 4 08QG
Proposition 6 — If f and g are two functions geqslant 0 (finite or not) defined on X such that f(x) = g(x) almost everywhere, then |μ|^*(f) = |μ|^*(g). ↩ 5 08QH
Proposition 7 — If f is a function geqslant 0 defined on X such that |μ|^*(f) < +∞, then f(x) is finite almost everywhere. ↩ 2 08QI
Proposition 8 Let (Fₙ) be a countable family (finite or infinite) of sets. For every index n, let fₙ, gₙ be two equivalent mappings of X into Fₙ; then, there exists a negligible set H such that… ↩ 1 08QJ
Corollary If φ is a mapping of ∏ₙ Fₙ into a set G, then the mappings φ((fₙ)) and φ((gₙ)) of X into G are equivalent. 08QK
Proposition 9 — Let f and g be two continuous mappings of X into a Hausdorff topological space F; for f and g to be equivalent, it is necessary and sufficient that f(x) = g(x) at every point of… ↩ 1 08QL
Proposition 10 — Let F be a Hausdorff locally convex space over R such that there exists in the dual F' of F a sequence (a'ₙ) that is dense for the weak topology σ(F', F) (TVS, II, §6, No. 2). In… 08QM

§ 3. LP spaces

Proposition 1 — For every real number p geqslant 1 and every pair of positive functions f, g (finite or not) defined on X, ↩ 1 08QN
Definition 1 Let X be a locally compact space, μ a measure on X. For every mapping f of X into a Banach space F, and for every number p such that 1 ≤ p < +∞, we denote by Nₚ(f, μ), or simply Nₚ… ↩ 1 08QO
Proposition 2 If f and g are two mappings of X into F, and α is any scalar ≠ 0, then, for 1 ≤ p < +∞, ↩ 1 08QP
Theorem 1 (countable convexity theorem) — Let (fₙ) be a sequence of functions ≥ 0 (finite or not) defined on X. For 1 ≤ p < +∞, ↩ 2 08QQ
Proposition 3 If f and g are two equivalent mappings of X into a Banach space F, then Nₚ(f - g) = 0 for 1 ≤ p < +∞; conversely, if Nₚ(f - g) = 0 for a value of p ≥ 1 then f and g are equivalent. ↩ 2 08QR
Remark — Let F be a Banach space over the field C of complex numbers; then, for every function f ∈ F_Fᵖ and every complex number α, α f belongs to F_Fᵖ and Nₚ(α f) = |α| Nₚ(f); in other w… 08QS
Proposition 4 Let B be a filter base on F_Fᵖ. Assume that there exists a compact set K ⊂ X such that, for every set M ∈ B, all of the mappings f ∈ M have their support contained in K. Under thes… ↩ 4 08QT
Proposition 5 The locally convex space F_Fᵖ is complete. ↩ 3 08QU
Proposition 6 Let (fₙ) be a sequence of functions in F_Fᵖ such that ∑ₙ₌₁^∞ Nₚ(fₙ) < +∞. Under these conditions, the series with general term fₙ(x) ∈ F is absolutely convergent almost everywhere… ↩ 4 08QV
Definition 2 Given a locally compact space X, a measure μ on X, and a Banach space F, we denote by L_Fᵖ(X, μ) (or simply L_Fᵖ(μ), or L_Fᵖ) the closure, in the locally convex space F_Fᵖ(X, μ), o… ↩ 4 08QW
Remark 1 As has already been signalled (§2, No. 5) the p-th power integrable functions with values in F in general do not form a vector space. 08QX
Remark 2 In general, the space F_Fᵖ is distinct from its subspace L_Fᵖ (§4, Exer. 8). 08QY
Proposition 7 For a function f to belong to L_Fᵖ, it is necessary and sufficient that, for every ε > 0, there exist a continuous function g with compact support such that Nₚ(f - g) ≤ ε. ↩ 3 08QZ
Proposition 8 Let f be a numerical function (finite or not) defined almost everywhere; if, for every ε > 0, there exists two p-th power integrable functions g, h such that g ≤ f ≤ h almost every… ↩ 2 08R0
Theorem 2 The space L_Fᵖ is complete; the space L_Fᵖ is a Banach space. ↩ 1 08R1
Theorem 3 Let (fₙ) be a Cauchy sequence in the space L_Fᵖ; there exists a subsequence (f_nₖ) of (fₙ) having the following properties: 1° the series with general term Nₚ(f_nₖ₊₁ - f_nₖ) is con… ↩ 8 08R2
Corollary 1 — If a Cauchy sequence (fₙ) in the space L_Fᵖ is such that the sequence (fₙ(x)) converges almost everywhere to f(x), then f is p-th power integrable and the sequence (fₙ) converges… ↩ 2 08R3
Corollary 2 — Let E be a dense subset of L_Fᵖ. For every function f ∈ L_Fᵖ, there exists a sequence (gₙ) of functions in E having the following properties: 08R4
Remark 3 A Cauchy sequence (fₙ) in L_Fᵖ can be such that the sequence (fₙ(x)) is not convergent at any point of X (Exer. 1). 08R5
Remark 4 If f belongs to L_Fᵖ, it is not always possible to find a sequence (fₙ) of continuous functions with compact support such that the sequence (fₙ(x)) converges everywhere in X to a f… 08R6
Theorem 4 — Let F and G be two Banach spaces, u a continuous linear mapping of F into G. For every function f ∈ L_Fᵖ, the composite function u ∘ f belongs to L_Gᵖ. ↩ 2 08R7
Corollary 1 — Let a' be a continuous linear form on F; if f ∈ L_Fᵖ, the numerical function x ↦ ⟨ f(x), a' ⟩ (denoted by ⟨ f, a' ⟩) belongs to Lᵖ. 08R8
Corollary 2 — Given n points aₖ of F (1 ≤ k ≤ n), and n numerical functions fₖ (1 ≤ k ≤ n) belonging to Lᵖ, the function f = ∑ₖ₌₁ⁿ aₖ fₖ belongs to L_Fᵖ. 08R9
Proposition 9 — Let F be an n-dimensional vector space over R, and let (eₖ)_1 ≤ k ≤ n be a basis of F. For a function f = ∑ₖ₌₁ⁿ eₖ fₖ to belong to L_Fᵖ, it is necessary and sufficient that each… 08RA
Proposition 10 — In the space L_Fᵖ, the linear subspace formed by the (finite) linear combinations ∑ₖ aₖ fₖ, where aₖ ∈ F and the fₖ are continuous numerical functions with compact support, is de… ↩ 2 08RB
Proposition 11 — If a function f belongs to L_Fᵖ, then the function |f| belongs to Lᵖ, and the mapping f ↦ |f| of L_Fᵖ into Lᵖ is uniformly continuous (for the topology of convergence in mean of… ↩ 6 08RC
Proposition 12 — For a numerical function f to belong to Lᵖ, it is necessary and sufficient that each of the functions f⁺ and f⁻ belong to Lᵖ. 08RD
Corollary — The upper (resp. lower) envelope of a finite family of functions in Lᵖ belongs to Lᵖ. ↩ 3 08RE
Proposition 13 In the Riesz space Lᵖ, equipped with the topology defined by the norm | tildef |ₚ, the mapping tildef ↦ | tildef | is uniformly continuous, and the set of elements tildef ≥ 0 is cl… 08RF
Proposition 14 Let H be a subset of the Riesz space Lᵖ, consisting of classes ≥ 0 and directed for the relation ≤. For H to have a supremum in Lᵖ, it is necessary and sufficient that ↩ 4 08RG
Lemma If f and g are two functions in Lᵖ such that 0 ≤ f ≤ g, then 08RH
Corollary 1 If tildeg is the supremum of H in Lᵖ, then 08RI
Corollary 2 The Riesz space Lᵖ is fully lattice-ordered. 08RJ
Theorem 5 Let (fₙ) be an increasing sequence of functions geqslant 0 in Lᵖ. For the upper envelope f of this sequence to be p-th power integrable, it is necessary and sufficient that supₙ Nₚ… ↩ 3 08RK
Corollary 1 Let (fₙ) be a decreasing sequence of functions geqslant 0 in Lᵖ; then, the lower envelope f of the sequence belongs to Lᵖ, the sequence (fₙ) converges in mean of order p to f, and ↩ 1 08RL
Corollary 2 Let (fₙ) be a sequence of functions in Lᵖ. For the upper envelope f of the sequence (fₙ) to be p-th power integrable, it is necessary and sufficient that there exist a function g g… ↩ 2 08RM
Corollary 3 — Let A be a countable set, 𝔉 a filter on A having a countable base, (f_α)_α ∈ A a family of functions ≥ 0 in Lᵖ. Assume that there exists a function g ≥ 0 such that Nₚ(g) < +∞ and… ↩ 1 08RN
Theorem 6 (Lebesgue) — Let F be a Banach space, (fₙ) a sequence of functions in L_Fᵖ such that: 1^∘ the sequence (fₙ(x)) converges almost everywhere to a limit f(x) ∈ F; 2^∘ there exists a numerical fu… ↩ 4 08RO
Corollary — Let A be a set of indices, filtered by a filter 𝔉 having a countable base. If (f_α)_α ∈ A is a family of functions in L_Fᵖ that, with respect to the filter 𝔉, converge pointwise… ↩ 2 08RP
Remark 1 Th. 6 no longer holds if the hypothesis |fₙ| ≤ g (with Nₚ(g) < +∞) is replaced by the weaker hypothesis supₙ Nₚ(fₙ) < +∞. Suppose, for example, that μ is Lebesgue measure on R; def… 08RQ
Remark 2 The Cor. of Th. 6 no longer holds if it is not assumed that the filter 𝔉 has a countable base (cf. §1, No. 3, Remark 1 following the Cor. of Th. 3). 08RR
Theorem 7 — Let p and q be two real numbers such that 1 ≤ p < +∞, 1 ≤ q < +∞. If a function f belongs to L_Fᵖ then the function |f|^(p/q)-1 · f belongs to L_F^q, and conversely. 08RS
Corollary 1 — For a function f to belong to L_Fᵖ, it is necessary and sufficient that the function |f|ᵖ⁻¹ · f belong to L_F¹. ↩ 1 08RT
Corollary 2 — For a positive numerical function f to belong to Lᵖ, it is necessary and sufficient that fᵖ belong to L¹. 08RU

§ 4. Integrable functions and sets

Definition 1 The functions belonging to L_F¹(X, μ) are said to be integrable with respect to the measure μ (or, again, that they are μ-integrable). The integral (with respect to μ) of the integ… 08RY
Example — Let X be a discrete space, μ a measure on X, and set α(x) = μ(φ_x) for every x ∈ X. The functions in F_F¹ are then integrable, in other words L_F¹ = F_F¹; moreover, for every fun… 08RZ
Proposition 1 For every positive μ-integrable numerical function f, ↩ 3 08S0
Corollary 1 For every integrable function f ∈ L¹_F, |f| is integrable and 08S1
Corollary 2 For every function f ∈ Lᵖ_F, the numerical function |f|ᵖ is integrable and 08S2
Proposition 2 — For every integrable function f, ↩ 1 08S3
Theorem 1 — Let F and G be two Banach spaces, u a continuous linear mapping of F into G. For every integrable function f with values in F, u ∘ f is integrable and ↩ 5 08S4
Corollary 1 — Let a' be a continuous linear form on F. If f is an integrable function with values in F, then the numerical function ⟨ f, a' ⟩ is integrable and ↩ 1 08S5
Corollary 2 — If the aₖ (1 ≤ k ≤ n) are vectors in F and the fₖ (1 ≤ k ≤ n) are integrable numerical functions, then the function f = ∑ₖ₌₁ⁿ aₖ fₖ is integrable and 08S6
Proposition 3 — Let B be a filter base on L_F¹. Assume that there exists a compact set K ⊂ X such that, for every set M ∈ B, all of the functions f ∈ M have their support in K. Under these condi… ↩ 2 08S7
Proposition 4 — Let (fₙ) be an increasing (resp. decreasing) sequence of integrable numerical functions. For the upper (resp. lower) envelope f of the sequence to be integrable, it is necessary… ↩ 7 08S8
Theorem 2 — Let A be a set of indices, filtered by a filter 𝔉 with a countable base. Let (f_α)_α ∈ A be a family of integrable functions that, with respect to the filter 𝔉, converge pointwis… ↩ 6 08S9
Corollary 1 — Let Ω be a topological space, t₀ a point of Ω admitting a countable fundamental system of neighborhoods, f a mapping of X × Ω into F having the following properties: a) for every… ↩ 1 08SA
Corollary 2 — Let (fₙ) be a sequence of integrable functions such that the series with general term fₙ(x) converges almost everywhere; if there exists a function g geqslant 0 such that ∫^* g d… ↩ 1 08SB
Proposition 5 — For a numerical function f geqslant 0 (finite or not), lower semi-continuous on X, to be integrable, it is necessary and sufficient that ∫^* f d|μ| < +∞. ↩ 3 08SC
Corollary 1 — For a finite numerical function f geqslant 0, upper semi-continuous on X, to be integrable, it is necessary and sufficient that ∫^* f d|μ| < +∞. ↩ 5 08SD
Corollary 2 — Let H be a nonempty set, directed for the relation leqslant (resp. geqslant), of lower (resp. upper) semi-continuous and integrable numerical functions; if ↩ 5 08SE
Corollary 3 — Let f be a bounded numerical function, upper semi-continuous on X and with compact support. Then, the mapping μ ↦ ∫ f dμ is upper semi-continuous on M₊(X) for the vague topology. ↩ 1 08SF
Theorem 3 — For a numerical function f ≥ 0 to be integrable, it is necessary and sufficient that, for every ε > 0, there exist an upper semi-continuous function g ≥ 0, with finite values and… ↩ 3 08SG
Corollary — For every integrable (resp. integrable and ≥ 0) numerical function f, there exist an increasing sequence (gₙ) of upper semi-continuous functions that are integrable (resp. integr… ↩ 1 08SH
Example For every positive measure μ on R, every step function with compact support is μ-integrable; for, the characteristic function of an open (resp. closed) interval is lower (resp. upp… 08SI
Remark 1 Let f be a regulated function on R that is integrable with respect to Lebesgue measure μ; then |f| is also integrable (No. 2, Cor. 1 of Prop. 1), and, setting Iₙ = [-n, n], |f| is… 08SJ
Remark 2 Applied to Lebesgue measure and to regulated functions, Prop. 3 of No. 3 yields anew the theorem on passage to the limit for integrals of regulated functions on a compact interval… 08SK
Definition 2 A subset A of a locally compact space X is said to be integrable with respect to a measure μ on X (or, that it is μ-integrable) if the characteristic function φ_A of A is integrabl… 08SL
Proposition 6 The union of a finite family (Aᵢ)_1 ≤ i ≤ n of integrable sets is integrable, and 08SM
Proposition 7 1° If A and B are two integrable sets such that B ⊂ A, then the set C = A - B is integrable and ↩ 2 08SN
Corollary If (Aₙ) is a decreasing sequence of integrable sets, then μ(⋂ₙ Aₙ) = lim_n → ∞ μ(Aₙ). ↩ 4 08SO
Proposition 8 Let (Aₙ) be an increasing sequence of integrable sets; for the union A = ⋃ₙ Aₙ to be integrable, it is necessary and sufficient that supₙ |μ|(Aₙ) < +∞, in which case, ↩ 1 08SP
Corollary Let (Aₙ) be a sequence of integrable sets such that ∑ₙ₌₁^∞ |μ|(Aₙ) < +∞; the union A = ⋃ₙ Aₙ is integrable, and ↩ 2 08SQ
Proposition 9 — Let (Aₙ) be a sequence of pairwise disjoint integrable sets such that ∑ₙ₌₁^∞ |μ|(Aₙ) < +∞; then 08SR
Proposition 10 — For an open (resp. closed) set A in X to be integrable, it is necessary and sufficient that |μ|^*(A) < +∞. ↩ 3 08SS
Corollary 1 — Every compact set is integrable; every relatively compact open set is integrable. 08ST
Corollary 2 — For every positive measure μ on X, A ↦ μ^*(A) is a capacity on X (cf. GT, IX, §6, No. 9, Example). 08SU
Example For Lebesgue measure μ on R, it follows from Prop. 10 that every bounded open interval [a, b[ is integrable and has measure b - a (\§1, No. 2, Prop. 9). Since every set reducing to… 08SV
Proposition 11 — Let G be a set, directed for the relation ⊂, of integrable open sets in X; for A = ⋃_G ∈ G G to be integrable, it is necessary and sufficient that sup_G ∈ G |μ|(G) < +∞, in which… ↩ 1 08SW
Corollary — Let 𝔉 be a set, directed for the relation ⊃, of integrable closed sets in X; then the closed set B = ⋂_H ∈ 𝔉 H is integrable, and one has μ(B) = lim_𝔉 μ(H) and |μ|(B) = inf_H ∈ 𝔉… 08SX
Theorem 4 — For a set A to be integrable, it is necessary and sufficient that, for every ε > 0, there exist an integrable open set G and a compact set K, such that K ⊂ A ⊂ G and |μ|(G - K) =… ↩ 10 08SY
Corollary 1 — For a set A to be integrable, it is necessary and sufficient that, for every ε > 0, there exist a compact set K ⊂ A such that |μ|^*(A - K) ≤ ε. The measure |μ|(A) is then the sup… ↩ 1 08SZ
Corollary 2 — For every integrable set A, there exist: ↩ 3 08T0
Corollary 3 — Every set of finite outer measure is contained in the union of a negligible set and a countable family of pairwise disjoint compact sets the sum of whose measures is finite. ↩ 2 08T1
Corollary 4 — For every open set U in X, |μ|^*(U) is the supremum of the measures |μ|(K) of the compact sets K ⊂ U. ↩ 2 08T2
Proposition 12 — For a measure μ on a locally compact space X to be bounded (Ch. III, §1, No. 8), it is necessary and sufficient that X be an integrable set with respect to μ (or, what comes to t… ↩ 3 08T3
Proposition 13 — Let μ be a bounded measure on X. Let B be a filter base on L_Fᵖ having the following properties: 1^∘ there exists a set M ∈ B such that the functions f ∈ M are uniformly bounded… 08T4
Corollary — For a bounded measure μ on X, every bounded continuous mapping f of X into F belongs to each of the L_Fᵖ (1 ≤ p < +∞). 08T5
Proposition 14 — Let X be a locally compact space, F a Banach space not reduced to 0; equip the space C(X; F) of all continuous mappings of X into F with the topology of compact convergence. For… ↩ 1 08T6
Proposition 15 — Let X be a locally compact space and, for every x ∈ X, let εₓ be the Dirac measure at the point x. The mapping x ↦ εₓ of X into C'(X; C) is continuous for the topology of compact… ↩ 2 08T7
Proposition 16 — Let K be a compact subset of X, L the vector space of measures μ on X with support contained in K. On L, the topologies induced by the topology T of compact convergence on C'(X… ↩ 1 08T8
Corollary 1 — For every compact subset K of X and every number a > 0, the set B of measures μ on X such that Supp(μ) ⊂ K and | μ | ≤ a is an equicontinuous subset of C'(X; C) that is compact f… 08T9
Corollary 2 — Every measure with compact support (resp. every positive measure with compact support) μ is in the closure in C'(X; C), for the topology T of compact convergence, of the set of m… 08TA
Definition 3 A nonempty set Φ of subsets of a set A is said to be a clan if there exists an algebra A (over R) consisting of real-valued functions defined on A, such that the relations M ∈ Φ an… 08TB
Example If μ is a measure on a locally compact space X then the linear combinations, with real coefficients, of the characteristic functions of integrable sets form an algebra A, because… 08TC
Proposition 17 In order that a nonempty set Φ of subsets of a set A be a clan, it is necessary and sufficient that it satisfy the following condition: ↩ 1 08TD
Lemma Let Φ be a nonempty set of subsets of A satisfying the axiom (CL). Given a finite family (Mᵢ)_1 ≤ i ≤ n of sets in Φ, there exists a 08TE
Definition 4 Given a clan Φ of subsets of a set A, and a Banach space F, one calls step function² over the sets of Φ (or Φ-step function), with values in F, every function of the form ∑ᵢ aᵢ φ_M… ↩ 3 08TF
Remark — It is easily seen that Def. 4 is equivalent to the following: a Φ-step function with values in F is a function f that takes on only a finite number of values and which is such th… 08TG
Definition 5 — A real-valued function λ defined on a clan Φ of subsets of a set A is said to be additive if, for every pair M, N of disjoint sets belonging to Φ, λ(M ∪ N) = λ(M) + λ(N). ↩ 1 08TH
Proposition 18 — Let λ be an additive set function defined on a clan Φ. There exists one and only one linear form (again denoted λ) on the vector space E(Φ) of real-valued Φ-step functions, such… ↩ 1 08TI
Proposition 19 — Let X be a locally compact space, Φ a clan of subsets of X, containing the set of compact subsets of X. For every continuous mapping f of X into a Banach space F (resp. every con… ↩ 3 08TJ
Corollary 1 Let μ be a measure on X; the space E_F(Φ) is dense in each of the spaces L_Fᵖ (1 ≤ p < +∞). ↩ 2 08TK
Corollary 2 For every closed subset S of X, every function f ∈ K(X, S; C) is the uniform limit of linear combinations ∑ᵢ λᵢ φ_Kᵢ, where the λᵢ belong to C and the Kᵢ are compact subsets of S. ↩ 1 08TL
Corollary 3 If μ and ν are two measures on X such that μ(K) = ν(K) for every compact subset K of X, then μ = ν. ↩ 2 08TM
Theorem 5 Let Φ be a set of subsets of a locally compact space X, satisfying (PCI) and (PC{II}), and let α be a real-valued function, defined on Φ, satisfying the conditions (PMI), (PM{II})… ↩ 2 08TN
Corollary — Let X be a locally compact space with a countable base, Ψ the set of Borel sets of X, β a mapping of Ψ into [0, +∞] satisfying the following conditions: 08TO

§ 5. Measurable functions and sets

Definition 1 Let X be a locally compact space, μ a measure on X. A mapping f of X into a topological space F is said to be measurable for the measure μ (or to be μ-measurable) if, for every com… ↩ 2 08UC
Proposition 1 For a mapping f of X into F to be measurable, it is necessary and sufficient that, for every compact set K ⊂ X and every number ε > 0, there exist a compact set K₁ ⊂ K such that |μ… ↩ 2 08UD
Proposition 2 Let (Fₙ) be a sequence of topological spaces and, for each n, let fₙ be a measurable mapping of X into Fₙ. For every compact set K ⊂ X and every ε > 0, there exists a compact set K… ↩ 4 08UE
Definition 2 A subset A of X is said to be measurable if its characteristic function φ_A is measurable. 08UF
Proposition 3 For a set A to be measurable, it is necessary and sufficient that, for every compact set K, A ∩ K be integrable. ↩ 3 08UG
Corollary 1 The open sets and the closed sets are measurable. 08UH
Corollary 2 — If X is metrizable, then every Souslin subset A of X (GT, IX, §6, No. 2) is μ-measurable for every measure μ on X. ↩ 2 08UI
Proposition 4 (Principle of localization) — Let f be a mapping of X into a topological space F. Suppose that for every x ∈ X, there exist an integrable neighborhood Vₓ of x and a measurable mapping gₓ of X into F such that… ↩ 3 08UJ
Definition 3 — A set A ⊂ X is said to be locally negligible (for the measure μ) if, for every x ∈ X, there exists a neighborhood V of x such that V ∩ A is negligible. 08UK
Proposition 5 — For a set A to be locally negligible, it is necessary and sufficient that, for every compact set K, A ∩ K be negligible. ↩ 4 08UL
Corollary 1 — For a set A to be negligible, it is necessary and sufficient that it be locally negligible and of finite outer measure. 08UM
Corollary 2 — Every locally negligible open set is negligible (and is therefore contained in the complement of the support of μ). ↩ 2 08UN
Corollary 3 — In a locally compact space X that is countable at infinity, every locally negligible set is negligible. 08UO
Corollary 4 — Let f be a mapping of X into a topological space F. If the set N of points of discontinuity of f is locally negligible, then f is measurable. ↩ 1 08UP
Proposition 6 Let f be a measurable mapping of X into a topological space F. Every mapping of X into F, equal to f locally almost everywhere, is measurable. ↩ 1 08UQ
Theorem 1 Let X be a locally compact space, μ a measure on X, (Fₙ) a sequence of topological spaces, F = ∏ₙ Fₙ their product. For each index n, let fₙ be a measurable mapping of X into Fₙ, a… ↩ 26 08UR
Remark — 1) The theorem does not extend to an \emph{arbitrary} product of topological spaces (Exer. 1). 2) If f is a continuous mapping of X into itself, and g is a measurable mapping of… 08US
Corollary 1 The upper envelope and the lower envelope of a finite number of measurable numerical functions (finite or not) are measurable. For, sup(u, v) and inf(u, v) are continuous on overli… ↩ 2 08UT
Corollary 2 For a numerical function f (finite or not) to be measurable, it is necessary and sufficient that f⁺ and f⁻ be measurable. The condition is necessary by Cor. 1; it is sufficient bec… 08UU
Corollary 3 — If f and g are two measurable mappings of X into a topological vector space F, then f + g and αf are measurable (α any scalar). ↩ 1 08UV
Corollary 4 — Let F be a vector space of dimension n over R and let (ek){1 \leq k \leq n} be a basis of F. In order that a function f = \sum{k=1}^{n} ek fk be measurable, it is necessary and s… 08UW
Corollary 5 — Let F, G, H be three topological vector spaces, and let (u, v) \to [u \cdot v] be a continuous bilinear mapping of F \times G into H. If f is a measurable mapping of X into F, an… ↩ 4 08UX
Corollary 6 — If f is a measurable mapping of X into a normed space F, then the numerical function f is measurable. ↩ 1 08UY
Theorem 2 (Egoroff) — Let X be a locally compact space, μ a measure on X, A a countable set, 𝔉 a filter on A having a countable base, and (f_α)_α ∈ A a family of measurable mappings of X into a metriz… ↩ 8 08UZ
Corollary 1 — Let (fₙ) be a sequence of numerical functions (finite or not). If the fₙ are measurable, then the functions supₙ fₙ, infₙ fₙ, limsup_n → ∞ fₙ and liminf_n → ∞ fₙ are measurable. ↩ 3 08V0
Corollary 2 — The measurable sets in X form a tribe (GT, IX, §6, No. 3). ↩ 3 08V1
Corollary 3 The Borel sets in X (GT, IX, §6, No. 3, Def. 4) are μ-measurable for every measure μ on X. ↩ 1 08V2
Theorem 3 For a mapping f of X into a metrizable space F to be measurable, it is necessary and sufficient that, for every compact set K ⊂ X, there exist a sequence (gₙ) of measurable step fu… ↩ 5 08V3
Corollary 1 Let f be a measurable mapping of X into a Banach space F; for every compact set K ⊂ X, there exists a sequence (gₙ) of measurable step functions, with supports contained in K, such… ↩ 1 08V4
Corollary 2 Let X be a locally compact space that is countable at infinity. If f is a measurable mapping of X into a metrizable space F, then there exists a sequence (gₙ) of measurable step fu… 08V5
Proposition 7 Let f be a measurable mapping of X into a topological space F; the inverse image under f of every closed (resp. open) set in F is measurable. ↩ 10 08V6
Theorem 4 — Let F be a metrizable space and let d be a metric on F compatible with the topology. For a mapping f of X into F to be measurable, it is necessary and sufficient that it satisfy… ↩ 8 08V7
Remark 1 Condition a) alone is not sufficient for f to be measurable (Exer. 7). 08V8
Remark 2 If the topology of F has a countable base then condition b) of Th. 4 is automatically satisfied for every mapping of X into F. The proof shows, moreover, that it suffices to assume… 08V9
Remark 3 The hypothesis a) can be replaced by the condition that the inverse image under f of every open ball of F is measurable. 08VA
Proposition 8 — Let D be a countable dense subset of R. In order that a numerical function f (finite or not) be measurable, it is sufficient (and necessary) that for every a ∈ D, the set of x ∈… 08VB
Corollary — Every lower (resp. upper) semi-continuous function is measurable. ↩ 3 08VC
Proposition 9 — If F is a metrizable compact space, then every measurable mapping f of X into F is the uniform limit (on all of X) of a sequence of measurable step functions. 08VD
Proposition 10 — Let F be a separable Banach space, F' its dual, and (a'ₙ) a weakly dense sequence in the unit ball of F' (TVS, III, § 3, No. 4, Cor. 2 of Prop. 6).¹ For a mapping f of X into F t… ↩ 5 08VE
Corollary 1 Let F be a Banach space. In order that a mapping f of X into F be measurable, it is necessary and sufficient that it satisfy the following two conditions: 08VF
Corollary 2 Let F be a locally convex space that is metrizable and separable, and let F' be its dual. For a mapping f of X into F to be measurable, it is necessary and sufficient that for ever… 08VG
Proposition 11 Let F be a locally convex space that is the direct limit of a sequence of separable metrizable locally convex spaces Fₙ, F being the union of the Fₙ. Let F' be the dual of F, equip… 08VH
Theorem 5 — In order that a mapping f of X into a Banach space F be p-th power integrable (1 ≤ p < +∞), it is necessary and sufficient that f be measurable and that Nₚ(f) be finite. ↩ 18 08VI
Lemma 1 Let g be a function with values in F, such that Np(g) < +\infty (in other words, a function in F_Fᵖ). The set A of points x \in X such that g(x) \neq 0 is contained in the union of… ↩ 2 08VJ
Corollary 1 For a set to be integrable, it is necessary and sufficient that it be measurable and have finite outer measure. 08VK
Corollary 2 Let (Fₙ) be a sequence of topological spaces; for every index n, let fₙ be a measurable mapping of X into Fₙ and let f(x) = (fₙ(x)) ∈ F = ∏ₙ Fₙ; finally, let u be a continuous mapp… 08VL
Corollary 3 For every integrable function f and every measurable set A, the function f φ_A is integrable. ↩ 1 08VM
Corollary 4 For every sequence (fₙ) of measurable functions geqslant 0 on X, ↩ 2 08VN
Corollary 5 For every sequence (Aₙ) of pairwise disjoint measurable sets, 08VO
Definition 4 — Given a locally compact subspace Y of a locally compact space X, we call measure induced on Y by a measure μ on X, and denote by μ_Y or μ|Y, the measure defined by the formula ↩ 2 08VP
Example — Let μ be Lebesgue measure on R, I any interval in R; I is a locally compact subspace of R, and the measure induced by μ on I is the linear form 08VQ
Lemma 2 — Let μ be a positive measure on X, and let K be a compact subset of X. ↩ 3 08VR
Lemma 3 Let Y be a locally compact subspace of X; for every measure μ on X, |μ_Y| = |μ|_Y. ↩ 5 08VS
Proposition 12 — Let μ be a measure on a locally compact space X, A a μ-measurable subset of X, and K a set of compact subsets of A satisfying the following conditions: (PLI) Every closed (hence… ↩ 8 08VT
Definition 6 — Let A be a μ-measurable subset of X. A set K of compact subsets of A is said to be μ-dense in A if it satisfies the conditions (PLI), (PLII), a), b), c), d) of Prop. 12. ↩ 1 08VU
Remark — Suppose that A is the union of a sequence (Lₙ) of compact sets and a μ-negligible (resp. locally μ-negligible) set, and let K be a set of compact subsets that is μ-dense in A. Ap… 08VV
Proposition 13 Let A be a μ-measurable subset of X, K a set of compact subsets that is μ-dense in A. Let ℌ be a set of compact subsets of A satisfying (PL_I) and (PL_II) and such that, for every… ↩ 2 08VW
Definition 7 A set 𝔄 of subsets of a topological space T is said to be locally countable if, for every t ∈ T, there exists a neighborhood V of t such that the set of A ∈ 𝔄 that intersect V is c… 08VX
Proposition 14 Let X be a locally compact space, μ a measure on X, A a μ-measurable subset of X, and K a set of compact subsets of A that is μ-dense in A. There exists a locally countable set ℌ ⊂… ↩ 6 08VY
Proposition 15 Let X be a locally compact space, μ a measure on X, A a μ-measurable subset of X, and f a mapping of A into a topological space F. The following conditions are equivalent: ↩ 14 08VZ
Definition 8 Let X be a locally compact space, μ a measure on X, and A a μ-measurable subset of X. A mapping f of A into a topological space F is said to be μ-measurable if it satisfies the equ… ↩ 1 08W0
Corollary 1 Let X be a locally compact space, μ a measure on X, A a μ-measurable subset of X, and f a μ-measurable mapping of A into a topological space F. Let K be a set of compact subsets of… 08W1
Corollary 2 Let K be a compact subspace of X; in order that a mapping f of K into a topological space F be μ-measurable, it is necessary and sufficient that it be μ_K-measurable. 08W2
Proposition 16 Let 𝔄 be a locally countable set of μ-measurable subsets of X and let B = ⋃_A ∈ 𝔄 A. For a mapping f of B into a topological space F to be μ-measurable, it is necessary and suffici… ↩ 5 08W3
Definition 9 The uniform structure on S(A, μ; F) of which the W(V, B, δ) form a fundamental system of entourages is called the uniform structure of convergence in measure in A.. 08W4
Proposition 17 — Let (A_λ)_λ ∈ L be a locally countable family of μ-measurable subsets of A, pairwise disjoint and such that A - ⋃_λ ∈ L A_λ is locally μ-negligible. If, for every class f ∈ S(A… ↩ 2 08W5
Proposition 18 — If F is metrizable, and A is the union of a locally μ-negligible set and a sequence (Aₙ) of μ-integrable sets, then the space S(A, μ; F) is metrizable. ↩ 1 08W6
Lemma 4 — Let F be a metrizable uniform space, and let B ⊂ A be a countable union of μ-integrable sets. Then, for every Cauchy sequence (fₙ) in S(A, μ; F), there exists a subsequence (f_nₖ… ↩ 2 08W7
Proposition 19 If the uniform space F is metrizable and complete, then the uniform space S(A, μ; F) is complete. 08W8
Corollary — Let F be a metrizable uniform space. 08W9
Proposition 20 — Let F be a Banach space, equipped with the uniform structure defined by its norm. ↩ 1 08WA
Remark — 1) The topological vector space S(X, μ; F) is not necessarily locally convex (Exer. 24). 2) The topology induced on the set of f such that Nₚ(f) ≤ 1 by the topology of convergenc… 08WB
Definition 10 Let X be a locally compact space, μ a measure on X, F a Banach space, and p ∈ [1, +∞[. A subset H of L_Fᵖ(X, μ) is said to be equi-integrable of order p (for μ) if it satisfies the… ↩ 1 08WC
Remark — Suppose μ is bounded. For every a > 0, the set of measurable mappings of X into F such that |f(x)| ≤ a almost everywhere is equi-integrable of order p, and this is true for any p… 08WD
Proposition 21 Let H be a subset of L_Fᵖ(X, μ) that is equi-integrable of order p. On H, the uniform structure of convergence in measure is equal to the uniform structure induced by that of L_Fᵖ(… ↩ 1 08WE
Lemma 5 Let X be a locally compact space, μ a bounded positive measure on X, F a Banach space, and f a bounded function on X with values in F. The following conditions are equivalent: ↩ 1 08WF
Proposition 22 — Let F be a Banach space, X a locally compact space, E the set of bounded positive measures on X, μ an element of E, and B a filter base on E. Assume that B converges vaguely to μ… ↩ 2 08WG
Remark The conditions (i) and (ii) of Proposition 22 are satisfied if f is continuous and bounded. 08WH
Example Let us take for X the compact space U of complex numbers of absolute value 1. On setting μ(f) = ∫₀¹ f(e^2iπ t) dt for every f ∈ H(U), one defines a positive measure of mass 1 on U… 08WI

§ 6. Convexity inequalities

Theorem 1 — Let X be a locally compact space, μ a positive measure on X, F a real Banach space, D a closed convex set in F, and f a function on X such that f(X) ⊂ D. For every non-negligible… ↩ 2 08XD
Corollary — If the positive measure μ has total mass equal to 1 and if f is integrable, then ∫ f dμ belongs to the closed convex envelope of f(X) in F. 08XE
Definition 1 — Let X be a locally compact space, μ a measure on X. Given a numerical function f (finite or not), defined locally almost everywhere in X, we call maximum in measure, or μ-maximum… 08XF
Proposition 1 (Inequality of the mean) — Let f be a measurable numerical function that is bounded in measure. For every integrable numerical function g geqslant 0, the function fg (defined almost everywhere) is integrab… ↩ 4 08XG
Definition 2 For every mapping f of X into a Banach space F, one sets N_∞(f) = M_∞(|f|); f is said to be bounded in measure (for the measure μ) if N_∞(f) is finite. The set of mappings of X int… ↩ 1 08XH
Proposition 2 The space L_F^∞ is complete; the space L_F^∞ is a Banach space. ↩ 1 08XI
Remark — 1) Every bounded continuous function f on X with values in F belongs to L_F^∞, and 08XJ
Theorem 2 (Hölder’s inequality) — Let f and g be two numerical functions that are finite almost everywhere and are such that f is equal almost everywhere to a function in Lᵖ and g to a function in L^q. Then, the… ↩ 3 08XK
Corollary 1 — Let F, G, H be three Banach spaces, and let (u, v) ↦ Φ(u, v) be a continuous bilinear mapping of F × G into H such that |Φ(u, v)| ≤ |u| · |v|. If f ∈ L_Fᵖ and g ∈ L_G^q, then the… ↩ 2 08XL
Corollary 2 — Let F be a real (resp. complex) Banach space, F' its strong dual (TVS, III, § 3, No. 1), and let (z, z') ↦ ⟨ z, z' ⟩ be the canonical bilinear form on F × F'. If f ∈ L_Fᵖ and g ∈… ↩ 1 08XM
Corollary 3 — Let μ be a positive measure on X, F a real (resp. complex) Hilbert space. On the space L_F², the symmetric (resp. Hermitian) form 08XN
Corollary 4 — Let F be a Banach space, f a function in L_Fᵖ, and g a numerical function belonging to L^q; then, the function fg is integrable and ↩ 1 08XO
Corollary 5 — Let f₁, f₂, …, fₙ be n positive integrable functions, and let α₁, α₂, …, αₙ be n numbers > 0 such that ∑ᵢ₌₁ⁿ αᵢ = 1; under these conditions, the function f₁^α₁ f₂^α₂ ⋯ fₙ^αₙ is i… 08XP
Proposition 3 — Let μ be a positive measure on X, F a real or complex Banach space, F' its strong dual, and (z, z') ↦ ⟨ z, z' ⟩ the canonical bilinear form on F × F'. ↩ 1 08XQ
Remark — 1) Let E be a dense linear subspace of L_F'^q; then the formula (9) holds when g runs over the intersection of E with the set B of functions in L_F'^q such that N_q(g) ≤ 1. For… 08XR
Corollary — Let μ be a positive measure on X, F a Banach space, F' its strong dual, and g any function in L^q_F'. Then, the linear form on Lᵖ_F, deduced from the linear form f ↦ ∫ ⟨ f, g ⟩ d… ↩ 1 08XS
Proposition 4 — Let f be a measurable function with values in a Banach space F; the set I of numbers p, such that 1 ≤ p ≤ +∞ and Nₚ(f) is finite, is either empty or is an interval of overlineR… ↩ 8 08XT
Corollary — If r, s, p are three numbers such that ↩ 2 08XU
Proposition 5 — Let μ be a bounded measure, and let f be a μ-measurable function with values in a Banach space F. The set I of numbers p such that 1 ≤ p ≤ +∞ and Nₚ(f) is finite, is either empty… ↩ 1 08XV
Corollary — If the measure μ is bounded, the relation r < s implies L_Fˢ ⊂ L_Fʳ; moreover, the topology of convergence in mean of order s is finer than the topology of convergence in mean of… 08XW
Proposition 6 — Let X be a discrete space, μ the measure on X defined by placing a mass +1 at each point of X. If f is a mapping of X into the Banach space F, the set I of numbers p such that 1… 08XX
Corollary — If X is discrete and the measure μ is defined by a mass +1 at each point of X, then the relation r < s implies L_Fʳ ⊂ L_Fˢ; moreover, the topology of convergence in mean of order… 08XY

§ 7. Barycenters

Definition 1 Let K be a compact subset of a Hausdorff locally convex space E. For every positive measure μ on K of total mass equal to 1, the vector b_μ = ∫ x dμ(x) (belonging to E'^*) is calle… 08YJ
Example — Let μ be a discrete measure on K, positive and of total mass 1; it is thus of the form μ = ∑ᵢ₌₁ⁿ λᵢ ε_xᵢ, where xᵢ ∈ K, and the λᵢ are real numbers such that λᵢ ≥ 0 for all i and… 08YK
Proposition 1 Let E be a Hausdorff locally convex space, K a compact subset of E, and C the closed convex envelope of K in E. The set C then consists of the points of E that are barycenters of a… ↩ 4 08YL
Corollary If the closed convex envelope C of K in E is compact, then the barycenter of every positive measure of total mass 1 on K belongs to E. ↩ 1 08YM
Remark The Cor. of Prop. 1 is applicable in particular when K is convex or when E is quasi-complete. 08YN
Proposition 2 — Let K be a compact convex subset of a Hausdorff locally convex space E, μ a positive measure of total mass 1 on K, b_μ its barycenter. For every convex numerical function f ≥ 0 t… ↩ 2 08YO
Corollary — For every convex numerical function function g on K that is bounded and lower semi-continuous, g(b_μ) ≤ ∫ g dμ. ↩ 2 08YP
Proposition 3 — Let K be a compact convex subset of a Hausdorff locally convex space E, x a point of K. Every measure μ on K, positive, of total mass 1, and admitting x as barycenter, is in the… ↩ 1 08YQ
Corollary — Let K' be a compact subset of K such that K is the closed convex envelope of K'. In order that x ∈ K' be an extremal point of K, it is necessary and sufficient that εₓ be the onl… ↩ 1 08YR
Theorem 1 (Choquet) — Let E be a Hausdorff locally convex space over R, K a metrizable compact convex subset of E, and M the set of extremal points of K. The set M is the intersection of a countable f… ↩ 5 08YS
Lemma 1 — Let overlineu be the lower envelope of the continuous affine linear functions on E that are ≥ u on K. Then S is the set of points (a, b) ∈ E × R such that a ∈ K and u(a) ≤ b ≤ ov… ↩ 1 08YT
Lemma 2 If u is strictly convex on K, then u(x) < overlineu(x) for every non-extremal point x of K. 08YU
Lemma 3 Let E be a Hausdorff locally convex space over R, and K a metrizable compact convex subset of E. Then, there exists a strictly convex numerical function on K. 08YV
Corollary Let E be a Hausdorff locally convex space over R, C a proper convex cone in E with vertex 0, that is complete and metrizable for the uniform structure induced by the weakened unifo… 08YW
Proposition 4 (i) The closed convex envelope C of i_H(X) in H' (for the weak topology σ(H', H)) is compact. ↩ 2 08YX
Proposition 5 Every function h ∈ H attains its supremum at at least one H-extremal point. ↩ 4 08YY
Proposition 6 For every point x ∈ X, the following properties are equivalent: ↩ 1 08YZ
Proposition 7 Let F be a closed subset of X. The following properties are equivalent: ↩ 2 08Z0
Proposition 8 Suppose X is metrizable. Then the set Ch_H(X) of H-extremal points of X is the intersection of a countable family of open sets in X, and for every x ∈ X, there exists a positive me… 08Z1
Example — Take X to be the unit ball |x| ≤ 1 in R³, and let H be a vector space of continuous functions on X, containing the restrictions to X of the affine linear functions on R³ and sati… 08Z2
Proposition 9 For every function f ∈ H, Ch_H(X) intersects the set of points where |f| attains its supremum. ↩ 3 08Z3
Proposition 10 Let F be a closed subset of X. The following properties are equivalent: a) F contains checkS_H(X). b) For every function f ∈ H, F intersects the set of points of X where |f| attain… 08Z4
Remark If f is a continuous real function, a point where |f| attains its supremum is a point where one of the functions f, -f attains its supremum. For a vector space H of continuous real… 08Z5
Lemma 4 Let X be a compact space, H a closed linear subspace of the Banach space C(X; C) (resp. C(X; R)). Let a be a point of X admitting a countable fundamental system of neighborhoods; a… ↩ 1 08Z6
Theorem 2 (E. Bishop) — Let X be a compact space, A a closed subalgebra of the complex Banach algebra C(X; C). Assume that A contains the constants and separates the points of X. Let a be a point of X… ↩ 1 08Z7
Corollary — Suppose in addition that X is metrizable. Then the following properties are equivalent: 08Z8
Example — Let X₁ be the set of points (z₁, z₂) ∈ C² such that |z₁|² + |z₂|² ≤ 1 (the unit ball in R⁴) and let A₁' be the set of restrictions to X₁ of the holomorphic functions, with values… 08Z9
Lemma 5 Assume that C is weakly complete. Let A be the set of restrictions to C of the continuous linear forms on E. Let (f_λ)_λ ∈ Λ be a finite family of elements of A, and f = sup(f_λ)… ↩ 2 08ZA
Lemma 6 We maintain the hypotheses and notations of Lemma 5, and assume the f_λ to be positive. The function overlinef is positively homogeneous, concave and upper semi-continuous in C. It… ↩ 1 08ZB
Theorem 3 (Choquet) Let E be a Hausdorff weak locally convex space, C a weakly complete proper convex cone with vertex 0 in E, G the union of the extremal generators of C, K a compact convex subset of… ↩ 1 08ZC
Corollary Let E be a Hausdorff locally convex space, C a proper convex cone with vertex 0 in E, admitting a compact sole M, and let G be the union of the extremal generators of C. Let x ∈ M… 08ZD

V. Integration of measures

§ 1. Essential upper integral

Definition 1 For every function f ∈ F₊(T) one calls essential upper integral of f with respect to μ, and denotes by μ^bullet(f), the supremum, finite or not, of the set of numbers μ^*(f φ_K), w… ↩ 1 08ZO
Proposition 1 a) If f and g are two numerical functions ≥ 0 that are equal locally almost everywhere, then μ^bullet(f) = μ^bullet(g). b) If f and g are two numerical functions ≥ 0 such that f ≤… ↩ 4 08ZP
Proposition 2 Let f, g, h be three elements of F₊; if g and h are measurable, then ↩ 2 08ZQ
Corollary Let f be a function ≥ 0, (gₙ) a sequence of measurable functions ≥ 0; then ∫^bullet f ( ∑ₙ gₙ ) dμ = ∑ₙ ( ∫^bullet f gₙ dμ ). ↩ 1 08ZR
Proposition 3 For every finite number α ≥ 0 and every pair of measures μ, ν on T, ↩ 2 08ZS
Proposition 4 — For every numerical function f geqslant 0 that is lower semi-continuous on T, μ^bullet(f) = μ^*(f). ↩ 5 08ZT
Proposition 5 — Let A be a subset of T; the following properties are equivalent: a) The set A is contained in the union of a sequence of μ-integrable open sets. b) The set A is contained in the… ↩ 4 08ZU
Definition 2 — A subset of T is said to be μ-moderated if it satisfies the equivalent conditions of Proposition 5. A function defined on T, with values in a vector space or in overlineR, is sai… 08ZV
Remark — 1) If θ is a complex measure on T, one says that a function f is θ-moderated (resp. that θ is moderated) if f is |θ|-moderated (resp. if |θ| is moderated). 2) Every bounded measu… 08ZW
Proposition 6 — Let f be a positive numerical function defined on T that is μ-measurable and μ-moderated. Then there exists a sequence (hₙ)_n ∈ N of elements of F₊(T), with sum equal to f, havin… ↩ 3 08ZX
Proposition 7 — Let f be an element of F₊(T). 1) If the function f is not μ-moderated, then μ^*(f) = +∞. 2) If the function f is μ-moderated, then μ^*(f) = μ^*(f). 3) If μ^*(f) < +∞ then there e… ↩ 5 08ZY
Corollary 1 For f to be negligible, it is necessary and sufficient that it be locally negligible and moderated. ↩ 2 08ZZ
Corollary 2 If μ is a moderated measure (in particular if μ is bounded, or if T is countable at infinity), then μ^* = μ^bullet. ↩ 1 0900
Proposition 8 a) Let H be a set of functions ≥ 0, lower semi-continuous, directed for the relation ≤; then 0901
Lemma Let g and g' be two μ-moderated mappings with values in F; if g and g' are equal locally almost everywhere to a same function f, then g = g' almost everywhere. 0902
Proposition 9 For a mapping f of T into F to belong to overlineF_Fᵖ (resp. to overlineL_Fᵖ), if is necessary and sufficient that (resp. that f be measurable and that) ↩ 5 0903
Corollary For f to belong to L_Fᵖ, it is necessary and sufficient that f belong to overlineL_Fᵖ and be moderated. ↩ 4 0904
Definition 3 The elements of overlineL_F¹ are called essentially μ-integrable functions with values in F. On composing the mapping tildef ↦ μ(f) of L_F¹ into F with the canonical mapping of ove… ↩ 1 0905
Proposition 10 Let 𝔎 be a μ-dense set of compact subsets of T. a) If f is a numerical function ≥ 0, then μ^bullet(f) = sup_K ∈ 𝔎 μ^*(f φ_K). b) If f is an essentially integrable function with val… ↩ 3 0906
Proposition 11 — Let (λ_α)_α ∈ A be a family of positive measures on T, directed for the relation ≤ and having a supremum λ in M(T). Then, for every numerical function f ≥ 0, ↩ 8 0907
Corollary 1 For a subset N of T to be locally λ-negligible, it is necessary and sufficient that N be locally λ_α-negligible for every α ∈ A. ↩ 1 0908
Corollary 2 For a mapping g of T into a topological space G to be λ-measurable, it is necessary and sufficient that it be λ_α-measurable for every α ∈ A. ↩ 2 0909

§ 2. Summable families of positive measures

Remark 1 It is immediate that, when the family (λ_α)_α ∈ A is summable, its sum is the supremum in M₊(X) of the finite partial sums ∑_α ∈ J λ_α, where J runs over the set of finite subsets… ↩ 1 090G
Remark 2 Let (θ_α)_α ∈ A be a family of complex measures on X; the family (θ_α) is said to be summable if the family (|θ_α|) of positive measures is summable; it is not sufficient for this… ↩ 1 090H
Proposition 1 — Let f be a positive numerical function defined on X. Then ν^bullet(f) = ∑_α ∈ A λ_α^bullet(f). ↩ 7 090I
Corollary 1 — For every compact (resp. open and relatively compact) subset M of X, ν(M) = ∑_α ∈ A λ_α(M). 090J
Corollary 2 — For a subset N of X to be locally ν-negligible, it is necessary and sufficient that, for every α ∈ A, N be locally λ_α-negligible. ↩ 2 090K
Corollary 3 — For every function f ∈ F₊(X), ν^*(f) ≥ ∑_α ∈ A λ_α^*(f). 090L
Proposition 2 — Let f be a mapping of X into a topological space G. For f to be ν-measurable, it is necessary and sufficient that f be λ_α-measurable for every α ∈ A. ↩ 6 090M
Proposition 3 — For a mapping f of X into a Banach space F to be essentially ν-integrable, it is necessary and sufficient that f be essentially λ_α-integrable for every α ∈ A and that ↩ 4 090N
Corollary 1 — Suppose that f is ν-integrable; then f is λ_α-integrable for every α ∈ A, and formula (6) holds. Conversely, if the set A is finite and f is λ_α-integrable for every α ∈ A, then… ↩ 1 090O
Corollary 2 — Let θ be a complex measure on X; set θ₁ = (Re θ)⁺, θ₂ = (Re θ)⁻, θ₃ = (Im θ)⁺, θ₄ = (Im θ)⁻. In order that a mapping f of X into a topological space G (resp. into a Banach space… 090P
Proposition 4 — Let μ be a positive measure on a locally compact space T, and let 𝔎 be a μ-dense set of compact subsets of T. There exists a summable family (μ_α)_α ∈ A of positive measures on T… ↩ 12 090Q
Remark A positive measure may be the sum of a sequence of measures with compact support, and not be moderated (see Exer. 4 a) of § 1). 090R

§ 3. Integration of positive measures

Example 1 Suppose that T is a discrete space, and that μ is the measure on T defined by placing a mass +1 at each point of T (Ch. III, §1, No. 3). Let h be a function geqslant 0 defined on T… 090W
Example 2 The mapping t ↦ εₜ of T into M₊(T) is vaguely continuous, scalarly essentially μ-integrable for every positive measure μ on T, and one has ∫ εₜ dμ(t) = μ. 090X
Proposition 1 Suppose that μ is the supremum of an increasing directed family (μᵢ)_i ∈ I of positive measures on T; in order that Λ : t ↦ λₜ be scalarly essentially μ-integrable, it is necessary… ↩ 1 090Y
Corollary — Suppose that μ is the sum of a summable family (μ_α)_α ∈ A of positive measures on T; in order that Λ : t ↦ λₜ be scalarly essentially μ-integrable, it is necessary and sufficien… ↩ 5 090Z
Definition 1 — Let X be a locally compact space, Λ : t ↦ λₜ a scalarly essentially μ-integrable mapping of T into M₊(X), and ν the integral of Λ. ↩ 6 0910
Proposition 2 — Let Λ : t ↦ λₜ be a scalarly essentially μ-integrable mapping of T into M₊(X), and let ν = ∫ λₜ dμ(t). ↩ 8 0911
Lemma Let X be a locally compact space having a countable base. Then, there exists in K(X) a countable subset S having the following property: for every function f ∈ K(X), there exist a… 0912
Proposition 3 — Let f be a numerical function geqslant 0 defined on X. a) The following inequalities hold: ↩ 6 0913
Corollary 1 — Let f be a function ≥ 0 defined on X, and let H be the set of t ∈ T such that f is not λₜ-negligible. a) If f ν-negligible, then H is locally μ-negligible. b) If f is ν-negligibl… ↩ 4 0914
Corollary 2 — Let f be a function ≥ 0 defined on X, ν-measurable and ν-moderated. The set of t ∈ T such that f is not λₜ-moderated is then locally μ-negligible (and even μ-negligible if Λ is v… ↩ 1 0915
Proposition 4 — Let f be a ν-measurable function defined on X, with values in a topological space G, and let M be the set of t ∈ T such that f is not λₜ-measurable. a) Suppose that f is constant… ↩ 4 0916
Proposition 5 Let f be a ν-measurable positive numerical function defined on X, and let N be the set of t ∈ T such that f is not both λₜ-measurable and λₜ-moderated. ↩ 9 0917
Corollary — Let f be a function defined on X, with values in a Banach space F or in overlineR, that is ν-measurable and ν-moderated. For f to be ν-integrable, it is necessary and sufficient… 0918
Theorem 1 — Let f be a function with values in a Banach space F or in overlineR, and let H be the set of t ∈ T for which f is not λₜ-integrable. ↩ 10 0919
Remark Let Λ : t ↦ λₜ be a μ-adequate mapping of T into M₊(X). If a mapping Λ' : t ↦ λ'ₜ of T into M₊(X) is equal to Λ locally μ-almost everywhere, it follows at once from the definitions… 091A
Definition 2 A mapping f of T into a topological space F is said to be universally measurable if it is μ-measurable for every positive measure μ on T. 091B
Proposition 6 For a mapping f of T into a topological space F to be universally measurable, it is necessary and sufficient that f be measurable for every positive measure on T with compact suppo… ↩ 1 091C
Proposition 7 Let μ be a positive measure on T, and let f be a μ-measurable mapping of T into a topological space F. Then, there exists a universally measurable mapping f' of T into F such that… ↩ 3 091D
Definition 3 Let X be a locally compact space, and let Λ : t ↦ λₜ be a mapping of T into M₊(X). The mapping Λ is said to be a diffusion of T in X if Λ is adequate for every positive measure on… 091E
Proposition 8 — For a mapping Λ : t ↦ λₜ of T into M₊(X) to be a diffusion, it is necessary and sufficient that the following conditions be satisfied: 1) For every lower semi-continuous function… ↩ 1 091F
Proposition 9 — Let Λ : t ↦ λₜ be a mapping of T into M₊(X), such that the function t ↦ λₜ(g) is universally measurable and locally bounded in T for every g ∈ K₊(X). One can affirm that Λ is a d… 091G
Proposition 10 Let f, g be two positive universally measurable functions on X, let a be a number ≥ 0, and let μ and ν be two positive measures on T. Then: 091H
Proposition 11 For a positive measure μ on T to belong to the domain of Λ, it is necessary and sufficient that Λ be scalarly essentially μ-integrable. 091I
Corollary 1 — If Λ is a bounded diffusion, then every bounded positive measure μ belongs to the domain of Λ, and | μ Λ | ≤ | μ | | Λ |. 091J
Corollary 2 — Suppose that μ is the sum of a summable family (μ_α)_α ∈ A of positive measures belonging to the domain of Λ. For μ to belong to the domain of Λ, it is necessary and sufficient t… 091K
Proposition 12 — Let μ be a positive measure on T that belongs to the domain of Λ, and let f be a universally measurable function ≥ 0 defined on X. If f is moderated for the measure μ Λ, or if th… 091L
Corollary — If X is countable at infinity, or if the measures λₜ are bounded, then the function Λ f is universally measurable on T for every universally measurable function f ≥ 0 defined on… ↩ 1 091M
Proposition 13 — Let T, X, Y be three locally compact spaces, Λ : t ↦ λₜ a bounded diffusion of T in X, and H : x ↦ ηₓ a bounded diffusion of X in Y. The mapping t ↦ λₜ H is then a bounded diffus… ↩ 2 091N
Definition 4 The notations being those of Proposition 13, the diffusion Λ H is called the composed diffusion (or the composition) of the bounded diffusions H and Λ. 091O
Example Let u be a universally measurable mapping of T into X, and v a universally measurable mapping of X into Y; by Prop. 2 b), one defines diffusions Λ and H by the formulas 091P

§ 4. Integration of positive point measures

Definition 1 Let μ be a positive measure on T. The pair (π, g) is said to be μ-adapted if the following conditions are satisfied: 1° The functions π and g are μ-measurable. 2° For every functio… 0921
Proposition 1 If the pair (π, g) is μ-adapted, then the mapping Λ : t ↦ λₜ = g(t)ε_π(t) of T into M₊(X) is scalarly essentially μ-integrable, vaguely μ-measurable and μ-adequate. Conversely, if… ↩ 1 0922
Lemma Let T and X be two topological spaces, π a proper continuous mapping (GT, I, §10, No. 1, Def. 1) of T into X. Let g be a lower semi-continuous numerical function defined on T. For… 0923
Theorem 1 Let (π, g) be a μ-adapted pair, and let ν = ∫ g(t)ε_π(t) dμ(t). For every numerical function f ≥ 0 defined on X, (1) ∫^bullet f(x) dν(x) = ∫^bullet f(π(t))g(t) dμ(t). ↩ 4 0924
Corollary — In order that a subset N of X be locally negligible for ν, it is necessary and sufficient that the intersection of π⁻¹(N) with the set of points t ∈ T where g(t) > 0 be locally n… ↩ 1 0925
Proposition 2 — Let π be a proper continuous mapping (GT, I, §10, No. 1) of T into X, and let g be a continuous, finite numerical function on T such that g(t) > 0 for all t ∈ T. The pair (π, g)… ↩ 5 0926
Proposition 3 — Let (π, g) be a μ-adapted pair, and let ↩ 7 0927
Remark If f is a mapping of X into a Banach space F, it comes to the same to say that the restriction of f ∘ π to S is μ-measurable or to say that the function (f ∘ π)g (defined on T) is… 0928
Theorem 2 Let (π, g) be a μ-adapted pair, and let ν = ∫ g(t) ε_π(t) dμ(t). Let f be a function defined on X, with values in a Banach space F or in overlineR. For f to be essentially ν-integr… ↩ 11 0929
Remark — Let (π, g) be a μ-adapted pair, π' a mapping of T into X, and g' a finite numerical function ≥ 0 defined on T, such that π' (resp. g') is equal to π (resp. g) locally almost ever… 092A

§ 5. Measures defined by numerical densities

Proposition 1 Let g be a function defined locally almost everywhere in T (for the positive measure μ), with values in a Banach space F (resp. in overlineR). The following properties are equivale… ↩ 2 092D
Definition 1 A function g, defined locally almost everywhere in T (for the positive measure μ), with values in a Banach space F (resp. in overlineR), is said to be locally integrable for μ (or… 092E
Definition 2 Let θ be a complex measure, and let u be a complex function (or a function with values in overlineR) defined locally θ-almost everywhere and locally θ-integrable. The complex measu… ↩ 1 092F
Proposition 2 Let θ be a complex measure, and u a locally θ-integrable complex function; then ↩ 13 092G
Lemma 1 — Let θ be a complex measure, and let f be an element of overlineL¹_C(T, θ). Then ↩ 1 092H
Corollary Let g₁ and g₂ be two locally μ-integrable numerical functions; then ↩ 4 092I
Proposition 3 For every numerical function f geqslant 0 defined on T, ↩ 10 092J
Corollary 1 In order that a function f, with values in a Banach space or in overlineR, be locally negligible for the measure u · θ, it is necessary and sufficient that u f be locally negligibl… ↩ 3 092K
Corollary 2 Let u₁ and u₂ be two locally θ-integrable complex functions. In order that u₁ · θ = u₂ · θ, it is necessary and sufficient that u₁ and u₂ be equal locally almost everywhere. ↩ 10 092L
Corollary 3 Let u be a complex function that is locally integrable for the positive measure μ. For u · μ to be a positive measure, it is necessary and sufficient that u(t) geqslant 0 locally a… ↩ 3 092M
Proposition 4 For a mapping f of T into a topological space G to be measurable for the measure u · θ, it is necessary and sufficient that the restriction of f, to the θ-measurable set S of the t… ↩ 2 092N
Corollary Let f be a function defined on T, with values in a Banach space F or in overlineR. For f to be (u · θ)-measurable, it is necessary and sufficient that u f be θ-measurable. ↩ 2 092O
Theorem 1 Let f be a function defined on T, with values in a Banach space F or in overlineR. In order that f be essentially integrable for the measure η = u · θ, it is necessary and sufficie… ↩ 10 092P
Corollary For the measure u · θ to be bounded, it is necessary and sufficient that u be essentially θ-integrable. ↩ 2 092Q
Example Let A be a subset of T; for φ_A to be locally μ-integrable, it is necessary and sufficient that A be μ-measurable. Assuming the condition to be fulfilled, set ν = φ_A · μ; for ever… 092R
Proposition 5 Let (λ_α)_α ∈ A be a family of positive measures on T, directed for the relation leqslant, admitting in M(T) a supremum λ. For a positive numerical function g to be locally λ-integ… ↩ 1 092S
Corollary — Suppose that μ is the sum of a family (μ_α)_α ∈ A of measures on T. For a positive numerical function g defined on T to be locally μ-integrable, it is necessary and sufficient th… 092T
Proposition 6 — Let (g_α)_α ∈ A be a locally countable family of locally μ-integrable positive numerical functions defined on T. In order that the function g = ∑_α ∈ A g_α be locally μ-integrabl… ↩ 2 092U
Corollary — Let (g_α) be a sequence of locally μ-integrable numerical functions, such that the sequence of measures gₙ · μ is increasing. In order that this sequence have an upper bound in t… ↩ 1 092V
Proposition 7 — Let X be a locally compact space that is countable at infinity, and let t ↦ λₜ be a μ-adequate mapping of T into M₊(X). ↩ 1 092W
Proposition 8 Let θ be a complex measure on T, g₁ a locally θ-integrable complex function, θ₁ the measure g₁ · θ. For a complex function g₂ defined on T to be locally θ₁-integrable, it is necess… ↩ 8 092X
Theorem 2 (Lebesgue–Nikodym) — Let μ and ν be two positive measures on T. The following properties are equivalent: ↩ 20 092Y
Lemma 2 — Let α be a bounded positive measure on T, and β a real measure on T such that |β| ≤ M α, where M is a positive constant. Then, there exists a real function u, α-integrable, such… ↩ 1 092Z
Lemma 3 — Suppose that the positive measure ν is such that every μ-negligible compact set is ν-negligible. Let 𝔎 be the set of compact subsets K of T having the following property: 0930
Corollary 1 — Let N be a set of positive measures with base μ, admitting a supremum ν in M(T); then ν is a measure with base μ. ↩ 1 0931
Corollary 2 — Let ν be a real measure on T. In order that ν belong to the band generated by μ in the fully lattice-ordered space M(T) (Ch. II, §1, No. 5), it is necessary and sufficient that ν… ↩ 3 0932
Corollary 3 — Let θ be a complex measure; there exists a universally measurable function v, with |v| = 1, such that θ = v · |θ|, |θ| = overlinev · θ. ↩ 8 0933
Remark 1 Suppose that λ is a positive measure, that v is a λ-measurable function such that |v| = 1 locally λ-almost everywhere (which implies that v is locally λ-integrable), and that θ = v… 0934
Remark 2 If |θ| ≤ a μ, where μ is a positive measure and a is a number ≥ 0, then θ is a measure with base μ. 0935
Corollary 4 — Let ρ and θ be two complex measures. In order that there exist a locally θ-integrable function u such that ρ = u · θ, it is necessary and sufficient that every θ-negligible compa… 0936
Corollary 5 — Let μ and ν be two positive measures on T. The conditions 1), 2), 3) of Th. 2 are also equivalent to the following ones: 0937
Scholium For every dotf ∈ L¹_loc(T, μ; R), set φ(dotf) = f · μ, where f ∈ dotf; the mapping φ is linear, increasing and injective (Cor. 2 of Prop. 3), and its image in M(T) is the band B ge… 0938
Proposition 9 Let θ be a bounded complex measure; for θ to be a positive measure, it is necessary and sufficient that | θ | = θ(1).. ↩ 2 0939
Proposition 10 Let μ and ν be two positive measures on T. The following conditions are equivalent: a) The locally negligible sets are the same for μ and ν. b) The bands generated by μ and ν in M(… ↩ 3 093A
Definition 3 Two complex measures θ, θ' on a locally compact space T are said to be equivalent if the measures |θ| and |θ'| satisfy the conditions a), b), c) of Prop. 10. ↩ 1 093B
Remark — If μ and ν are two equivalent positive measures then the measurable functions defined on T, with values in any topological space G, are the same for μ and ν, as follows at once f… 093C
Proposition 11 Let μ be a positive measure on T. If T is countable at infinity, then there exists a continuous function h such that h(t) > 0 for all t ∈ T and such that the measure ν = h · μ (equ… ↩ 5 093D
Proposition 12 Let (μₙ) be a sequence of bounded positive measures on T; there exists a bounded positive measure μ on T such that the relation μ^*(N) = 0 is equivalent to «μₙ^*(N) = 0 for every n… 093E
Definition 4 One says that a complex measure θ on T is concentrated on a subset M of T, or that M carries θ, if CM is locally negligible for θ. ↩ 2 093F
Proposition 13 In order that two complex measures ρ and σ on T be alien to each other, it is necessary and sufficient that there exist in T two disjoint sets R and S such that ρ is concentrated o… ↩ 2 093G
Corollary 1 For every real measure ν on T, there exist two disjoint sets M, N carrying ν⁺ and ν⁻, respectively. 093H
Corollary 2 — Let ρ and σ be two alien complex measures, and let ρ' and σ' be two complex measures admitting densities relative to ρ and σ, respectively; then ρ' and σ' are alien. ↩ 1 093I
Corollary 3 — Let ρ and σ be two alien complex measures; then |ρ + σ| = |ρ| + |σ|. ↩ 1 093J
Theorem 3 (Lebesgue) — Every complex measure θ on T may be written in one and only one way in the form θ = g · μ + θ', where g is locally μ-integrable and θ' is a measure alien to μ. Then |θ| = |g| · μ… ↩ 2 093K
Corollary — There exists a projector p of the space M_C(T) onto the space L¹_loc(T, μ; C), whose kernel overlinep⁻¹(0) is the set of complex measures alien to μ, such that 093L
Theorem 4 — Let p and q be two conjugate exponents such that 1 ≤ p < +∞. Every continuous linear form on Lᵖ(T, μ) is of the type f ↦ ∫ fg dμ, where g is a function in L^q(T, μ) whose class i… ↩ 4 093M
Corollary — For every number p such that 1 < p < +∞, the Banach space Lᵖ(T, μ) is reflexive. 093N
Proposition 14 — In order that a positive linear form θ on L^∞ be of the type f ↦ ∫ f g dμ, where g ∈ L¹, it is necessary and sufficient that, for every increasing directed family (f_α)_α ∈ A of… ↩ 2 093O
Lemma 4 1° Let f be a lower semi-continuous and bounded positive function on T. Then its class dotf in L^∞ is the supremum of the set of classes dotφ, where φ runs over the set of function… 093P
Definition 5 A measure θ on T is said to be diffuse if |θ|(t) = 0 for every t ∈ T. 093Q
Example — Lebesgue measure on R is diffuse (Ch. IV, §1, No. 3, Remark 1). 093R
Proposition 15 Every complex measure σ on T may be written in one and only one way in the form ρ + θ, where ρ is an atomic measure and θ is a diffuse measure; one then has |σ| = |ρ| + |θ|.. ↩ 3 093S

§ 6. Images of a measure

Proposition 1 Let π be a μ-measurable mapping of T into a locally compact space X. The following two properties are equivalent: ↩ 3 094O
Definition 1 Let μ be a positive measure on a locally compact space T. A mapping π of T into a locally compact space X is said to be μ-proper (or proper for the measure μ) if the pair (π, 1) is… ↩ 2 094P
Remark — 1) If μ is bounded (in particular, if μ has compact support) then every μ-measurable mapping of T into X is μ-proper (Ch. IV, § 5, No. 3, Th. 1 and No. 6, Th. 5). ↩ 1 094Q
Proposition 2 — For every numerical function f geqslant 0 defined on X, ∫^bullet f(x) dν(x) = ∫^bullet f(π(t)) dμ(t). (2) ↩ 3 094R
Corollary 1 — For every subset A of X, ν^bullet(A) = μ^bullet(π⁻¹(A)). ↩ 1 094S
Corollary 2 — For a subset A of X to be locally negligible for ν, it is necessary and sufficient that π⁻¹(A) be locally negligible for μ. ↩ 6 094T
Corollary 3 — If the measure μ is concentrated on a set M, then π(μ) is concentrated on π(M). ↩ 5 094U
Corollary 4 — Let S be the support of μ. If π is continuous, then the support of π(μ) is π(S). 094V
Proposition 3 — For a mapping f of X into a topological space G to be ν-measurable, it is necessary and sufficient that f ∘ π be μ-measurable. ↩ 6 094W
Corollary — For a subset A of X to be ν-measurable, is is necessary and sufficient that π⁻¹(A) be μ-measurable. ↩ 1 094X
Theorem 1 — Let f be a function defined on X with values in overlineR or in a Banach space F. For f to be essentially ν-integrable, it is necessary and sufficient that f ∘ π be essentially μ… ↩ 9 094Y
Corollary — For a subset A of X to be essentially ν-integrable, it is necessary and sufficient that π⁻¹(A) be essentially μ-integrable, in which case ν(A) = μ(π⁻¹(A)). ↩ 1 094Z
Proposition 4 — Let T, T', T'' be three locally compact spaces, μ a positive measure on T, π a μ-measurable mapping of T into T', π' a mapping of T' into T'', and π'' = π' ∘ π. ↩ 5 0950
Corollary Let T and T' be two locally compact spaces, μ a positive measure on T, π a bijective mapping of T onto T', and π⁻¹ the inverse mapping. Suppose that π is μ-proper, and let μ' = π(μ… 0951
Proposition 5 Let T and X be two locally compact spaces, μ a positive measure on T, π a μ-proper mapping of T into X, g a finite numerical function geqslant 0, defined on X and such that g ∘ π i… ↩ 1 0952
Proposition 6 — Let T and X be two locally compact spaces, (λ_α)_α ∈ A a family of positive measures on T, directed for the relation ≤, admitting in M(T) a supremum μ. In order that a mapping π… 0953
Corollary 1 — Let (μ_α)_α ∈ A be a summable family of positive measures on T, such that μ = ∑_α ∈ A μ_α; for a mapping π of T into a locally compact space X to be μ-proper, it is necessary and… ↩ 1 0954
Corollary 2 — Let T and X be two locally compact spaces, (λᵢ)_1 ≤ i ≤ n a finite sequence of positive measures on T, and let μ = ∑ᵢ₌₁ⁿ λᵢ. For a mapping π of T into X to be μ-proper, it is nec… ↩ 2 0955
Definition 2 Let θ be a complex measure on a locally compact space T. A mapping π of T into a locally compact space X is said to be θ-proper if it is |θ|-proper. The measure f ↦ ∫ (f ∘ π) dθ is… 0956
Proposition 7 Let θ be a complex measure on T, π a θ-proper mapping of T into a locally compact space X, ν the image measure π(θ). ↩ 2 0957
Proposition 8 Suppose g is μ-integrable. If J denotes the open interval in R with end-points G(a+) and G(b-), the image under G of the measure g · μ is the measure φ_J · ν if G(a+) ≤ G(b-) and t… ↩ 2 0958
Corollary If a function f defined on G(I), with values in overlineR or in a Banach space, is such that the function t ↦ f(G(t))g(t) is integrable on I for Lebesgue measure, then f is integra… 0959
Proposition 9 Suppose g ≥ 0 and locally μ-integrable. Then, the image under G of the positive measure g · μ is Lebesgue measure on G(I). For a function f, defined on G(I), with values in overlin… 095A
Proposition 10 With the above notations, suppose that π is ν-measurable. Let g be the function t ↦ λₜ^*(1) on T. For π to be ν-proper, it is necessary and sufficient that g be locally μ-integrabl… ↩ 3 095B
Corollary 1 — Suppose that π is ν-measurable. ↩ 4 095C
Corollary 2 Suppose that π is ν-proper; let f be a mapping defined on T, with values in a Banach space or in overlineR. For f ∘ π to be essentially ν-integrable, it is necessary and sufficient… 095D
Example — Let X and T be two locally compact spaces, and let π be a local homeomorphism of X into T. In other words (GT, I, §11, Exer. 25), we assume that every point x ∈ X admits a neighb… 095E
Lemma — a) Let f be an element of K₊(X); the function t ↦ λₜ(f) is positive, upper semi-continuous, with compact support, and its restriction to π(X) is continuous. 095F
Proposition 11 Let π be a local homeomorphism of a locally compact space X into a locally compact space T, and let μ be a positive measure on T. Let n be the numerical function that associates to… 095G

§ 7. Integration with respect to an induced measure

Proposition 1 — For every numerical function g geqslant 0 defined on X, ↩ 7 0963
Corollary 1 — For every subset B of X, μ_X^bullet(B) = μ^bullet(B); for B to be locally μ_X-negligible, it is necessary and sufficient that B be locally μ-negligible. 0964
Corollary 2 — Let M be a subset of T. If μ is concentrated on M, then μ_X is concentrated on M ∩ X. ↩ 1 0965
Corollary 3 — For the measure μ_X to be zero, it is necessary and sufficient that X be locally μ-negligible. 0966
Remark — If S is the support of μ, then S ∩ X (which is closed in X) contains the support of μ_X by Cor. 2, but may be distinct from it. For example, if μ is a diffuse measure and X is a… 0967
Proposition 2 — For a mapping g of X into a topological space to be μ_X-measurable, it is necessary and sufficient that g be μ-measurable in X (§5, No. 3, Example). ↩ 1 0968
Corollary — For a subset B of X to be μ_X-measurable, it is necessary and sufficient that B be μ-measurable. 0969
Theorem 1 — Let g be a function defined on X, with values in overlineR or in a Banach space. For g to be essentially μ_X-integrable, it is necessary and sufficient that g be essentially μ-in… ↩ 6 096A
Corollary 1 For a subset B of X to be essentially μ_X-integrable, it is necessary and sufficient that it be essentially μ-integrable, in which case μ_X(B) = μ(B). 096B
Corollary 2 Let g be a complex function defined on T and locally μ-integrable; the restriction g_X of g to X is then locally μ_X-integrable, and ↩ 1 096C
Corollary 3 Let θ be a complex measure on T; then ↩ 2 096D
Remark — a) Cor. 3 has already been proved by another method (Ch. IV, § 5, No. 7, Lemma 3). b) By virtue of Cor. 3, the corollaries 1,2,3 of Prop. 1, Prop. 2, Theorem 1 and its corollarie… 096E
Scholium — For every function f (resp. g) defined on X (resp. T) with values in the Banach space F or in overlineR, let us denote by ζ(f) (resp. ρ(g)) the extension by 0 of f to T (resp. th… 096F
Proposition 3 — Let X be a locally compact subspace of T, and λ a complex measure on X. The following properties are equivalent: a) the canonical injection i : X → T is λ-proper; b) for every co… ↩ 2 096G
Corollary 1 — If X is closed, then every complex measure λ on X is induced by a measure on T. 096H
Corollary 2 — Let θ be a complex measure on T, π a θ-proper mapping of T into a locally compact space Y, and πX its restriction to X. Then πX is θX-proper, and πX(θX) = π(φX · θ). 096I
Proposition 4 — Let X and Y be two locally compact subspaces of T such that Y ⊂ X. If θ is a complex measure on T, then the measure (θX)Y induced by θX on Y is equal to θY ('transitivity of indu… ↩ 2 096J
Proposition 5 — Let (λ_α)_α ∈ A be an increasing directed family of positive measures on T, admitting a supremum λ, and let X be a locally compact subspace of T. The family of induced measures λ… 096K
Corollary — Let (μᵢ)_i ∈ I be a summable family of positive measures on T, with sum μ. The family of induced measures μᵢ|X is then summable, and 096L
Proposition 6 — Let Λ : t ↦ λₜ be a μ-adequate mapping of T into M₊(X), where X is a locally compact space that is countable at infinity, and let Y be a locally compact subspace of X. Set ∫ λₜ d… 096M

§ 8. Products of measures

Lemma 1 For every numerical function f geqslant 0 defined on X, ↩ 4 096Y
Lemma 2 Let f be a mapping of X into a topological space. For f to be λ'ₜ-measurable, it is necessary and sufficient that fₜ be μ'-measurable. This is a consequence of Prop. 3 of §6, No. 2… 096Z
Lemma 3 Let f be a function defined on X, with values in overlineR or in a Banach space. For f to be λ'ₜ-integrable, it is necessary and sufficient that fₜ be μ'-integrable, in which case 0970
Remark Lemmas 1,2,3 can be proved very simply without making use of the results of §§4 and 6, by a direct argument. For example, the relation (2) is obvious by definition if f ∈ K(T × T')… 0971
Proposition 1 — Let (μ_α)_α ∈ A (resp. (μ'_β)_β ∈ B) be a summable family of positive measures on T (resp. on T'), with sum denoted by μ (resp. by μ'). The family (μ_α ⊗ μ'_β)_(α, β) ∈ A × B is… ↩ 1 0972
Proposition 2 — Let f be a ν-measurable function defined on T × T', with values in a topological space G, and let M be the set of t ∈ T such that the mapping t' ↦ f(t, t') is not μ'-measurable. ↩ 4 0973
Corollary — Let A be a ν-measurable subset of T × T', and let M be the set of t ∈ T such that the section A(t) of A at t is not μ'-measurable. 0974
Proposition 3 — Let f be a mapping of T into a topological space F. If f is μ-measurable, then the mapping (t, t') ↦ f(t) is ν-measurable. Conversely, if μ' ≠ 0, and if this mapping is ν-measura… ↩ 3 0975
Corollary 1 — Let F, F' and G be three topological spaces, and let u be a continuous mapping of F × F' into G. Let f (resp. f') be a function defined on T (resp. T') with values in F (resp. F'… ↩ 4 0976
Corollary 2 — If A ⊂ T and A' ⊂ T' are measurable (for μ and μ', respectively), then A × A' is measurable for μ ⊗ μ'. This follows at once from Cor. 1. 0977
Corollary 3 — Consider two positive numerical (resp. complex-valued) functions f defined on T and f' defined on T'. If these functions are measurable for μ and μ', respectively, then the funct… ↩ 1 0978
Proposition 4 — Let A be a subset of T. If A is locally μ-negligible, then A × T' is locally ν-negligible. Conversely, if A × T' is locally ν-negligible and if μ' ≠ 0, then A is locally μ-neglig… ↩ 6 0979
Corollary — If the measure μ (resp. μ') is concentrated on M (resp. M'), then μ ⊗ μ' is concentrated on M × M'. 097A
Proposition 5 — Let f be a numerical function geqslant 0, lower semi-continuous on T × T'. Then, the function ↩ 2 097B
Corollary 1 — Let f (resp. f') be a lower semi-continuous function geqslant 0 defined on T (resp. T'); the function f ⊗ f' : (t, t') ↦ f(t)f'(t') is then lower semi-continuous on T × T', and ↩ 1 097C
Corollary 2 — Let A be a μ-moderated subset of T, and A' a μ'-moderated subset of T'; then A × A' is ν-moderated in T × T'. ↩ 3 097D
Corollary 3 — Let A be a μ-negligible subset of T, and let B' be a μ'-moderated subset of T'; then A × B' is ν-negligible. 097E
Proposition 6 — Let f be a numerical function geqslant 0 defined on T × T'. Then ↩ 3 097F
Proposition 7 — Let f be a ν-measurable positive numerical function defined on T × T'. ↩ 6 097G
Corollary 1 — Let H be a subset of T × T', and let A be the set of t ∈ T such that the section H(t) of H at t is not μ'-negligible. ↩ 1 097H
Corollary 2 — If a set A ⊂ T × T' is ν-integrable, then the section A(t) of A at t is μ′-integrable for almost every t ∈ T, the function t ↦ μ'(A(t)) is μ-integrable, and ↩ 2 097I
Proposition 8 — For every pair of numerical functions f ≥ 0, f' ≥ 0 defined on T and T', respectively, ↩ 4 097J
Corollary 1 With notations as in Prop. 8, 097K
Corollary 2 Let f and f' be two functions with values in C or in overlineR, defined on T and T', respectively, and essentially integrable (resp. integrable) for the measures μ and μ', respecti… ↩ 3 097L
Theorem 1 (Lebesgue–Fubini) — Let f be a function defined on T × T', with values in a Banach space F or in overlineR; let N be the set of t ∈ T such that the function t' ↦ f(t,t') is not μ'-integrable. ↩ 3 097M
Scholium — Let f be a function defined on T × T', with values in overlineR or in a Banach space, that is ν-measurable and ν-moderated. For the three integrals 097N
Remark — 1) When the measure μ' is not moderated, it can happen that f is essentially ν-integrable and the function t' ↦ f(t, t') is not essentially μ'-integrable for any value of t ∈ T (… 097O
Proposition 9 Let F, F' and G be three Banach spaces, and let (x, y) ↦ [x · y] be a continuous bilinear mapping of F × F' into G. Let f (resp. f') be a function defined on T (resp. T') with valu… 097P
Proposition 10 — Let g (resp. g') be a complex function (or a function with values in overlineR) defined on T (resp. T'). a) If g (resp. g') is locally integrable for μ (resp. μ'), then the funct… ↩ 2 097Q
Proposition 11 Let π (resp. π') be a mapping of T (resp. T') into a locally compact space T₁ (resp. T'₁). ↩ 2 097R
Proposition 12 — Let X (resp. X') be a locally compact subspace of T (resp. T'). Then, the induced measure (μ ⊗ μ')_X × X' on the locally compact subspace X × X' of T × T' is equal to the product… 097S

VI. Vectorial integration

§ 1. Integration of vector-valued functions

Definition 1 One calls integral of f with respect to μ, and denotes by ∫ f dμ or ∫ f(t) dμ(t), the element of F'^* defined by ⟨ z', ∫ f dμ ⟩ = ∫ ⟨ z', f ⟩ dμ for all z' ∈ F'. 098E
Example — Let X be a locally compact space, t ↦ λₜ a mapping of T into the space M(X) of measures on X. To say that the family t ↦ λₜ is μ-adequate means that it consists of positive measu… 098F
Remark 1 If F is finite-dimensional, then every scalarly essentially integrable mapping of T into F is essentially integrable (Ch. V, §1, No. 3). However, in the general case, a scalarly ne… 098G
Remark 2 It is clear that the integral of f depends only on the class of f modulo the space of mappings of T into F that are scalarly locally μ-negligible. Note that a scalarly locally negl… ↩ 3 098H
Proposition 1 — If f is a mapping of T into F that is scalarly essentially μ-integrable, then the mapping u ∘ f is scalarly essentially μ-integrable and ∫ (u ∘ f) dμ = u ( ∫ f dμ ). For every z'… ↩ 4 098I
Proposition 2 — Let f be a scalarly essentially μ-integrable mapping of T into F. For every numerical function g geqslant 0 that is μ-measurable and bounded, the mapping t ↦ g(t)f(t) (denoted gf… ↩ 2 098J
Proposition 3 — Let X be a locally compact space, t ↦ λₜ a μ-adequate³ family (Ch. V, §3, No. 1, Def. 1) of positive measures on X, and let ν = ∫ λₜ dμ(t). Let f be a mapping of X into F; assume… ↩ 2 098K
Proposition 4 — Let T and T' be two locally compact spaces, μ (resp. μ') a positive measure on T (resp. T'), ν = μ ⊗ μ' the product measure on X = T × T'. Let f be a mapping of X into F. Assume… ↩ 2 098L
Proposition 5 — Let μ be a bounded positive measure on T, S a μ-measurable set carrying μ (Ch. V, §5, No. 7, Def. 4), f a scalarly μ-integrable () function with values in F. Let D be the closed… ↩ 2 098M
Corollary — Let μ be a bounded positive measure on T, S a μ-measurable set carrying μ, and f a mapping of T into F, scalarly μ-measurable and such that f(S) is contained in a weakly compact… ↩ 4 098N
Proposition 6 — Let f be a scalarly essentially μ-integrable function with values in F, such that ∫ f dμ ∈ F. For every lower semi-continuous semi-norm q on F, ↩ 1 098O
Proposition 7 — Let f be a mapping of T into F, scalarly essentially μ-integrable, such that for every compact subset K of T, f(K) is contained ↩ 1 098P
Lemma 1 The closure in F'^* (for the topology σ(F'^*, F')) of every bounded subset of F is contained in the bidual F''. ↩ 1 098Q
Corollary Suppose F is semi-reflexive, and let f be a scalarly essentially μ-integrable mapping of T into F such that, for every compact subset K of T, f(K) is bounded. Then ∫ f dμ belongs t… ↩ 1 098R
Proposition 8 Let μ be a bounded positive measure on T, S a μ-measurable set carrying μ, f a μ-measurable mapping of T into F, such that f(S) is contained in a complete, bounded, balanced convex… ↩ 9 098S
Proposition 9 Let U be a scalarly essentially μ-integrable mapping of T into F = Lₛ(G; H). In order that ∫ U dμ ∈ F, it is necessary and sufficient that the following two conditions be satisfied… ↩ 2 098T
Corollary — The condition b) of Prop. 9 is satisfied in each of the following two cases: 098U
Proposition 10 — Let U be a mapping of T into F = Lₛ(G; H). In each of the following three cases, U is scalarly essentially μ-integrable and ∫ U dμ ∈ Lₛ(G; H): ↩ 1 098V
Proposition 11 Every Hausdorff locally convex space F with the property (GDF) is barreled. ↩ 2 098W
Theorem 1 (Gelfand–Dunford)—Let F be a Hausdorff locally convex space having the property (GDF), F'ₛ its weak dual. For every mapping f of T into F'ₛ that is scalarly essentially μ-integrabl… ↩ 7 098X
Lemma 2 Let f be a mapping of T into F'ₛ, such that for every z ∈ F, the numerical function ⟨ z, f ⟩ belongs to overlineLᵖ(μ) (1 ≤ p ≤ +∞); let θ(z) be the class of this function in Lᵖ(μ)… ↩ 1 098Y
Corollary 1 Let Gᵢ (1 ≤ i ≤ n) be n Hausdorff locally convex spaces having the property (GDF), and let F be the space of separately continuous multilinear forms on ∏ᵢ₌₁ⁿ Gᵢ, equipped with the… ↩ 2 098Z
Corollary 2 Let G be a Hausdorff locally convex space having the property (GDF), and H a semi-reflexive space whose strong dual H_b' has the property (GDF) (cf. App., No. 2, Prop. 3). Let F be… 0990
Proposition 12 If F is a separable, metrizable locally convex space, then every scalarly μ-measurable mapping f of T into F is μ-measurable. ↩ 2 0991
Proposition 13 Let F be a locally convex space that is the direct limit of a sequence of separable, metrizable locally convex spaces Fₙ, F being the union of the Fₙ. Let F' be the dual of F, equi… ↩ 6 0992
Proposition 14 Let T be a locally compact space, F a quasi-complete, Hausdorff locally convex space, and f a continuous mapping of T into F. If the space C'(T) of measures on T with compact suppo… ↩ 2 0993
Remark — 1) The mapping t ↦ εₜ is a homeomorphism of T into the space C'(T); for, if L is a compact subset of C(T), and t₀ ∈ T, there exists (GT, X, §2, No. 5, Cor. 3 of Th. 2) a neighbor… ↩ 1 0994
Proposition 15 Let T be a locally compact space, F a Hausdorff locally convex space that is quasi-complete, and f a continuous mapping of T into F such that f(T) is bounded. If M¹(T) is equipped… 0995
Lemma 3 Let G, H be two locally convex spaces, U a mapping of a locally compact space T into L(G; H). The following conditions are equivalent: a) The mapping (t, x) ↦ U(t) · x of T × G int… ↩ 1 0996
Proposition 16 — Let G, H be two Hausdorff locally convex spaces, with H assumed to be quasi-complete. Let U be a mapping of T into L(G; H) such that (t, x) ↦ U(t) · x is a continuous mapping of… ↩ 2 0997
Proposition 17 — Let G, H be two Hausdorff locally convex spaces, with H assumed to be quasi-complete. Let U be a mapping of T into L(G; H) such that (t, x) ↦ U(t) · x is continuous on T × G, and… ↩ 1 0998
Proposition 18 — Let G1, G2, H1, H2 be four Hausdorff locally convex spaces, with H1 and H2 assumed to be quasi-complete. Let A : G1 → G2 and B : H1 → H2 be two continuous linear mappings. Let U1… 0999
Remark 1 Suppose that G and H are Banach spaces, and let U be a mapping of T into L(G; H) such that (t, x) ↦ U(t) · x is continuous on T × G. Note that this implies that the finite function… 099A
Remark 2 Suppose that G = H is quasi-complete and that U satisfies the hypotheses of Prop. 16. Let M be a dense subset of the space C'(T), for the weak topology σ(C'(T), C(T)), and let X be… 099B

§ 2. Vectorial measures . "

Definition 1 Let F be a Hausdorff locally convex space over R. One calls vectorial measure on T with values in F every continuous linear mapping of the space K(T) into F. 09A3
Example — 1) The identity mapping of K(T) is a vectorial measure on T with values in K(T). 09A4
Remark — 1) For a linear mapping m of K(T) into F to be a vectorial measure, it is necessary and sufficient that, for every compact subset K of T, the image under m of the unit ball |f| ≤… 09A5
Proposition 1 Let m be a vectorial measure on T with values in F, f a numerical function on T that is essentially integrable for m. The mapping z' ↦ ∫ f d(z' ∘ m)⁺ - ∫ f d(z' ∘ m)⁻ is a linear f… ↩ 4 09A6
Definition 2 Let m be a vectorial measure on T with values in F, f a numerical function on T that is essentially integrable for m. One calls integral of f with respect to m, and denotes by m(f)… 09A7
Example — Let us take for m the identity mapping of K(T) onto itself. Since the dual of K(T) is the space M(T) of scalar measures on T, the functions f ∈ L(m) are those that are essentiall… 09A8
Proposition 2 Every numerical function f essentially integrable for m is essentially integrable for u ∘ m, and ∫ f d(u ∘ m) = u(∫ f dm). 09A9
Proposition 3 If the image under m of the set of f ∈ K(T) such that sup_t ∈ T |f(t)| ≤ 1 is weakly relatively compact in F, then ∫ f dm ∈ F for every bounded numerical function f essentially int… ↩ 8 09AA
Corollary 1 If, for every compact subset K of T, the image under m of the set of f ∈ H(T, K) such that sup_t ∈ T |f(t)| ≤ 1 is weakly relatively compact in F, then ∫ f dm ∈ F for every functio… ↩ 1 09AB
Corollary 2 If F is semi-reflexive, then ∫ f dm ∈ F for every numerical function f essentially integrable for m. ↩ 2 09AC
Definition 3 Let m be a vectorial measure on T with values in F. If q is a lower semi-continuous semi-norm on F, m is said to be q-majorizable if there exists a positive measure μ such that |z'… ↩ 2 09AD
Proposition 4 Let (Fᵢ)_1 ≤ i ≤ n be a finite family of Hausdorff locally convex spaces, F = ∏ᵢ₌₁ⁿ Fᵢ their product, qᵢ (1 ≤ i ≤ n) a lower semi-continuous semi-norm on Fᵢ, and q the semi-norm on… 09AE
Corollary — If the space F is finite-dimensional, then every vectorial measure m with values in F is majorizable. In order that a numerical function be essentially integrable for m, it is ne… ↩ 1 09AF
Proposition 5 — Let q be a lower semi-continuous semi-norm on F. Let m be a q-majorizable measure, and let f be a function essentially integrable for m and such that ∫ f dm ∈ F. Then ↩ 2 09AG
Proposition 6 — Let F be a quasi-complete, Hausdorff locally convex space, m a majorizable measure on T with values in F. If f is a numerical function essentially integrable for all of the measu… ↩ 2 09AH
Lemma 1 — When L₀ and K(T) are equipped with the topology induced by that of L¹((μ_ι)_ι ∈ I): a) each element of L₀ is in the closure of some bounded subset of K(T); b) each element of L¹(… ↩ 1 09AI
Definition 4 Let μ be a positive measure on T. A vectorial measure m on T, with values in F, is said to be a measure with base μ if there exists a mapping f of T into F, scalarly locally μ-inte… ↩ 1 09AJ
Proposition 7 — Let f be a mapping scalarly locally integrable with respect to a positive measure μ on T, such that, for every function g ∈ K(T), one has ∫ gf dμ ∈ F. Then the mapping g ↦ ∫ gf d… 09AK
Proposition 8 — Let μ be a positive measure on T, m a measure on T with values in F, with base μ and density f with respect to μ. Let q be a lower semi-continuous semi-norm on F. ↩ 2 09AL
Remark Suppose that there exists in A_q' a countable subset D that is dense for σ(F', F); then, the function q ∘ f is always μ-measurable, because q(f(t)) = sup_z' ∈ D |⟨ z', f(t) ⟩| (Ch… 09AM
Proposition 9 — Let F be a finite-dimensional Banach space. Every measure m on T with values in F is a measure with base |m|. If m = f · |m|, then |f(t)| = 1 locally almost everywhere for |m|. F… ↩ 2 09AN
Remark If z = ∑ᵢ₌₁ⁿ zᵢ eᵢ then ψ(z₁, …, zₙ) = |z| is a positively homogeneous continuous function on Rⁿ. Setting μᵢ = e'ᵢ ∘ m = hᵢ · |m|, by definition (Ch. V, §5, No. 9) ψ(μ₁, …, μₙ) = ψ… ↩ 1 09AO
Example — Let us take for m the identity mapping of H(T). To say that m is scalarly of base μ means that every real measure on T has base μ. In particular, the point measure εₜ (t ∈ T) mus… 09AP
Lemma 2 Every separable subspace G of the Banach space L^∞(T, μ) has the lifting property. ↩ 1 09AQ
Definition 5 — Let F be a Hausdorff locally convex space, F'ₛ its dual equipped with the topology σ(F', F). We denote by L_F'ₛ^∞ the vector space of mappings f of T into F'ₛ, such that f is sca… 09AR
Lemma 3 Let F be a Hausdorff locally convex space, f an element of L_F'ₛ^∞. For every z ∈ F, one has ⟨ z, f ⟩ ∈ L^∞, and the linear mapping z ↦ π(⟨ z, f ⟩) of F into L^∞ is continuous; if… ↩ 1 09AS
Lemma 4 Let F be a Hausdorff locally convex space, f an element of L_F'ₛ^∞. For every numerical function g ∈ overlineL¹, the function gf is scalarly essentially μ-integrable and ∫ gf dμ ∈… ↩ 1 09AT
Theorem 1 — Let F be a Hausdorff locally convex space that contains a countable dense subset. For every function f ∈ L_F'ₛ^∞ and every z ∈ F, let v_f(z) = π(⟨ z, f ⟩) ∈ L^∞; the mapping f ↦… ↩ 5 09AU
Corollary 1 Let F be a Hausdorff locally convex space containing a countable dense subset. For every function f ∈ L_F'^∞, every z ∈ F and every function g ∈ L¹, let Φ_f(z, tildeg) = ∫ ⟨ z, f(t… ↩ 2 09AV
Corollary 2 (Dunford–Pettis theorem) — Let F be a Hausdorff locally convex space containing a countable dense subset. For every function f ∈ L_F'^∞ and every function g ∈ L¹, let w_f(tildeg) = ∫ gf dμ ∈ F' (Lemma 4)… ↩ 3 09AW
Remark — It is clear that the mappings u ∈ R(L¹, F') are continuous for every G-topology on F' (G a covering of F by bounded subsets). Conversely, if F is assumed moreover to be barreled… 09AX
Corollary 3 Let F be a Hausdorff locally convex space containing a countable dense subset, m a vectorial measure on T with values in the weak dual F' of F. If the image under m of the set B of… ↩ 2 09AY
Corollary 4 Let F be a separable normed space, and m a measure on T with values in the strong dual F', majorizable for the norm of F'. Then m is a measure with base |m|, and if m = f · |m| the… ↩ 1 09AZ
Proposition 10 — Let F be a separable Banach space. For every function f ∈ overlineL¹_F and every function g ∈ L_F'ₛ^∞, the numerical function ⟨ f, g ⟩ : t ↦ ⟨ f(t), g(t) ⟩ is essentially μ-integ… ↩ 2 09B0
Proposition 11 — Let F, G, H be three Banach spaces, Φ a continuous bilinear mapping of F × G into H. Let m be a majorizable vectorial measure on T, with values in G. Then there exists one and on… ↩ 1 09B1
Corollary If m and m' are two majorizable measures on T, with values in G, then I_Φ,m+m' = I_Φ,m + I_Φ,m' and I_Φ,λ m = λ I_Φ,m for every scalar λ. ↩ 1 09B2
Remark — When m is of the form bμ, where b ∈ G and μ is a real measure on T, 09B3
Definition 6 One calls complex measure on T every continuous linear form on the complex vector space K_C(T).\footnote{Cf. Ch. III, §1, No. 3, Def. 2.} 09B4
Proposition 12 Let m be a complex measure on T, p and q conjugate exponents (Ch. IV, §6, No. 4). The bilinear form (f,g) ↦ ∫ fg dm is defined and continuous on the product L_Cᵖ(m) × L_C^q(m); the… ↩ 2 09B5
Lemma 5 Let m be a complex measure on T, f an m-integrable complex function. Then ∫ |f| d|m| = sup |∫ fh dm|, as h runs over the set of functions in H_C(T) such that |h(t)| ≤ 1 for all t ∈… ↩ 1 09B6
Proposition 13 — Let m be a complex measure, and μ = |m|. For m to be bounded, it is necessary and sufficient that μ be bounded, in which case |m| = |μ|. 09B7
Corollary — Let m be a bounded complex measure. Every function f ∈ L_F^∞(m) is then m-integrable, and | ∫ f dm | ≤ N_∞(f)|m|. 09B8

§ 3. Disintegration of measures

Lemma 1 If Y is a locally compact space with a countable base, the space H(Y) contains a countable dense subset. More precisely, there exists in H(Y) a countable subset S consisting of fun… ↩ 4 09C0
Lemma 2 If T is a locally compact space with a countable base, then the Banach space K(Y) of continuous numerical functions tending to 0 at the point at infinity is separable. ↩ 1 09C1
Lemma 3 — Let T and X be two locally compact spaces with countable bases, μ a positive measure on T, and t ↦ λₜ (t ∈ T) a family of positive measures on X. If the mapping t ↦ λₜ is scalarl… ↩ 1 09C2
Theorem 1 Let T and B be two locally compact spaces having countable bases, μ a positive measure on T, p a μ-proper mapping (Ch. V, §6, No. 1, Def. 1) of T into B, and ν = p(μ) the image of… ↩ 3 09C3
Definition 1 Let T and B be two locally compact spaces, μ a positive measure on T, and p a μ-measurable mapping of T into B. A positive measure ν on B is said to be a pseudo-image measure of μ… ↩ 2 09C4
Example — 1) If p is μ-proper and ν = p(μ), then ν is a pseudo-image measure of μ under p (Ch. V, §6, No. 2, Cor. 2 of Prop. 2). 09C5
Proposition 1 — Let T be a locally compact space countable at infinity, μ a positive measure on T, and p a μ-measurable mapping of T into a locally compact space B. Then there exists a pseudo-im… 09C6
Theorem 2 — Let T and B be two locally compact spaces having countable bases, μ a positive measure on T, p a μ-measurable mapping of T into B, and ν a pseudo-image measure of μ under p. Then… ↩ 4 09C7
Definition 2 Let T and B be two Polish locally compact spaces. Given a positive measure μ on T, a μ-measurable mapping p of T into B, and a pseudo-image measure ν of μ under p, every ν-adequate… 09C8
Remark — Th. 1 of Ch. V, §3, No. 4 shows (taking into account the fact that T and B have countable bases) that for every function f defined on T, with values in overlineR or in a Banach s… 09C9
Definition 3 Let T be a locally compact space, μ a positive measure on T. An equivalence relation R in T is said to be μ-measurable if there exists a μ-dense set of compact subsets K of T for w… 09CA
Proposition 2 Let T be a locally compact space countable at infinity, μ a positive measure on T. ↩ 7 09CB
Proposition 3 — Let T be a Polish locally compact space, μ a positive measure on T, and R an equivalence relation in T. The following properties are equivalent: ↩ 1 09CC
Remark If R is a μ-measurable equivalence relation in T, the saturation for R of a compact subset of T is not necessarily μ-measurable (Exer. 5). 09CD
Theorem 3 — Let T be a locally compact space with a countable base, μ a positive measure on T, and R a μ-measurable equivalence relation in T. Then, there exists a μ-measurable subset S of T… ↩ 2 09CE
Theorem 4 Let T be a Polish locally compact space, μ a positive measure on T, and R a μ-measurable equivalence relation in T. Let B, B' be two Polish locally compact spaces, p, p' two μ-meas… ↩ 1 09CF

App. 0. Complements on topological vector spaces

Proposition 1 For a bilinear form Φ ∈ B(E, F) to belong to B(E, F), it is necessary and sufficient that there exist a neighborhood of 0 in E whose image under ˡ Φ is an equicontinuous subset of… ↩ 1 09CS
Corollary If Φ is a continuous bilinear form on E × F, then ˡ Φ is a continuous linear mapping of E into the strong dual F'_b of F. If, moreover, E and F are normed spaces, then |ˡ Φ| = |Φ|. 09CT
Proposition 2 — Let E be a vector space, (F_α)_α ∈ A a family of locally compact spaces having the property (GDF), and for each α ∈ A let h_α be a linear mapping of F_α into E. If E is equipped… ↩ 1 09CU
Corollary — Every quotient space of a locally convex space having the property (GDF) has the property (GDF). 09CV
Proposition 3 — The strong dual of a reflexive Fréchet space has the property (GDF). 09CW
Lemma Let F be a Fréchet space, F' its strong dual, F'' its bidual. If every subset of F'', bounded for σ(F'', F'), is contained in the closure (for σ(F'', F')) of a bounded subset of F… 09CX

VII. Haar measure

§ 1. Construction of a Haar measure

Definition 1 Let μ be a measure on X. a) μ is said to be invariant under G if γ(s)μ = μ for every s ∈ G. b) μ is said to be relatively invariant under G if γ(s)μ is proportional to μ for every… 09CY
Remark — 1) Assume μ invariant. Then |μ|, R(μ), I(μ) are invariant. If μ is real, then μ⁺ and μ⁻ are invariant. 2) Assume μ relatively invariant and nonzero. There exists, for every s ∈ G… 09CZ
Lemma 1 — Let X, Y, Z be three topological spaces, with Y locally compact. Let (x, y) ↦ xy be a continuous mapping of X × Y into Z, which defines a mapping x ↦ uₓ of X into F(Y; Z) by the… ↩ 7 09D0
Proposition 1 — Assume that G is locally compact. Let μ be a nonzero relatively invariant measure on X. Then its multiplier χ is a continuous function on G. ↩ 3 09D1
Definition 2 Let G be a locally compact group. A nonzero positive measure on G that is left (resp. right) invariant is called a left (resp. right) Haar measure on G. 09D2
Theorem 1 On every locally compact group, there exists a left (resp. right) Haar measure, and, up to a constant factor, there exists only one. ↩ 7 09D3
Corollary — Every left-invariant (resp. right-invariant) measure on G is proportional to a left (resp. right) Haar measure. ↩ 1 09D4
Example — 1) On the additive group R, the Lebesgue measure dx is a Haar measure (Ch. III, §1, No. 3, Example). 09D5
Proposition 2 — Let G be a locally compact group, μ a left or right Haar measure on G. For G to be discrete, it is necessary and sufficient that μ(e) > 0. For G to be compact, it is necessary an… ↩ 6 09D6
Definition 3 The function Δ_G on G is called the modulus (or modular function) of G. If Δ_G = 1, the group G is said to be unimodular. 09D7
Remark — If φ is an isomorphism of G onto a locally compact group G', then Δ_G' ∘ φ = Δ_G. In particular: 1) Since x ↦ x⁻¹ is an isomorphism of G onto the opposite group G⁰, one has Δ_G⁰… 09D8
Proposition 3 If there exists in G a compact neighborhood V of e invariant under the inner automorphisms, then G is unimodular. ↩ 1 09D9
Corollary If G is discrete, or compact, or commutative, then G is unimodular. 09DA
Definition 4 The number a > 0 such that φ⁻¹(μ) = a μ is called the modulus of the automorphism φ and is denoted mod_G φ or simply mod φ. 09DB
Proposition 4 Let G be a locally compact group, Γ a topological group, and γ ↦ u_γ a homomorphism of Γ into the group G of automorphism of G, such that (γ, x) ↦ u_γ(x) is a continuous mapping of… ↩ 1 09DC
Proposition 5 — Let (G_ι)_ι ∈ I be a family of locally compact groups. For every ι ∈ I let μ_ι be a left (resp. right) Haar measure on G_ι. Assume that there exists a finite subset J of I such t… 09DD
Example — 1) Lebesgue measure on Rⁿ is a Haar measure of the additive group Rⁿ. 09DE
Proposition 6 For every α ∈ A, let μ_α be a positive measure on G_α. Suppose that φ_βα(μ_α) = μ_β for α ≥ β. Then, there exists one and only one positive measure μ on G such that φ_α(μ) = μ_α fo… ↩ 2 09DF
Definition 5 The measure μ is said to be the inverse limit (or projective limit) of the μ_α. 09DG
Proposition 7 We retain the notations of Prop. 6. If each μ_α is a left (resp. right) Haar measure on G_α, then μ is a left (resp. right) Haar measure on G. 09DH
Proposition 8 a) Let μ and μ' be two positive measures on G such that, for every α and for every coset C of K_α, μ(C) = μ'(C). Then μ = μ'. ↩ 1 09DI
Example — Let Qₚ be the p-adic field, the completion of Q for the p-adic absolute value |x|ₚ = p^-vₚ(x) (GT, IX, §3, No. 2, Example 3). The elements of Qₚ are called p-adic numbers. We den… 09DJ
Proposition 9 Let G be a locally compact group, V an open subset of G, and μ a nonzero positive measure on V having the following property: if U is an open subset of V and if s ∈ G is such that… ↩ 2 09DK
Corollary Let G, G' be two locally compact groups, V (resp. V') an open neighborhood of the neutral element of G (resp. G'), and φ a local isomorphism of G' with G (GT, III, §1, No. 3, Def… ↩ 1 09DL
Example — The Haar measure on T obtained in No. 2, Example 3 may be deduced from the Lebesgue measure on R by a local isomorphism of R with T. 09DM
Proposition 10 Let G be a locally compact group, μ a relatively left-invariant measure on G with multiplier χ. If χ₁ is a continuous representation of G in C^*, the measure χ₁ · μ is relatively l… ↩ 2 09DN
Corollary 1 Let μ be a left Haar measure on G. For a nonzero measure ν on G to be relatively left-invariant, it is necessary and sufficient that it be of the form a χ · μ, where a ∈ C^* and χ… ↩ 4 09DO
Corollary 2 Every relatively left-invariant measure is relatively right-invariant. 09DP
Proposition 11 Let G be a locally compact group, μ a left Haar measure on G. For a measure ν ≠ 0 on G to be left quasi-invariant, it is necessary and sufficient that ν be equivalent to μ. ↩ 2 09DQ
Definition 6 — Let K be a locally compact field⁽¹⁾. For a ∈ K^*, one calls modulus of a, and denotes by mod_K(a) or simply mod(a), the modulus of the automorphism x ↦ ax of the additive group K… 09DR
Example — 1) Let K = R. If s > 0 then s · [0,1] = [0,s]; if s < 0, s · [0,1] = [s,0]. Thus mod_R t = |t| for all t ∈ R. 09DS
Proposition 12 — The function mod is continuous on K, and mod(ab) = mod(a)mod(b) for all a, b in K. ↩ 2 09DT
Proposition 13 — For every M > 0, let V_M be the set of x ∈ K such that mod(x) ≤ M. If K is nondiscrete, the V_M form a fundamental system of compact neighborhoods of 0 in K. 09DU
Corollary — The topology of a nondiscrete locally compact field admits a countable base. ↩ 3 09DV
Proposition 14 — Let α be a Haar measure on K⁺. Then the measure β = (mod_K)⁻¹ · α on K^* is a left Haar measure on the multiplicative group K^*. ↩ 2 09DW
Corollary — Let f be a function defined on K^*, with values in overlineR or in a Banach space. For f to be β-integrable, it is necessary and sufficient that (mod_K)⁻¹ f be α-integrable, in w… 09DX
Proposition 15 — Assume K to be commutative. Let u be an automorphism of the vector space E = K^n. Then ↩ 5 09DY
Corollary 1 — Let K be a commutative locally compact field, E a finite-dimensional vector space over K, and u an endomorphism of the vector space E. Then mod_E(u) = mod_K(det u). ↩ 3 09DZ
Corollary 2 — Let E be a real vector space of finite dimension n, (e₁, e₂, …, eₙ) a basis of E, P the set of x = ∑ᵢ₌₁ⁿ ξᵢ eᵢ ∈ E such that 0 ≤ ξᵢ ≤ 1 for all i, and μ the unique Haar measure o… 09E0
Proposition 16 — Let α be a Haar measure of the additive group of A. The measures ↩ 1 09E1
Proposition 17 — Suppose A is a field (locally compact). For every a ∈ A, mod_A(a) = mod_K N_A/K(a). ↩ 1 09E2
Example — 1) Take K = R, A = C. Taking into account Alg., chap. VIII, §12, n° 2, prop. 4, we obtain mod_C(z) = |z|² for all z ∈ C. (2) 09E3

§ 2. Quotient of a space by a group; homogeneous spaces

Proposition 1 Let f be a continuous real-valued function on X whose support S has compact intersection with the saturation of every compact subset of X. ↩ 5 09EY
Proposition 2 The mapping f ↦ f^χ of K(X) into K^χ(X) is linear, and the image of K(X) (resp. K₊(X)) is K^χ(X) (resp. K₊^χ(X)). ↩ 6 09EZ
Lemma 1 — Let K be a compact subset of X, u a function of K₊(X) with u(x) > 0 for x ∈ K. Let g ∈ K^χ(X) be such that Supp g ⊂ KH. a) One has inf_x ∈ KH u¹(x) > 0. b) The function h equal t… ↩ 2 09F0
Proposition 3 Let μ be a measure on X. The following conditions are equivalent: ↩ 3 09F1
Remark 1 — We are going to show that the mapping f ↦ fᵇ is a strict morphism (GT, III, §2, No. 8) of K(X) onto K(X/H). 09F2
Proposition 4 a) Let λ be a measure on X/H. There exists one and only one measure λ^# on X such that ∫_X/H fᵇ dλ = ∫_X f dλ^# for all f ∈ K(X). One has δ(ξ)λ^# = Δ_H(ξ)λ^# for all ξ ∈ H. ↩ 5 09F3
Definition 1 With hypotheses and notations as in Prop. 4, λ is called the quotient of μ by β and is denoted μ/β or μ/β. 09F4
Remark 2 — Let E be a locally convex vector space and let m be a vectorial measure on X/H with values in E. The mapping f ↦ m(fᵇ) of K(X) into E is then a vectorial measure on X, with value… 09F5
Lemma 2 Let f be a function on X, with values in a topological space. ↩ 1 09F6
Proposition 5 — Let λ be a positive measure on X/H. ↩ 6 09F7
Proposition 6 — Let λ be a positive measure on X/H. ↩ 6 09F8
Corollary 1 — Let λ, λ' be two positive measures on X/H. For λ' to have base λ, it is necessary and sufficient that λ'^# have base λ^#. For λ and λ' to be equivalent, it is necessary and suffi… ↩ 3 09F9
Corollary 2 — Let λ be a positive measure on X/H, and f a λ^#-measurable numerical function on X. Suppose that, for every ξ ∈ H, δ(ξ)f = f locally λ^#-almost everywhere. Then, there exists a λ… ↩ 1 09FA
Corollary 3 — a) Let (λ_ι)_ι ∈ I be a family of real measures on X/H. For the family (λ_ι) to be bounded above in M(X/H), it is necessary and sufficient that the family (λ_ι^#) be bounded abov… ↩ 2 09FB
Remark 1 Prop. 6 a) may be expressed by saying that λ is a pseudo-image measure of λ^# under π (Ch. VI, §3, No. 2, Def. 1). 09FC
Remark 2 Suppose H is compact and β is normalized. The saturation of every compact subset of X is compact. Therefore, if f ∈ K(X/H) then f ∘ π ∈ K(X); and, for every positive measure λ on X… ↩ 1 09FD
Remark 3 Cor. 3 c) of Prop. 6 shows at once that the results of this subsection remain valid in the case of complex measures (except for those that involve the upper integral). 09FE
Remark 4 Let m be a vectorial measure on X/H with values in E, and let q be a lower semi-continuous semi-norm on E. For m to be q-majorizable (Ch. VI, §2, No. 3, Def. 3), it is necessary an… 09FF
Proposition 7 Assume X/H paracompact. Let χ be a continuous representation of H in R₊^*. ↩ 2 09FG
Proposition 8 Assume X/H paracompact. There exists a continuous function h geqslant 0 on X, whose support has compact intersection with the saturation of every compact subset of X, and is such t… ↩ 5 09FH
Remark 1 In particular, let X be a locally compact space on which a discrete group D operates continuously and properly on the right; suppose X/D is paracompact. Then, there exists a contin… 09FI
Remark 2 Let us conserve the hypotheses and notations of Prop. 8. The mapping g ↦ h · (g ∘ π) is a continuous mapping of K(X/H) into K(X) that is a right inverse of the mapping f ↦ fᵇ. Cons… 09FJ
Proposition 9 — We conserve the hypotheses and notations of Prop. 8. Let λ be a positive measure on X/H. ↩ 3 09FK
Corollary — The continuous linear mapping f ↦ fᵇ of L¹(X, λ^#) into L¹(X/H, λ) defined by Prop. 5 is surjective. 09FL
Remark 3 — Suppose X/H paracompact, and let us retain the notations of Proposition 9. The mapping k ↦ h · (k ∘ π) of L¹(X/H, λ) into L¹(X, λ^#) is then isometric by (14) and is a right inve… 09FM
Lemma 3 — Let G be a locally compact group, μ a left Haar measure on G, ν and ν' two nonzero quasi-invariant measures on G. If, for every s ∈ G, the densities of γ(s)ν with respect to ν an… ↩ 1 09FN
Lemma 4 Let λ be a measure ≠ 0 on G/H, and μ a left Haar measure on G. The following properties are equivalent: a) λ is quasi-invariant under G; b) for a subset A of G/H to be locally λ-ne… ↩ 1 09FO
Theorem 1 Let G be a locally compact group, H a closed subgroup of G. 09FP
Lemma 5 Let μ be a left Haar measure on G, and ρ a locally μ-integrable function. For ρ · μ to be of the form λ^#, it is necessary and sufficient that, for every ξ ∈ H, 09FQ
Theorem 2 Let G be a locally compact group, H a closed subgroup of G, μ a left Haar measure on G, and β a left Haar measure on H. ↩ 2 09FR
Remark 1 One deduces from Remark 1 of No. 3 that the nonzero quasi-invariant measures on G/H are none other than the pseudo-images under π of a Haar measure on G. 09FS
Remark 2 If G is a Lie group, we shall see later that the function ρ of Th. 2 can be chosen to be infinitely differentiable. 09FT
Lemma 6 Let λ be a measure on G/H, χ a continuous representation of G in C^*. The following properties are equivalent: a) λ is relatively invariant on G/H with multiplier χ; b) λ^# is rela… ↩ 2 09FU
Theorem 3 Let G be a locally compact group, H a closed subgroup of G, μ (resp. β) a left Haar measure on G (resp. H), χ a continuous representation of G in C^*. ↩ 6 09FV
Corollary 1 For there to exist on G/H a nonzero positive measure relatively invariant under G, it is necessary and sufficient that there exist a continuous representation of G in R₊^* extendin… ↩ 1 09FW
Corollary 2 — For there to exist on G/H a nonzero positive measure invariant under G, it is necessary and sufficient that Δ_G coincide with Δ_H on H. ↩ 1 09FX
Corollary 3 — Suppose that H is unimodular and that there exists on G/H a nonzero bounded positive measure ν relatively invariant under G. Then ν is invariant, and G is unimodular. ↩ 1 09FY
Proposition 10 — Let G be a locally compact group, G' a closed normal subgroup, G'' the group G/G', π the canonical mapping of G onto G/G', and α, α', α'' left Haar measures on G, G', G''. ↩ 7 09FZ
Proposition 11 — We maintain the notations of Prop. 10. Let u be an automorphism of G such that u(G') = G'. Let u' be the restriction of u to G', and u'' the automorphism of G'' deduced from u by… ↩ 2 09G0
Corollary — For every x ∈ G, Δ_G(x) = Δ_G/G'(dotx) mod(iₓ), where dotx denotes the canonical image of x in G/G', and iₓ the automorphism s ↦ x⁻¹sx of G'. 09G1
Lemma 7 Let K be a compact subset of X/H', and L a compact subset of X. Then ⋃_x ∈ L ωₓ⁻¹(K) is relatively compact in H/H'. 09G2
Proposition 12 — With the foregoing notations, ↩ 1 09G3
Corollary 1 — a) Let f be a function on X/H', with values in a Banach space or in overlineR, integrable for μ/β'. There exists a (μ/β)-negligible subset N of X/H having the following property… ↩ 1 09G4
Corollary 2 — Let G be a locally compact group, A and B closed subgroups of G such that A ⊃ B. Suppose that there exists, on the homogeneous space G/B of left cosets with respect to B, a nonze… 09G5
Proposition 13 — Suppose in addition that Z is compact. Let μ_G, μ_X, μ_Y be left Haar measures on G, X, Y, and Λ the restriction of Δ_G to Y. Then the restriction μ of μ_G to Ω is, up to a const… ↩ 5 09G6
Corollary — Let f be a function defined on Ω, with values in a Banach space or in overlineR. For f to be μ-integrable, it is necessary and sufficient that the function (x, y) ↦ f(xy) Δ_G(y)… 09G7
Proposition 14 — Suppose that the conditions of Prop. 13 are fulfilled and that, in addition, Y is normal. ↩ 3 09G8
Remark Prop. 14 applies in particular when G is the topological semi-direct product of X by Y (GT, III, §2, No. 10). In this case, Z = e and Ω = G. Since yx = xiₓ(y) for x ∈ X,\ y ∈ Y, th… 09G9
Definition 2 Let F ⊂ X. One says that F is a fundamental domain (for H) if the restriction of π to F is a bijection of F onto X/H (in other words, F is a system of representatives for the equiv… 09GA
Lemma 8 — Let F be a fundamental domain. For every x ∈ X, ∑_s ∈ H φ_Fs(x) = n(x). (24) ↩ 2 09GB
Proposition 15 Assume that X is countable at infinity. Let μ be a measure ≥ 0 on X. Let F be a fundamental domain such that Fs is μ-measurable for every s ∈ H. Let f be a μ-integrable function on… ↩ 1 09GC
Theorem 4 Let X be a locally compact space countable at infinity, H a discrete group operating continuously and properly on the right in X, π the canonical mapping of X onto X/H, μ a positiv… ↩ 2 09GD
Corollary — We maintain the hypotheses and notations of Th. 4. Let F' be a second μ-measurable fundamental domain. Let u be a function on X, with values in a Banach space or in overlineR, in… 09GE

§ 3. Applications and examples

Proposition 1 — Let G be a subgroup of Aut(E). The following three properties are equivalent: ↩ 2 09GS
Lemma 1 Let F be a Hilbert space, K a compact group, and s ↦ U(s) a representation of K in the group of invertible elements of L(F; F), continuous for the topology of pointwise convergence… 09GT
Remark Let Φ be a nondegenerate positive hermitian form on E. The unitary group U(Φ) is closed in Aut(E), hence is compact (Prop. 1). Prop. 1 also shows that every compact subgroup of Aut… 09GU
Proposition 2 Let X be a locally compact space in which a locally compact group H acts on the right, continuously and properly, by (x, ξ) ↦ xξ. Assume that X/H is paracompact. Let g be a continu… ↩ 2 09GV
Corollary Let X be a locally compact space in which a finite-dimensional real vector space V operates on the right, continuously and properly, by (x, v) ↦ xv. Let π be the canonical mapping… ↩ 1 09GW
Remark — Let E be a finite-dimensional real affine space, T a compact space, μ a measure on T of total mass 1, and f a continuous mapping of T into E. If an origin a in E is chosen, E bec… 09GX
Lemma 2 Let G be a compact group, μ the normalized Haar measure of G, E a finite-dimensional real affine space, A the affine group of E, and ρ a homomorphism of G into A. Assume that, for… ↩ 3 09GY
Proposition 3 Let G be a locally compact group. Let H be a closed normal subgroup of G, isomorphic to Rⁿ and such that G/H is compact. ↩ 1 09GZ
Proposition 4 Let G be a locally compact group and H a closed normal subgroup of G such that K = G/H is compact. Then every continuous representation u of H in R, such that u(sξ s⁻¹) = u(ξ) for… ↩ 1 09H0
Lemma 3 Let G be a topological group generated by a compact neighborhood of e. Let H be a closed subgroup of G such that the homogeneous space G/H is compact. Then H is generated by a comp… 09H1
Lemma 4 Let G be a connected topological group, D a totally disconnected normal subgroup of G. Then D is contained in the center of G. 09H2
Proposition 5 Let G be a connected topological group admitting a discrete normal subgroup D such that K = G/D is compact, and such that the commutator subgroup of K is dense in K. Then D is fini… 09H3
Example 1 General linear group. ↩ 3 09H4
Lemma 5 The continuous representations of GL(n, K) in C^* are the mappings of the form X ↦ χ(det X), where χ is a continuous representation of K^* in C^*. ↩ 1 09H5
Example 2 Affine group. For every X ∈ GL(n, K) and every x ∈ Kⁿ, let (X, x) be the affine linear mapping ξ ↦ Xξ + x in Kⁿ. The set of (X, x) is the affine group G of Kⁿ (A, II, §9, No. 4). T… 09H6
Example 3 Strict triangular group. ↩ 1 09H7
Lemma 6 Let L be a totally ordered finite set. For every λ ∈ L, let V_λ be a free module of finite dimension over a commutative ring k; for λ, μ in L such that λ ≤ μ, let f_λ μ ∈ Homₖ(V_μ… ↩ 2 09H8
Example 4 — Large triangular group. ↩ 2 09H9
Remark The group T(n, K)^* is a closed subgroup of GL(n, K), and Δ_T(n, K)^*((zᵢⱼ)) = ∏ᵢ₌₁ⁿ (mod zᵢᵢ)^2i-n-1. We saw in Example 1 that Δ_GL(n, K) = 1. If n > 1, the function Δ_T(n, K)^*/Δ… 09HA
Example 5 — Special triangular group. Let us take up again the notations at the beginning of Example 4, and consider the subgroup G₁ = G ∩ SL(n, K). This subgroup is the topological semi-dir… 09HB
Example 6 — Special linear group. The closed subgroups T₁(n, K) and ᵗ(T(n, K)^*) of GL(n, K) have intersection e. Thus the mapping (M, N) ↦ M · N is a continuous bijection φ of T₁(n, K) × ᵗ(… ↩ 1 09HC
Lemma 7 — a) Let U = (uᵢⱼ) ∈ GL(n, K). In order that U ∈ Ω, it is necessary and sufficient that det(uᵢⱼ)_k ≤ i, j ≤ n ≠ 0 for k = 2, 3, …, n. b) Ω is an open subset of GL(n, K). c) The map… ↩ 2 09HD
Lemma 8 a) Let U = (uᵢⱼ) ∈ SL(n, K). In order that U ∈ Ω', it is necessary and sufficient that det(uᵢⱼ)_k ≤ i, j ≤ n ≠ 0 for k = 2, 3, …, n. ↩ 1 09HE
Proposition 6 a) The group SL(n, K) is unimodular. ↩ 1 09HF
Lemma 9 Let ψ be a polynomial ≠ 0 of K[X₁, …, Xᵣ]. In the space Kʳ, the set N defined by ψ(x₁, …, xᵣ) = 0 is negligible for the Haar measure. 09HG
Example 7 Iwasawa decomposition of GL(n, K). 09HH
Lemma 10 Let (f₁, f₂, …, fₙ) be a basis of E. a) There exists one and only one orthonormal basis (e₁, e₂, …, eₙ) of E such that fᵢ = e₁ αᵢ₁ + e₂ αᵢ₂ + ⋯ + eᵢ αᵢᵢ (i = 1, 2, …, n) with αᵢᵢ >… ↩ 1 09HI
Proposition 7 Let D₊^* be the group of diagonal matrices with diagonal elements > 0. The mapping (U, D, T) ↦ UDT is a homeomorphism of U(n, K) × D₊^* × T₁(n, K) onto GL(n, K). 09HJ
Example 8 Spaces of hermitian forms. 09HK

App. 1.

Lemma 1 Let X be a locally compact space, R an open equivalence relation in X, such that the quotient space X/R is paracompact; let π be the canonical mapping of X onto X/R. There exists a… 09HX
Lemma 2 Let G be a locally compact group countable at infinity, M a Baire space. Suppose that G operates on the left continuously and transitively in M. For every x ∈ M, let Hₓ be the stab… 09HY

App. 2.

Lemma 1 Let X, B be two locally compact spaces, π a mapping of X into B, ν a positive measure on B. Let b ↦ λb (b ∈ B) be a ν-adequate family of positive measures on X such that, for every… 09HZ

VIII. Convolution and representations

§ 1. Convolution

Definition 1 Let X₁, …, Xₙ be locally compact spaces, μᵢ a measure on Xᵢ (1 ≤ i ≤ n); let X be the product of the Xᵢ, μ that of the μᵢ. Let φ be a mapping of X into a locally compact space Y. O… 09I0
Example — 1) For any φ, the measures ε_xᵢ, where xᵢ ∈ Xᵢ for 1 ≤ i ≤ n, are always convolvable and have convolution product ε_y, with y = φ(x₁, x₂, …, xₙ). Consequently, if each of the μᵢ… 09I1
Lemma 1 For 1 ≤ i ≤ n, let Xᵢ, Yᵢ be two locally compact spaces, μᵢ a measure on Xᵢ, and φᵢ a continuous mapping of Xᵢ into Yᵢ. Let X = ∏ᵢ Xᵢ, Y = ∏ᵢ Yᵢ, μ = ⨂ᵢ μᵢ, and φ the mapping of X… ↩ 1 09I2
Lemma 2 Let T, T', T'' be three locally compact spaces, μ a measure on T, π a μ-measurable mapping of T into T', π' a continuous mapping of T' into T'', and π'' = π' ∘ π. If π'' is μ-prope… ↩ 1 09I3
Proposition 1 Let Xᵢⱼ (1 ≤ i ≤ m, 1 ≤ j ≤ nᵢ), Yᵢ (1 ≤ i ≤ m), and Z be locally compact spaces; for each i, let φᵢ be a mapping of Xᵢ = ∏ⱼ Xᵢⱼ into Yᵢ; let φ be the product of the φᵢ, mapping X… ↩ 2 09I4
Corollary — Let Xᵢ, X'ᵢ (1 ≤ i ≤ n), Y, Y' be locally compact spaces; let φ, φ' be continuous mappings of X = ∏ᵢ Xᵢ into Y and of X' = ∏ᵢ X'ᵢ into Y', respectively; let fᵢ be continuous mapp… ↩ 1 09I5
Proposition 2 — Let X₁, …, Xₙ, Y be locally compact spaces, μᵢ a bounded measure on Xᵢ (1 ≤ i ≤ n), μ the product of the μᵢ, φ a μ-measurable mapping of ∏ᵢ₌₁ⁿ Xᵢ into Y. Then the μᵢ are φ-convol… ↩ 3 09I6
Proposition 3 — Let X₁, …, Xₙ, Y be locally compact spaces, φ a continuous mapping of ∏ᵢ₌₁ⁿ Xᵢ into Y. Then the mapping (μ, …, μₙ) ↦ *_φ (μᵢ) of ∏ᵢ₌₁ⁿ M¹(Xᵢ) into M¹(Y) is a continuous multiline… 09I7
Proposition 4 — Let X₁, …, Xₙ, Y be locally compact spaces, μᵢ a measure on Xᵢ (1 ≤ i ≤ n), Sᵢ its support, and φ a continuous mapping of ∏ᵢ Xᵢ into Y such that the restriction of φ to ∏ᵢ Sᵢ is… ↩ 6 09I8
Proposition 5 Let X₁, …, Xₙ, Y be locally compact spaces, μᵢ a measure on Xᵢ (1 ≤ i ≤ n), μ the product of the μᵢ, φ a μ-proper mapping of ∏ᵢ Xᵢ into Y, and Sᵢ the support of μᵢ. 09I9
Corollary If φ is continuous and the μᵢ have compact support, then the μᵢ are convolvable and *ᵢ μᵢ has compact support. ↩ 2 09IA
Proposition 6 Let X, Y, Z be locally compact spaces, φ a continuous mapping of X × Y into Z, and λ, μ measures on X, Y. For λ and μ to be φ-convolvable, it is necessary and sufficient that the m… ↩ 1 09IB
Proposition 7 Let X, Y, Z be locally compact spaces, φ a continuous mapping of X × Y into Z, and λ, μ measures on X, Y. Assume that for every x ∈ X, εₓ and μ are φ-convolvable. For λ and μ to be… ↩ 3 09IC

§ 2. Linear representations of groups

Definition 1 — (i) U is said to be separately continuous if, for every s ∈ G, U(s) is a continuous endomorphism of E, and if, for every x ∈ E, the mapping s ↦ U(s)x of G into E is continuous. 09IF
Remark 1 To say that U is separately continuous means that s ↦ U(s) is a continuous mapping of G into the space L(E; E) of continuous endomorphisms of E, equipped with the topology of point… ↩ 1 09IG
Remark 2 To say that U is continuous is equivalent to the following set of three conditions: ↩ 4 09IH
Remark 3 Suppose that U is a continuous linear representation of G in E. For every s ∈ G, let widehatU(s) be the continuous extension of U(s) to the completion widehatE of E. Then widehatU… 09II
Remark 4 When E is a normed space, U is said to be isometric if |U(s)| = 1 for every s ∈ G. For this, it suffices that |U(s)| ≤ 1 for all s ∈ G, because one then has 1 = |1| ≤ |U(s)| · |U(s… 09IJ
Proposition 1 — If G is a locally compact group and E is barreled, then every separately continuous linear representation U of G in E is continuous. ↩ 3 09IK
Lemma 1 Let G be a locally compact group, ρ a lower semi-continuous finite numerical function geqslant 0 on G such that ρ(st) leqslant ρ(s)ρ(t) for all s,t ∈ G. Then ρ is bounded above on… 09IL
Lemma 2 Let G be a topological group, U a linear representation of G in a normed space E, and A a dense subset of E. Assume that for every s ∈ G, U(s) is continuous, and that, for every x… 09IM
Proposition 2 Let G be a locally compact group, U a linear representation of G in a normed space E. Let A be a dense subset of E. Assume that for every s ∈ G, U(s) is continuous and that, for ev… ↩ 2 09IN
Lemma 3 Let X be a locally compact space, Y and Z topological spaces, φ a continuous mapping of X × Y into Z, and φₓ the mapping y ↦ φ(x,y) of Y into Z. The spaces C(Y), C(Z) being equippe… ↩ 2 09IO
Proposition 3 (i) If U is separately continuous, then ᵗU is separately continuous when E' is equipped with the weak topology σ(E', E). (ii) If G is locally compact and U is continuous, then ᵗU i… ↩ 1 09IP
Proposition 4 The linear representations γ_χ of G in C(X) and K(X) are continuous. 09IQ
Proposition 5 Suppose that each function χ(s, ·) is bounded. Then γ_χ leaves overlineK(X) stable, and the linear representation γ_χ of G in overlineK(X) is continuous. 09IR
Proposition 6 If one equips M(X) (resp. C'(X)) with the topology of uniform convergence in the compact subsets of K(X) (resp. C(X)), then the linear representation γ_χ of G in M(X) (resp. C'(X))… ↩ 1 09IS
Proposition 7 Suppose that each function χ(s, ·) is bounded. Then γ_χ leaves stable M¹(X) and, if M¹(X) is equipped with the topology of uniform convergence in the compact subsets of K(X), then… 09IT
Proposition 8 The linear representation γ_χ,p of G in L_Cᵖ(X, β) is continuous and isometric. ↩ 2 09IU
Proposition 9 Suppose that each function χ(s, ·) is bounded. Then γ_χ leaves L_Cᵖ(X, β) stable, and the linear representation γ_χ of G in L_Cᵖ(X, β) is continuous. ↩ 3 09IV
Lemma 4 Let T be a locally compact space, a a point of T, M a subset of M(T), and 𝔉 a filter on M. Assume that: (i) for every compact subset K of T, the numbers |μ|(K), for μ ∈ M, are boun… ↩ 2 09IW
Corollary 1 With hypotheses as in Lemma 4, suppose in addition that there exists a compact subset K₀ of T containing the supports of all of the measures μ ∈ M. Then 𝔉 also converges to εₐ in C… ↩ 3 09IX
Corollary 2 With hypotheses as in Cor. 1, let f be a continuous mapping of T into a quasi-complete locally convex space E. Then ↩ 1 09IY
Corollary 3 Let G be a locally compact group, E a quasi-complete locally convex space, and U a continuous linear representation of G in E. Let β be a positive measure on G, a an element of G… ↩ 4 09IZ
Proposition 10 Let G be a locally compact group, E a quasi-complete locally convex space, U a continuous linear representation of G in E, and β a positive measure on G with support G. 09J0

§ 3. Convolution of measures on groups

Proposition 1 — Let G be a locally compact group, λ, μ, ν measures ≠ 0 on G. ↩ 1 09JB
Proposition 2 (i) Any two elements of M^ρ(G) are convolvable. (ii) For convolution, and for the norm | λ |, M^ρ(G) is a complete normed algebra, admitting εₑ as unity element. (iii) C'(G) is a s… 09JC
Proposition 3 Let μ₁, …, μₙ be measures on G. If all of the μᵢ, except at most one, have compact support, then the μᵢ are convolvable. 09JD
Proposition 4 The mapping (λ, μ) ↦ λ * μ (resp. (λ, μ) ↦ μ * λ), where λ ∈ C'(G), μ ∈ M(G), defines on M(G) the structure of a left (resp. right) module over the algebra C'(G). 09JE
Proposition 5 Let λ be a left (resp. right) Haar measure on G, and μ ∈ M¹(G). Then μ and λ (resp. λ and μ) are convolvable, and μ * λ = μ(1)λ (resp. λ * μ = μ(1)λ). ↩ 1 09JF
Proposition 6 Let μ and ν be two convolvable measures on G. Let χ be a continuous representation of G in C^*. Then χ · μ and χ · ν are convolvable and (χ · μ) * (χ · ν) = χ · (μ * ν). ↩ 2 09JG
Proposition 7 Let G and G' be two locally compact groups, u a continuous representation of G in G', and μ₁, …, μₙ measures on G, all ≠ 0. Then the following assertions are equivalent: (i) u is μ… ↩ 1 09JH
Corollary Let G be a locally compact group, μ₁, …, μₙ measures on G. For the sequence (μᵢ)_1 ≤ i ≤ n to be convolvable, it is necessary and sufficient that the sequence (checkμₙ₋ᵢ)_0 ≤ i ≤ n… 09JI
Proposition 8 — Let μ be a measure on G, ν a measure on X. (i) If μ has compact support, then μ and ν are convolvable. (ii) If ν has compact support, and if G operates properly in X, then μ and… ↩ 5 09JJ
Proposition 9 — For convolution, M¹(X) is a left module over M¹(G), while M(X) and C'(X) are left modules over C'(G). This follows from Prop. 8, and from §1, Props. 1, 3 and the Cor. of Prop. 5. ↩ 1 09JK
Proposition 10 — Let μ be a measure on G, ν a measure on X, μ and ν being convolvable. Suppose in addition that there exists a positive measure β on X such that γ(s)ν has base β for every s ∈ G… ↩ 2 09JL
Proposition 11 — Let G be a locally compact group, E a quasi-complete locally convex space, U a continuous representation of G in E. (i) If λ ∈ C'(G), μ ∈ C'(G), then U(λ * μ) = U(λ)U(μ). (ii) Su… ↩ 1 09JM
Proposition 12 If λ ∈ C'(G) and μ ∈ M(X), then ↩ 2 09JN
Corollary The mapping (λ, μ) ↦ λ * μ of C'(G) × M(X) into M(X) is hypocontinuous relative to the equicontinuous subsets of C'(G) and the compact subsets of M(X) (C'(G) and M(X) being equippe… ↩ 1 09JO
Remark — 1) Let λ₀ ∈ C'(G). The mapping μ ↦ λ₀ * μ of M(X) into M(X) is vaguely continuous. For, let f ∈ H(X). One has ↩ 2 09JP
Proposition 13 The mapping (s, μ) ↦ γ(s)μ of G × M₊(X) into M₊(X) is continuous when the set M₊(X) of positive measures on X is equipped with the vague topology. ↩ 2 09JQ

§ 4. Convolution of measures and functions

Definition 1 If μ and f · β are convolvable, μ and f are said to be convolvable relative to β. Every density of μ * (f · β) with respect to β is called a convolution product of μ and f relative… 09K4
Lemma 1 Let μ be a measure on G. Then χ is locally (μ ⊗ β)-integrable, and the image of μ ⊗ β under the homeomorphism (s, x) ↦ (s, s⁻¹x) of G × X onto G × X is χ · (μ ⊗ β). ↩ 6 09K5
Proposition 1 Let μ be a measure on G, f a locally β-integrable complex function on X. Suppose that the function s ↦ f(s⁻¹x)χ(s⁻¹, x) is essentially μ-integrable except for a locally β-negligibl… ↩ 2 09K6
Proposition 2 — Let μ be a measure on G, f a locally β-integrable complex function on X. Assume that one of the following conditions is satisfied: (i) f and χ are continuous; (ii) G operates pro… ↩ 6 09K7
Remark The hypothesis (iii) of Prop. 2 is satisfied notably when μ is bounded. For, for every n > 0, there then exists a compact subset Kₙ of G such that |μ|(G - Kₙ) ≤ 1/n (Ch. IV, §4, No… 09K8
Proposition 3 — Assume χ continuous. Let μ ∈ C'(G), f ∈ C(X). Then: (i) μ and f are convolvable relative to β. (ii) Formula (3) of No. 1 defines for every x ∈ X a convolution product μ *_β f tha… ↩ 3 09K9
Proposition 4 Assume that G operates properly in X and that χ is continuous. Let μ ∈ M(G) and f ∈ K(X). ↩ 2 09KA
Remark 1 — Under the hypotheses of Prop. 4, the mapping μ ↦ μ *^β f of M₊(G) into C(X) is continuous when M₊(G) is equipped with the vague topology, for every f ∈ K(X). For, let K be a comp… ↩ 2 09KB
Proposition 5 Assume χ a continuous multiplier and each function χ(s, ·) bounded. (i) The function s ↦ ρ(s) = sup_x ∈ X χ(s⁻¹, x) on G is lower semi-continuous > 0 and satisfies ρ(st) ≤ ρ(s)ρ(t)… ↩ 3 09KC
Proposition 6 Assume χ a continuous multiplier and each function χ(s, ·) bounded. Let ρ(s) = sup_x ∈ X χ(s⁻¹, x). Let p and q be two conjugate exponents (1 ≤ p < +∞). Let μ ∈ M^ρ^1/q(G) and f ∈… ↩ 4 09KD
Corollary Under the hypotheses of Prop. 6, the mapping (μ, f) ↦ μ *^β f defines on Lᵖ(X, β) the structure of a left module over M^ρ^1/q(G) (1 ≤ p ≤ +∞). 09KE
Remark 2 — Let X be a locally compact space on which a locally compact group G operates continuously on the right by (x, s) ↦ xs. Let β be a positive measure on X. Let χ be a function > 0 o… 09KF
Proposition 7 Let f be a locally β-integrable function on X, ν a measure on X, and μ a measure on G. Assume that: (i) μ and f are convolvable and formula (3) of No. 1 defines locally β-almost ev… ↩ 1 09KG
Example — 1) One can take f ∈ C(X), ν ∈ C'(X) and μ ∈ C'(G) by Prop. 3, and the Cor. of Prop. 5 of §1, No. 4. The formula (7) then means that the endomorphism ν ↦ (χ · checkμ) * ν of C'(X)… 09KH
Proposition 8 Let f and g be two locally β-integrable functions on X and let μ ∈ M(G). Assume that: 09KI
Example — 4) One can take f ∈ C(X), g ∈ K(X) and μ ∈ C'(G) (with ψ = 1). ↩ 1 09KJ
Proposition 9 Let f, g be in L(G). Assume that the function s ↦ g(s⁻¹x)f(s)χ(s⁻¹) is essentially β-integrable except for a locally β-negligible set of values of x, and that the function ↩ 1 09KK
Proposition 10 Let f, g be in L(G). Assume that one of these two functions is continuous or is zero on the complement of a countable union of compact sets. If f and g are convolvable, then the fu… 09KL
Proposition 11 — Let f, g be in L(G). Assume that one of the functions f, g is continuous, and that one of the functions f, g has compact support. Then f and g are convolvable. The formula (15) d… ↩ 1 09KM
Proposition 12 — Let p and q be two conjugate exponents (1 ≤ p ≤ +∞). If f χ^-1/q ∈ L¹(G, β) and g ∈ Lᵖ(G, β), then f and g are convolvable, f * g is equal locally β-almost everywhere to a functi… 09KN
Proposition 13 — If f χ⁻¹ ∈ L¹(G, β) and g ∈ overlineK(G), or if f ∈ K(G) and g χ'⁻¹ ∈ L¹(G, β), then f and g are convolvable, and (15) defines for every x ∈ G a product f * g that belongs to K(G… 09KO
Proposition 14 — If f χ⁻¹ ∈ L¹(G, β) and g ∈ L^∞(G, β), then the formula (15) defines for every x ∈ G a product f * g that is bounded and is uniformly continuous for the right uniform structure o… 09KP
Proposition 15 — Let p and q be two conjugate exponents (1 < p < +∞). Assume that β is left-invariant. Let f ∈ Lᵖ(G, β), g ∈ L^q(G, checkβ). Then f and g are convolvable. The formula (15) defines… ↩ 3 09KQ
Corollary — Let f ∈ L²(G, β), g ∈ L²(G, β). Then f and dotg are convolvable. One of the convolutions f * dotg belongs to overlineK(G) and its value at e is ∫_G f(s)g(s) dβ(s). ↩ 1 09KR
Proposition 16 — Let U be a continuous representation of G in a Banach space E. Set ρ(s) = |U(s)| for all s ∈ G. For every f ∈ L^ρ(G, β), set U(f) = U(f · β). Then f ↦ U(f) is a linear representa… 09KS
Proposition 17 — Let G be a locally compact group, A a subset of G, measurable and not locally negligible for a Haar measure. Then A · A⁻¹ is a neighborhood of e. ↩ 3 09KT
Corollary 1 — Let H be a subgroup of G measurable for a Haar measure β. Then H is either open or locally β-negligible. ↩ 1 09KU
Corollary 2 — Let L be a subset of G stable for multiplication and whose complement is locally negligible for a Haar measure β. Then L = G. ↩ 1 09KV
Proposition 18 — Let G be a locally compact group, Γ a set equipped with a multiplication (u, v) ↦ uv and a Hausdorff topology such that: 1) the topology of Γ is invariant under the translations… ↩ 2 09KW
Corollary 1 Let G be a locally compact group, β a Haar measure on G, E a Hausdorff barreled locally convex space, U a linear representation of G in E, such that U(s) ∈ L(E;E) for all s ∈ G, β-… ↩ 1 09KX
Corollary 2 Let G be a locally compact group, β a Haar measure on G, E a separable Banach space, and U a linear representation of G in E such that U(s) ∈ L(E;E) for all s ∈ G. Let (aₘ) be a to… ↩ 1 09KY
Proposition 19 — Let G be a locally compact group, β a relatively invariant positive measure ≠ 0 on G, B a base for the filter of neighborhoods of e in G, consisting of compact neighborhoods. For… ↩ 4 09KZ
Remark 1 We thus see that every measure on G is the limit of measures admitting a continuous density with respect to every Haar measure (for the topology indicated in Prop. 19 and a fortior… 09L0
Remark 2 If G is metrizable, B can be taken to be a sequence (Vₙ) of neighborhoods. Then μ is the limit of the sequence of measures (μ * f_Vₙ) · β with continuous densities. If G is a real… 09L1
Proposition 20 — We conserve the hypotheses and notations of Prop. 19. Let p ∈ [1, +∞[ and g ∈ Lᵖ(G, β). Then 09L2
Remark 3 By Prop. 15, the functions g * f_V, f_V * g belong to K(G). ↩ 1 09L3
Corollary — Let W be a closed linear subspace of L¹(G, β). For W to be a left (resp. right) ideal of L¹(G, β), it is necessary and sufficient that W be invariant under the left (resp. right)… 09L4
Example We take G = R. Let us define a function Fₙ ∈ K(R) by Fₙ(x) = (1 - x²)ⁿ if x ∈ [-1, 1] \\ Fₙ(x) = 0 if x ∉ [-1, 1]. Let Aₙ = ∫₋₁⁺¹ Fₙ(x) dx, and Gₙ = Aₙ⁻¹ Fₙ. It is immediate that t… 09L5

§ 5. The space of closed subgroups

Lemma 1 Let α be a positive measure ≠ 0 on G, S its support; the following two conditions are equivalent: a) S is a closed subgroup of G and the measure induced by α on S is a right Haar m… ↩ 1 09LY
Proposition 1 The set Γ is closed in the space M₊(G) - 0 equipped with the vague topology. We first prove the following lemmas: ↩ 2 09LZ
Lemma 2 Let X be a locally compact space and for every measure α ∈ M₊(X) - 0, let S_α be the support of α. Let Φ be a filter on M₊(X) - 0 that converges vaguely to a measure α₀ ≠ 0. Then… ↩ 3 09M0
Lemma 3 Let E be a set filtered by a filter Φ, and let ξ ↦ α(ξ) be a mapping of E into Γ that converges vaguely with respect to Φ to a measure α₀ ≠ 0. On the other hand, let ξ ↦ t_ξ be a m… ↩ 1 09M1
Proposition 2 — Let φ be a function in K₊(G) such that φ(e) > 0. Then the set Γ_φ of measures α ∈ Γ such that ∫ φ(x) dα(x) = 1 is compact for the vague topology. ↩ 4 09M2
Proposition 3 — Under the hypotheses of Prop. 2, the mapping α ↦ ( ⟨ φ, α ⟩, α/(⟨ φ, α ⟩) ) is a homeomorphism of Γ onto the product space R₊^* × Γ_φ. ↩ 2 09M3
Proposition 4 — Let Γ⁰ be the set of measures α ∈ Γ such that H_α is unimodular, and for every α ∈ Γ⁰ set μ_α = μ / α; then the mapping α ↦ | μ_α | of Γ⁰ into overlineR is lower semi-continuous… ↩ 2 09M4
Proposition 5 Let g be a μ-integrable positive numerical function and let Γ⁰(g) be the set of measures α ∈ Γ⁰ such that ∫^* g(xs) dα(s) ≥ 1 for all x ∈ G. Then the mapping α ↦ |μ_α| of Γ⁰(g) int… ↩ 3 09M5
Corollary 1 — Let K be a compact subset of G, V a symmetric compact neighborhood of e in G, c a real number > 0. The restriction of the mapping α ↦ | μ_α | to the set of α ∈ Γ⁰ such that G = K… 09M6
Corollary 2 Let A be a μ-integrable subset of G. The restriction of the mapping α ↦ | μ_α | to the set N_A of normalized Haar measures of the discrete subgroups H of G such that G = AH, is vag… 09M7
Theorem 1 Let G be a locally compact group. The space Σ of closed subgroups of G is compact. Moreover, one has the following properties: ↩ 3 09M8
Proposition 6 Suppose that the locally compact group G is generated by a compact neighborhood of e. Then the set Γ_c⁰ of measures α ∈ Γ⁰ such that G/H_α is compact is open in Γ⁰, and the restric… ↩ 1 09M9
Proposition 7 Let N ⊂ Γ⁰ be the subspace of normalized Haar measures on the discrete subgroups of G, and for every relatively compact open neighborhood U of e in G, let N_U be the subset of N fo… ↩ 2 09MA
Corollary — The subspace N of Γ⁰ is locally compact. ↩ 1 09MB
Theorem 2 — Let G be a locally compact group satisfying the following condition: (L) There exists a neighborhood of e in G that contains no finite subgroup of G not reduced to e. The followi… 09MC
Proposition 8 The canonical bijection of N onto D is a homeomorphism. ↩ 2 09MD
Lemma 4 If the group G satisfies the condition (L), the mapping α ↦ α(e) of Γ_d into R₊^* is vaguely continuous. 09ME
Proposition 9 Let G be a locally compact abelian group, generated by a compact neighborhood of e. For a subset A of N_c to be relatively compact in N_c, it is necessary and sufficient that it sa… ↩ 1 09MF
Corollary Let G be a locally compact abelian group, generated by a compact neighborhood of e and satisfying the condition (L) of No. 4. ↩ 2 09MG
Example Let us take G = Rⁿ and for μ the Lebesgue measure; all of the hypotheses of the Cor. of Prop. 9 are satisfied. The discrete subgroups H of G such that G/H is compact are none other… 09MH

IX. Measures on Hausdorff topological spaces

§ 1. Premeasures and measures on a topological space

Definition 1 — Let T be a set. One calls encumbrance on T any mapping p of F₊(T) into overlineR₊ that has the following properties: a) If f and g are two elements of F₊ such that f ≤ g, then p(… ↩ 7 09MO
Proposition 1 Let (p_α)_α ∈ A be a family of encumbrances on T. The sum and upper envelope of the family (p_α) (in F₊(F₊(T))) are then encumbrances. ↩ 2 09MP
Definition 2 Let p be an encumbrance on a set T. One says that p is bounded if p(T) < +∞. If T is a topological space, p is said to be locally bounded provided that every x ∈ T admits a neighbo… ↩ 2 09MQ
Lemma 1 — If the encumbrance p is concentrated on A ⊂ T, then p(f) = p(f φ_A) for every f ∈ F₊(T). ↩ 1 09MR
Definition 3 One calls premeasure on a topological space T every mapping w that associates, to every compact subset K of T, a measure w_K on K, and that has the following property: 09MS
Definition 4 Let w be a positive premeasure on T. We shall set, for every function f ∈ F₊(T), ↩ 2 09MT
Proposition 2 a) Let w be a positive premeasure. For every compact subset K of T, the encumbrance (w^bullet)_K induced by w^bullet on K is equal to (w_K)^bullet. For every function f ∈ F₊(T), on… ↩ 15 09MU
Corollary — Let v and w be two positive premeasures on T, such that v^bullet(L) = w^bullet(L) for every compact subset L of T; then v = w. In particular, the relation v^bullet = w^bullet imp… ↩ 6 09MV
Definition 5 — Let w be a premeasure on a topological space T. One says that w is a measure (resp. a bounded measure) if the encumbrance |w|^bullet is locally bounded (resp. bounded) (cf. No. 1… ↩ 1 09MW
Remark 2 If T is locally compact, then every premeasure w on T is a measure. For, every x ∈ T admits a compact neighborhood K, and |w|^*(K) = |w_K| < +∞, so that the encumbrance |w|^* is lo… 09MX
Remark 3 For every Borel subset A of T (in particular for A = T) and every positive measure μ on T, the number μ^*(A) is the supremum of the measures μ^*(K) of the compact subsets of A. Ind… ↩ 3 09MY
Example 1 — Measures on a locally compact space. 09MZ
Proposition 3 Let T be a locally compact space, and let μ be a measure on T. Denote by W(μ) the mapping that associates to each compact subset K of T the induced measure μ_K. Then W(μ) is a prem… ↩ 2 09N0
Example 2 — Measures with compact support on a topological space. ↩ 5 09N1
Lemma 2 — Let T be a topological space, L a compact subset of T, and λ a positive measure on L. There exists a unique positive measure μ on T such that, for every function f ∈ F₊(T), 09N2
Remark — 1) When T is locally compact, μ is the image of λ under the injection of L into T. We shall see in §2, No. 3, Example, when image measures will have been treated, that this inter… 09N3
Definition 6 A function f ∈ F₊ (resp. a subset A of T) is said to be locally negligible for the measure μ if μ^*(f) = 0 (resp. μ^*(A) = 0). One says that μ is concentrated on a subset A of T if… 09N4
Remark — 1) The concepts so defined coincide, when T is locally compact, with the usual concepts. 2) After we have defined negligible sets, we will see that the locally negligible sets ar… 09N5
Example — Let L be a compact subset of T, λ a measure on L, and μ the measure on T defined by λ (No. 3, Example 2). The formula (3) implies at once that a function f ∈ F₊(T) is locally μ-n… 09N6
Definition 7 A function f defined on T, with values in a topological space F (Hausdorff or not) is said to be measurable for the measure μ (or to be μ-measurable) if, for every compact subset K… ↩ 1 09N7
Example — Let L be a compact subset of T, λ a measure on L, and μ the measure on T defined by λ (No. 3, Example 2). A function f defined on T is μ-measurable if and only if f_L is λ-measur… 09N8
Proposition 4 — Let f be a positive function and (gₙ)_n ≥ 1 a sequence of μ-measurable positive functions on T. Setting g = ∑_n ≥ 1 gₙ, one has ↩ 3 09N9
Corollary — Let (Aₙ) be a sequence of pairwise disjoint measurable subsets, with union A. For every subset B of T, ↩ 2 09NA
Proposition 5 a) Let H be an increasing directed set of functions geqslant 0 that are lower semi-continuous on every compact subset of T. Then ↩ 5 09NB
Corollary — a) Let (U_α)_α ∈ I be an increasing directed family of open subsets of T, with union U. Then μ^bullet(U) = sup_α ∈ I μ^bullet(U_α). ↩ 3 09NC
Definition 8 — The support of a measure μ on T is defined to be the complement of the largest locally μ-negligible open set in T. 09ND
Remark 1 If μ is a complex measure, the support of μ is defined to be the support of the positive measure |μ|; it is again the complement of the largest locally μ-negligible open set. 09NE
Remark 2 Let us show that the measures introduced in Example 2 of No. 3 are measures with compact support in T. Let μ be a positive measure on T whose support is a compact set K, and let ν… ↩ 1 09NF
Proposition 6 — Let (λ_α)_α ∈ A be an increasing directed family of measures on T, and let p = sup_α λ_α^bullet. For the family (λ_α) to be bounded above in M(T), it is necessary and sufficient… ↩ 2 09NG
Definition 9 Let (μᵢ)_i ∈ I be a family of measures on T. Let A be the set of finite subsets of I; for every α ∈ A let λ_α = ∑_i ∈ α μᵢ. If the family (λ_α) admits a supremum μ in M(T), the fam… ↩ 1 09NH
Proposition 7 For the family (μᵢ)_i ∈ I to be summable, with sum μ, it is necessary and sufficient that the encumbrance p = ∑_i ∈ I μᵢ^bullet be locally bounded, in which case p = μ^bullet. For… ↩ 8 09NI
Proposition 8 Let μ be the sum of a summable family (μᵢ)_i ∈ I of measures on T. In order that a mapping f of T into a topological space F (Hausdorff or not) be μ-measurable, it is necessary and… ↩ 4 09NJ
Definition 10 One calls crushing of T for μ, or μ-crushing, any locally countable family (K_α)_α ∈ A of pairwise disjoint compact subsets of T such that the set N = T - ⋃_α ∈ A K_α is locally μ-… 09NK
Proposition 9 a) There exists a crushing (K_α)_α ∈ A of T for μ. b) Let (K_α)_α ∈ A be a crushing of T for μ. If μ_α is the measure on T defined by μ_K_α (No. 3, Example 2), then the family (μ_α… ↩ 5 09NL
Proposition 10 — If g is a μ-measurable mapping with values in a topological space G (Hausdorff or not), there exists a μ-crushing (L_β)_β ∈ B of T such that the restrictions g_L_β are continuous… ↩ 5 09NM
Scholium — Let (K_α)_α ∈ A be a crushing of T, and let N = T - ⋃_α K_α. We denote by T' the locally compact space obtained by equipping T' with the sum topology of the topologies of the sub… 09NN
Proposition 11 Let X be a μ-measurable subset of T. There exists a locally countable family (L_α)_α ∈ A of compact subsets of X, pairwise disjoint, such that X - ⋃_α ∈ A L_α is locally μ-negligib… ↩ 2 09NO
Definition 11 For every function f ∈ F₊(T), one defines the upper integral of f (with respect to the measure μ) to be the finite or infinite positive number 09NP
Proposition 12 The function μ^* is an encumbrance on T. ↩ 1 09NQ
Corollary A function f, with values in a Banach space or in overlineR, is negligible if and only if f(t) = 0 almost everywhere. 09NR
Proposition 13 For every subset A of T, μ^*(A) is the infimum of the outer measures of the open sets containing A. ↩ 2 09NS
Definition 12 Let f be a function defined on T, with values in a Banach space or in overlineR. One says that f is moderated for the measure μ, or μ-moderated, if f is zero on the complement of a… 09NT
Remark 1 On a Lindelöf space T (TG, IX, Appendix I, Def. 1),¹ and in particular on a Souslin space (ibid., Cor. of Prop. 1), every measure is moderated. For, the open sets of finite measure… ↩ 1 09NU
Remark 2 Beware, however, that the existence of a sequence of Borel sets of finite measure for μ, with union T, does not necessarily imply the existence of a sequence of open sets of finite… 09NV
Proposition 14 Let f ∈ F₊(T). If f is μ-moderated, then μ^*(f) = μ^bullet(f); if f is not μ-moderated, then μ^*(f) = +∞. ↩ 3 09NW
Corollary 1 For a function to be negligible, it is necessary and sufficient that it be locally negligible and moderated. ↩ 2 09NX
Corollary 2 For a function f to be locally negligible, it is necessary and sufficient that every x ∈ T possess a neighborhood V such that f φ_V is negligible. ↩ 1 09NY
Corollary 3 Let f be a moderated function defined on T. There exists a sequence (Kₙ) of pairwise disjoint compact sets, and a negligible set H, such that f = f φ_H + ∑ₙ f φ_Kₙ. ↩ 2 09NZ
Corollary 4 If μ and ν are two measures on T such that μ^* = ν^*, then μ = ν. 09O0
Corollary 5 If μ is a moderated measure on T, there exists a sequence (μₙ)_n ∈ N of measures with compact support such that μ = ∑_n ∈ N μₙ. ↩ 1 09O1
Definition 13 — Let p ∈ [1, +∞[; one denotes by overlineLᵖ(T, μ) (resp. overlineL_Fᵖ(T, μ) if F is a Banach space) the set of mappings f of T into overlineR (resp. into F) that are μ-measurable… 09O2
Proposition 15 The set of functions f with values in F, such that Supp(f) is compact and such that the restriction of f to Supp(f) is continuous, is dense in overlineL_Fᵖ(μ) and in L_Fᵖ(μ), for 1… ↩ 1 09O3
Proposition 16 There exists one and only one continuous linear mapping f ↦ ∫ f dμ, of the space overlineL_F¹(μ) into F, having the following property: If f is of the form t ↦ g(t)a, where a ∈ F… 09O4

§ 2. Operations on measures

Definition 1 Let X be a μ-measurable subset of T. The restriction of μ : K ↦ μ_K to the set of compact subsets of X is called the measure induced by μ on the subspace X, and is denoted by μ_X o… ↩ 2 09OF
Proposition 1 Let X be a μ-measurable subset of T. The encumbrance (μ_X)^bullet is equal to the encumbrance (μ^bullet)_X induced by μ^bullet on X (§1, No. 1). In other words, (μ_X)^bullet(g) = μ… ↩ 4 09OG
Remark 1 The relation (μ_X)^bullet = (μ^bullet)_X permits using the notation μ_X^bullet without ambiguity; we shall do so henceforth. The preceding Prop. 1 and Prop. 2 of §1, No. 2 show tha… 09OH
Remark 2 Def. 1 may be extended to the case that μ is a complex measure on T. To show in this case that the premeasure μ_X is a measure, it suffices to observe that |μ_K| = |μ|_K for every… 09OI
Definition 2 A function f defined on T, with values in overlineR or in a Banach space, is said to be locally μ-integrable if f is μ-measurable and every point x ∈ T admits a neighborhood V such… ↩ 1 09OJ
Proposition 2 If f is a positive locally μ-integrable function, then, for every function g ∈ F₊(T), one has the relation ↩ 4 09OK
Definition 3 Let f be a locally μ-integrable positive function. The measure f · μ : K ↦ f_K · μ_K is called the measure with density f with respect to μ, or the product measure of μ by the func… 09OL
Remark 1 The definition of f · μ extends to the case that f is a complex locally integrable function; one then has |f · μ| = |f| · μ, which implies at once that f · μ is a measure, not just… 09OM
Remark 2 Similarly, if θ is a complex measure, f is said to be locally θ-integrable if it is locally |θ|-integrable, and one defines the measure f · θ : K ↦ f_K · θ_K. One has |f · θ| = |f|… 09ON
Proposition 3 — Let ν be a measure on T. For ν to be of the form f · μ, where f is a locally μ-integrable positive function, it is necessary and sufficient that every μ-negligible compact set be… ↩ 2 09OO
Definition 4 Let π be a mapping of T into a topological space X. One says that π is μ-proper if π is μ-measurable, and if every point x of X admits a neighborhood V such that μ^bullet(overlineπ… ↩ 1 09OP
Remark 1 When T and X are locally compact, this definition is equivalent to that of Ch. V, §6, No. 1. 09OQ
Remark 2 A proper continuous mapping (GT, I, §10, No. 1, Def. 1) of T into X is μ-proper for every measure μ. For, let x ∈ X; since overlineπ(x) is compact (loc. cit., No. 2, Th. 1), the se… ↩ 1 09OR
Remark 3 If μ is bounded, every μ-measurable mapping of T into X is μ-proper. ↩ 3 09OS
Remark 4 If θ is a complex measure on T, π is said to be θ-proper if π is proper for the positive measure |θ|. 09OT
Proposition 4 Let π be a μ-proper mapping of T into a topological space X. There exists one and only one measure ν on X such that ν^bullet is equal to the image encumbrance π(μ^bullet) (§1, No… ↩ 3 09OU
Definition 5 If π is a μ-proper mapping of T into a topological space X, the unique measure ν on X such that ν^bullet(g) = μ^bullet(g ∘ π) for all g ∈ F₊(X) is called the image measure of μ und… 09OV
Example — Let K be a compact subspace of T, i the canonical injection of K into T, and λ a measure on K; since i is continuous and λ is bounded, i is λ-proper. Formula (3) of §1, No. 3 sho… 09OW
Proposition 5 Let π be a μ-proper mapping of T into a topological space X, and let f be a mapping of X into a topological space F (Hausdorff or not). For f to be π(μ)-measurable, it is necessary… ↩ 4 09OX
Corollary — Let X and Y be two topological spaces, π a μ-proper mapping of T into X, and π' a π(μ)-proper mapping of X into Y. The mapping π'' = π' ∘ π is then μ-proper, and π''(μ) = π'(π(μ)… ↩ 1 09OY
Proposition 6 — Let π be a μ-proper mapping of T into a topological space X, and let B be a π(μ)-measurable subset of X. Set A = overlineπ⁻¹(B), and denote by π' the mapping of A into B that coi… 09OZ
Proposition 7 — Let T be a subspace of a topological space X, and let i be the injection of T into X. ↩ 6 09P0
Proposition 8 Let T and X be two topological spaces, π a mapping of T into X. ↩ 9 09P1
Lemma 1 Let U and V be two compact spaces, h a continuous mapping of U onto V. The mapping λ ↦ h(λ) of M₊(U) into M₊(V) is then surjective. 09P2
Remark Suppose that π is injective. Let θ be a complex measure such that π is θ-proper and π(θ) = 0; then θ = 0. Indeed, by separating θ into its real and imaginary parts, one can reduce… 09P3
Proposition 9 — Let T be a Souslin space (GT, IX, §6, No. 2, Def. 2), X a Hausdorff space, π a continuous mapping of T onto X, and ν a bounded measure on X. Then there exists a bounded measure μ… ↩ 3 09P4
Definition 6 The premeasure ν defined by (3) is called the product premeasure of λ and μ, and is denoted λ ⊗ μ. 09P5
Proposition 10 Let ν be the product premeasure of λ and μ; for every function f ∈ F₊(S) and every function g ∈ F₊(T), ↩ 3 09P6
Corollary 1 — If λ and μ are measures, then ν is a measure. ↩ 1 09P7
Corollary 2 — If A is a subset of S locally negligible for λ, then A × T is locally ν-negligible. 09P8
Corollary 3 — Suppose that λ (resp. μ) is the sum of a summable family (λ_α)_α ∈ A (resp. (μ_β)_β ∈ B) of measures on S (resp. T). The family (λ_α ⊗ μ_β)_(α, β) ∈ A × B is then summable, and i… ↩ 1 09P9
Lemma 2 Let f be a ν-measurable positive function on S × T; for every compact subset L of T, let I_fᴸ be the function s ↦ μ^bullet(fₛ φ_L) on S. Then the function I_fᴸ is λ-measurable, and ↩ 2 09PA
Proposition 11 Let f be a lower semi-continuous function geqslant 0 defined on X = S × T. ↩ 1 09PB
Lemma 3 — Let X be a topological space (Hausdorff or not), f a lower semi-continuous function geqslant 0 defined on X; then f is the limit of an increasing sequence (fₙ)_n ∈ N of lower sem… ↩ 1 09PC
Corollary 1 Let f be a positive numerical function defined on X = S × T; then iint_X^* f(s,t) dν(s,t) ≥ ∫_S^* dλ(s) ∫_T^* f(s,t) dμ(t). ↩ 1 09PD
Corollary 2 — Let f be a numerical function defined on S × T and ν-negligible. Then the function fₛ : t ↦ f(s, t) is μ-negligible for λ-almost every s ∈ S. ↩ 1 09PE
Proposition 12 — Let f be a ν-measurable positive function defined on X = S × T. Assume that f is ν-moderated (resp. that μ is moderated). Then: a) The set N of s ∈ S such that the function fₛ… ↩ 1 09PF
Remark — Let (K_α)_α ∈ A be a crushing of S for λ and let M = S - ⋃_α ∈ A K_α; define in an analogous way (L_β)_β ∈ B and N for the measure μ on T. We denote by S' the locally compact spa… 09PG
Proposition 13 Let X be a topological space, ν a moderated measure on X, p a ν-proper mapping of X into a topological space T, and μ = p(ν). Assume that every compact subspace of X is metrizable… ↩ 2 09PH
Lemma 4 Let X be a topological space, ν a measure on X, and f a ν-measurable mapping of X into a topological space F (Hausdorff or not). There exists a universally measurable mapping f' of… 09PI
Remark — 1) If X is a Souslin space, then every compact subspace of X is a Souslin space, hence is metrizable (TG, IX, Appendix I, Cor. 2 of Prop. 3),⁽²⁾ 09PJ

§ 3. Measures and additive set functions

Theorem 1 — Let T be a topological space, and I a mapping of K(T) into R₊. In order that there exist a measure μ on T such that I(K) = μ^bullet(K) for all K ∈ K(T), it is necessary and suffi… ↩ 5 09PN
Lemma 1 Assume that T is compact, and set l = I(T). For every A ⊂ T, set 09PO
Remark 1 The condition 4) may be replaced, in the statement of Theorem 1, by the following condition ('right-continuity'): 4') For every K ∈ K(T) and every ε > 0, there exists an open set U… 09PP
Remark 2 The set of conditions 2) and 3) may be replaced, in the statement of Theorem 1, by the following condition: If K and L are compact subsets of T, then I(K ∪ L) + I(K ∩ L) = I(K) + I… 09PQ
Definition 1 Let T be a topological space, and let B(T) be the Borel tribe of T; let I be a mapping of B(T) into overlineR₊. a) I is said to be countably additive if, for every sequence (Aₙ) of… 09PR
Remark 1 The condition a) clearly implies that I is an increasing mapping of B(T) (ordered by inclusion) into overlineR₊. 09PS
Remark 2 Suppose that I is countably additive; let (Aₙ)_n ∈ N be an increasing sequence of Borel sets, and let A = ⋃_n ∈ N Aₙ. The sets D₀ = A₀, Dₙ = Aₙ - Aₙ₋₁ being pairwise disjoint, and… ↩ 1 09PT
Remark 3 Let (Aₙ) be any sequence of Borel sets of T. If I is countably additive, then I(⋃ₙ Aₙ) ≤ ∑ₙ I(Aₙ). By the preceding remark, it suffices to establish this inequality for a finite se… 09PU
Remark 4 If I is a countably additive and locally bounded function, the preceding remark implies at once that I(K) < +∞ for every compact set K ⊂ T. 09PV
Remark 5 One can show that if I is additive, that is, satisfies (4) for finite sequences, and if I is inner regular, then I is countably additive (Exer. 7). The reader can also ascertain th… 09PW
Theorem 2 — Let T be a topological space, and let I be a function defined on B(T), with values in overlineR₊. In order that there exist a measure μ on T such that μ^*(A) = I(A) for every A ∈… ↩ 1 09PX
Definition 2 Let T be a topological space. One says that T is a Radon (resp. strongly Radon) space if T is Hausdorff and if every function defined on the Borel tribe B(T) of T, with values in o… 09PY
Proposition 1 Every Lindelöf⁽¹⁾ Radon space is strongly Radon. ↩ 1 09PZ
Proposition 2 — Let X be a topological space and T a subspace of X. ↩ 2 09Q0
Corollary — If X is a Radon space, then every Borel subset T of X is Radon. ↩ 1 09Q1
Proposition 3 — Every Souslin space (in particular, every Polish or Lusin space) is strongly Radon. ↩ 2 09Q2
Remark — Let X be a Lusin space (in particular, any Polish space), and f a bijective continuous mapping of X onto a (Lusin) regular space Y. One knows (TG, IX, §6, No. 7, Prop. 14) that t… 09Q3

§ 4. Inverse limits of measures

Proposition 1 — Let X and Y be two topological spaces and f a continuous mapping of X into Y. Let (K_α)_α ∈ A be a decreasing directed family of compact subsets of X, with intersection K. Then f… ↩ 3 09QM
Proposition 2 — Let there be given an inverse system (Tᵢ, pᵢⱼ) of topological spaces indexed by I, a topological space T, and a coherent and separating family of continuous mappings pᵢ : T → Tᵢ… ↩ 2 09QN
Definition 1 — Let T = (Tᵢ, pᵢⱼ) be an inverse system of topological spaces indexed by I. One calls inverse system (resp. sub-inverse system) of measures on T a family (μᵢ)_i ∈ I, where μᵢ is a… ↩ 1 09QO
Proposition 3 Let there be given an inverse system of topological spaces T = (Tᵢ, pᵢⱼ) indexed by I, a topological space T, a coherent and separating family of continuous mappings pᵢ : T → Tᵢ (f… ↩ 3 09QP
Theorem 1 (Prokhorov) — Let T = (Tᵢ, pᵢⱼ) be an inverse system of topological spaces indexed by I, T a topological space and (pᵢ)_i ∈ I a coherent and separating family of continuous mappings pᵢ : T → T… ↩ 5 09QQ
Remark — Let (μᵢ)_i ∈ I be an inverse system of measures on the inverse system of spaces T = (Tᵢ, pᵢⱼ). Assume given a topological space T' and continuous mappings p'ᵢ : T' → Tᵢ; assume t… 09QR
Theorem 2 — Assume that the directed preordered set I has a countable cofinal subset. Let T = (Tᵢ, pᵢⱼ) be an inverse system of topological spaces, T = lim_← Tᵢ and pᵢ the canonical mapping… ↩ 1 09QS

§ 5. Measures on completely regular spaces

Lemma Let T be a completely regular space, K a compact subset of T, and U an open subset of T containing K. a) There exists an open subset U' of T such that K ⊂ U' ⊂ overlineU' ⊂ U. b) L… 09QV
Proposition 1 — Let T be a completely regular space. a) Let μ be a positive measure on T, and f a numerical function ≥ 0 defined on T and lower semi-continuous (resp. upper semi-continuous ous… ↩ 2 09QW
Remark 1 Let f be a positive lower semi-continuous function, and let J_f be the set of positive bounded continuous functions zero outside a μ-integrable open set and bounded above by f. One… 09QX
Remark 2 If the measure μ is bounded, the formula μ^*(f) = inf_g ∈ S_f μ(g) is obviously valid for every function f that is upper semi-continuous, positive and bounded. 09QY
Proposition 2 — Let η and η' be two complex measures on a completely regular space T, such that η(f) = η'(f) for every function f ∈ Cᵇ(T) that is integrable for |η| and |η'|. Then η = η'. ↩ 2 09QZ
Proposition 3 — Let μ be a positive measure on a completely regular space T, and let p ∈ [1, +∞[. The space H of functions f ∈ Cᵇ(T), whose support is contained in a μ-integrable open set, is de… ↩ 1 09R0
Proposition 4 — In order that a bounded complex measure θ on a completely regular space T be positive, it is necessary and sufficient that θ(f) ≥ 0 for every function f ∈ C₊ᵇ(T). ↩ 1 09R1
Proposition 5 — Let T be a completely regular space, and I a continuous complex linear form on the normed space Cᵇ(T; C). In order that there exist a bounded complex measure θ on T such that θ(f… ↩ 2 09R2
Corollary — With notations as in Prop. 5, suppose that there exists a bounded positive measure μ on T such that |I(f)| ≤ μ(|f|) for all f ∈ Cᵇ(T; C); then there exists a complex measure θ on… 09R3
Definition 1 The weak topology on Mᵇ(T) associated with the preceding duality between Cᵇ(T) and Mᵇ(T) is called the topology of tight convergence (or the tight topology) on Mᵇ(T). 09R4
Remark 1 If T is locally compact, the tight topology is finer than the topology induced on Mᵇ(T) by the vague topology, and these two topologies coincide only when T is compact. For, if T i… 09R5
Remark 2 It follows at once from Prop. 4 that M₊ᵇ(T) is closed in Mᵇ(T). 09R6
Remark 3 If T is completely regular, the mapping t ↦ εₜ of T into Mᵇ(T) is a homeomorphism (GT, IX, §1, No. 5). ↩ 2 09R7
Proposition 6 — Let T be a completely regular space. ↩ 7 09R8
Proposition 7 — Let T be a completely regular space. Let μ be a bounded positive measure on T, and let f be a bounded positive function on T, such that the set of points of T where f is not cont… 09R9
Proposition 8 — Let X be a completely regular space, T a subspace of X, and i the canonical injection of T into X. Denote by W the set of bounded positive measures on X that are concentrated on… ↩ 5 09RA
Proposition 9 — Let T be a locally compact space, and let 𝔉 be a filter on Mᵇ₊(T) that converges vaguely to a bounded measure μ. For 𝔉 to converge tightly to μ, it is necessary and sufficient th… ↩ 3 09RB
Proposition 10 Let T be a Polish space; the space M₊ᵇ(T) is then Polish for the tight topology. ↩ 1 09RC
Corollary 1 If T is a metrizable space of countable type, then M₊ᵇ(T) is metrizable of countable type for the tight topology. 09RD
Corollary 2 If T is a completely regular Souslin (resp. Lusin) space, then the space M₊ᵇ(T) is Souslin (resp. Lusin). 09RE
Definition 2 Let T be a topological space, and let H be a subset of Mᵇ(T); one says that H satisfies Prokhorov’s condition if a) sup_μ ∈ H |μ|(1) < +∞; b) for every number ε > 0, there exists a… ↩ 1 09RF
Proposition 11 Let T be a completely regular space, and let H be a subset of Mᵇ(T) that satisfies Prokhorov’s condition; then its closure overlineH in Mᵇ(T) satisfies Prokhorov’s condition. ↩ 2 09RG
Theorem 1 (Prokhorov) — Let T be a completely regular space, and let H be a subset of Mᵇ(T) that satisfies Prokhorov’s condition; then H is relatively compact in Mᵇ(T) for the tight topology. ↩ 3 09RH
Corollary — Let T be a subspace of a completely regular space X, and let H be a subset of Mᵇ(T) that satisfies Prokhorov’s condition. If i denotes the canonical injection of T into X, then t… ↩ 1 09RI
Theorem 2 — Let T be a locally compact space, or a Polish space, and let H be a relatively compact subset of M₊ᵇ(T); then H satisfies Prokhorov’s condition. ↩ 2 09RJ
Proposition 12 — Let T be a completely regular space, and let B be the unit ball of the normed space Cᵇ(T; C). Let I be a linear form on Cᵇ(T; C). In order that there exist a bounded complex meas… ↩ 2 09RK
Proposition 13 — Let T be a locally compact space, and H a bounded subset of the normed space Cᵇ(T; C). The mapping (μ, f) ↦ μ(f) of M₊ᵇ(T) × H into C is then continuous, when M₊ᵇ(T) is equipped… ↩ 1 09RL
Remark Let T be a completely regular space, M a subset of Mᵇ(T) that satisfies Prokhorov’s condition, H a bounded subset of Cᵇ(T). An argument very close to the one just made may be used… 09RM
Corollary Let T be a completely regular space, X a topological space, and f a complex-valued function defined on T × X, continuous and bounded. For every bounded measure μ on T, let F_μ be t… ↩ 2 09RN
Proposition 14 For every bounded complex measure μ on M, the function Lμ on X is continuous and bounded. If M₊ᵇ(M) is equipped with the tight topology and Cᵇ(X; C) with the topology of compact co… ↩ 1 09RO
Theorem 3 Let S be a full submonoid of X. a) If μ and μ' are two bounded complex measures on M, such that Lμ and Lμ' have the same restriction to S ∩ X₀, then μ = μ'. b) Let 𝔉 be a filter on… ↩ 5 09RP
Corollary — Let S be a full submonoid of X, such that the closure of S ∩ X₀ contains 1. Let L be the subset of Cᵇ(S; C) consisting of the restrictions to S of the Laplace transforms of the m… 09RQ
Proposition 15 Let A be a set equipped with a filter 𝔉. For every α ∈ A, let (u_α,n)_n ∈ N be a summable sequence of positive numbers, and let Φ_α be the function defined on the interval ]0, 1] o… ↩ 2 09RR
Proposition 16 Let A be a set equipped with a filter F. For every α ∈ A, let μ_α be a bounded positive measure on R₊, and let Φ_α be the function defined on R₊ by Φ_α(p) = ∫₀^+∞ e⁻ᵖˣ dμ_α(x). In… 09RS

§ 6. Promeasures and measures on a locally convex space

Definition 1 Let E be a locally convex space. One calls promeasure on E every inverse system⁽¹⁾ of measures (§ 4, No. 2, Def. 1) on the inverse system of finite-dimensional quotients of E. 09S6
Proposition 1 Let E be a locally convex space. The mapping λ ↦ widetildeλ is a bijection of the set of bounded measures on E onto the set of promeasures (μ_V)_V ∈ F(E) on E satisfying the follow… ↩ 1 09S7
Proposition 2 — Let T be a countable set, and E the vector space of real functions on T, equipped with the topology of pointwise convergence. Every promeasure on E is a measure. ↩ 3 09S8
Proposition 3 Let E be a locally convex space. The mapping μ ↦ Fμ of the set of promeasures on E into the set of functions on E' is injective. ↩ 4 09S9
Remark — When E is finite-dimensional, every character of E is of the form x ↦ e^i⟨ x, x' ⟩ with x' ∈ E' (Théor. spect., Ch. II, §1, No. 9, Cor. 3 of Prop. 12). In this case, Prop. 3 foll… 09SA
Lemma 1 For every integer n geqslant 0, ↩ 2 09SB
Lemma 2 For every complex number y, ↩ 3 09SC
Proposition 4 Let E be a locally convex space. For every positive quadratic form Q on E', there exists one and only one promeasure Γ_Q on E such that F Γ_Q = e^-Q/2. The total mass of Γ_Q is equ… ↩ 2 09SD
Definition 2 Let E be a locally convex space. For every positive quadratic form Q on E', the promeasure on E whose Fourier transform is equal to e^-Q/2 is called the Gaussian promeasure on E wi… ↩ 3 09SE
Remark — 1) Let E be a finite-dimensional vector space, and let μ be a positive measure on E of mass 1, such that every linear form on E belongs to L²(E, μ). One defines an element m of E… 09SF
Proposition 5 Let E and E₁ be two locally convex spaces, and u a continuous linear mapping of E into E₁. Let Q be a positive quadratic form on E', and Q₁ the positive quadratic form Q ∘ ᵗ u on E… ↩ 1 09SG
Corollary Let E be a locally convex space and Q a positive quadratic form on E'. For every x' ∈ E', the image of Γ_Q under x' is the Gaussian measure on R with variance Q(x'). ↩ 1 09SH
Proposition 6 Let E be a locally convex space, and μ a Gaussian measure on E, with variance Q. For every integer n ≥ 0 and every x' ∈ E', one has the relations ↩ 6 09SI
Proposition 7 Let E be a locally convex space and μ a promeasure on E. Suppose that x'(μ) is a Gaussian measure on R for every x' ∈ E'. Then μ is a Gaussian promeasure on E. ↩ 1 09SJ
Lemma 3 For any t, t' in D, ∫_Ω | X( (t + t')/2 ) - (X(t) + X(t'))/2 |³ dm = frac1(8π)^1/2 |t - t'|^3/2. Note that (t + t')/2 belongs to D. One knows (No. 6, Example 2) that the family (X(… ↩ 1 09SK
Lemma 4 — For every integer n geqslant 0, 09SL
Lemma 5 For every measure μ ∈ D, ↩ 1 09SM
Lemma 6 Let μ ∈ M¹. There exists a sequence of measures μₙ ∈ D such that μ(f) = lim_n → ∞ μₙ(f) for all f ∈ C and W(μ) = lim_n → ∞ W(μₙ). 09SN
Remark — For every semi-open interval J = ]a, b] contained in T, let us set l(J) = b - a (the length of J) and denote by A_J the linear form f ↦ f(b) - f(a) on C. It can be shown that the… 09SO
Proposition 8 — Let E be a locally convex space, μ a promeasure on E, and Φ the Fourier transform of μ. One has the inequalities ↩ 5 09SP
Corollary — Equip E' with a topology compatible with its vector space structure. For Φ to be continuous, it is necessary and sufficient that its real part Re Φ be continuous at the origin, i… ↩ 1 09SQ
Proposition 9 — If F is barreled, then the Fourier transform of every bounded measure on F' is a uniformly continuous function on F. ↩ 2 09SR
Proposition 10 — Under the hypothesis Φ(0) - Re Φ ≤ ε + q, we have ↩ 2 09SS
Theorem 2 (Minlos) — Let F be a locally convex space, T a locally convex topology on F' intermediate to Tₛ and T_c, and μ a promeasure on F'_T. Assume that the Fourier transform Φ of μ is continuous… ↩ 2 09ST
Corollary — Let F be a barreled nuclear space, T a locally convex topology on F' intermediate to Tₛ and T_c, μ a promeasure on F'_T, and Φ the Fourier transform of μ. For μ to be a measure… ↩ 1 09SU
Remark — Let F be a barreled space and T a locally convex topology on F' intermediate to Tₛ and T_c. Every subset of F' compact for T is compact for the coarser topology Tₛ. Conversely, l… 09SV
Theorem 3 (Prokhorov–Sazonov) Let E be a Hilbert space and Es the space E equipped with the weakened topology. Let μ be a promeasure on E, and Φ its Fourier transform. The following conditions are equivalent: ↩ 4 09SW
Corollary 1 — Let E₁ and E₂ be two Hilbert spaces, u a Hilbert–Schmidt mapping of E₁ into E₂, and μ a promeasure on E₁. Assume that the Fourier transform Φ of μ is continuous on E₁. Then the p… ↩ 4 09SX
Corollary 2 — Let Q be a nuclear positive quadratic form on the Hilbert space E. The Gaussian promeasure Γ_Q on E with variance Q is a measure on Eₛ. ↩ 1 09SY
Remark — 1) Let E be a Hilbert space, Eₛ the space E equipped with the weak topology, and j the identity mapping of E into Eₛ. One knows that j defines a bijection of the set of promeasur… 09SZ
Definition 3 — Let G be a group. A complex-valued function Φ on G is said to be of positive type if the inequality ↩ 2 09T0
Proposition 11 — Let E be a finite-dimensional vector space, μ a bounded (positive) measure on E, and Φ the Fourier transform of μ. The function Φ is continuous and of positive type on E'. ↩ 2 09T1
Theorem 4 — Let E be a locally convex space. The Fourier transformation is a bijection of the set of promeasures on E onto the set of functions of positive type on E' whose restriction to ev… ↩ 2 09T2
Corollary — Let F be a barreled nuclear space; equip F' with a locally convex topology T intermediate to the weak topology σ(F', F) and the topology of uniform convergence on the compact con… 09T3

App. 0. Complements on Hilbert spaces

Remark — 1) For every subspace F of E, denote by Q_F the restriction of Q to F, and by H_F that of H. Then Tr(Q_F/H_F) ≤ Tr(Q/H) and Tr(Q/H) is the supremum of the numbers Tr(Q_F/H_F) for… ↩ 1 09T5
Proposition 1 — Assume that E is finite-dimensional and H is nondegenerate. ↩ 3 09T6
Proposition 2 — Assume that E is finite-dimensional. There exist a basis (e₁, …, eₙ) of E and an integer m with 0 ≤ m ≤ n such that 09T7
Remark 3 — Assume E finite-dimensional and H nondegenerate. Let (e₁, …, eₙ) be a basis of E. Set q = ((eᵢ|eⱼ)_Q)_1 ≤ i,j ≤ n and h = ((eᵢ|eⱼ)_H)_1 ≤ i,j ≤ n. With notations as in Prop. 1, t… ↩ 2 09T8
Proposition 3 Suppose u is continuous. Let (eᵢ)_i ∈ I be an orthonormal basis of E₁ and (fⱼ)_j ∈ J an orthonormal basis of E₂. Then ↩ 2 09T9