§ 7. Integration with respect to an induced measure

1. Integration with respect to an induced measure

Let $X$ be a locally compact subspace of $T$ , $\mu$ a positive measure on $T$ , and $\mu_X$ the measure induced on $X$ by $\mu$ (Ch. IV, §5, No. 7). For every $t \in T$ , let us define a measure $\lambda_t$ on $X$ in the following way: $\lambda_t = \varepsilon_t$ if $t \in X$ , $\lambda_t = 0$ if $t \in \mathbf{C}X$ . For every finite numerical function $g$ defined on $X$ , $\int g(x) d\lambda_t(x) = g(t)$ if $t \in X$ and $\int g(x) d\lambda_t(x) = 0$ if $t \in \mathbf{C}X$ . If $g$ is a function in $\mathscr{K}(X)$ we therefore have, by the definition of $\mu_X$ ,

\mu_X(g) = \int \langle g, \lambda_t \rangle d\mu(t) .

This means that one can write

\mu_X = \int \lambda_t d\mu(t)

(§3, No. 1).

Let us now define a mapping $\pi$ of T into X by setting $\pi(t) = t$ for $t \in X$ , and $\pi(t) = t_0$ for $t \in \mathbf{C}X$ , $t_0$ being an arbitrary point of X; one can write $\lambda_t = \varphi_X(t)\varepsilon_{\pi(t)}$ for every $t \in T$ . The mapping $\pi$ is $\mu$ -measurable, because its restrictions to X and $\mathbf{C}X$ are (Ch. IV, §5, No. 10, Prop. 16); it follows at once that the pair $(\pi, \varphi_X)$ is $\mu$ -adapted (§4, No. 1). We therefore have the following results:

Proposition 1

— For every numerical function $g \geqslant 0$ defined on X,

\int^\bullet g\,d\mu_X = \int_X g\,d\mu

(cf. §5, No. 3, Example, for the notation $\int_X^\bullet$ ).

Taking into account the preceding remarks and (2), the relation (3) follows from Th. 1 of §4.

Corollary 1

— For every subset B of X, $\mu_X^\bullet(B) = \mu^\bullet(B)$ ; for B to be locally $\mu_X$ *-negligible, it is necessary and sufficient that* B be locally $\mu$ *-negligible*.

Corollary 2

— Let M be a subset of T. If $\mu$ is concentrated on M, then $\mu_X$ is concentrated on $M \cap X$ .

Corollary 3

— For the measure $\mu_X$ to be zero, it is necessary and sufficient that X be locally $\mu$ *-negligible*.

Remark

— If S is the support of $\mu$ , then $S \cap X$ (which is closed in X) contains the support of $\mu_X$ by Cor. 2, but may be distinct from it. For example, if $\mu$ is a diffuse measure and X is a subspace reduced to a point, then the induced measure $\mu_X$ is zero, hence its support is empty. Note, however, that the support of $\mu_X$ is equal to $S \cap X$ if X is open in T.

Proposition 2

— For a mapping g of X into a topological space to be $\mu_X$ *-measurable, it is necessary and sufficient that* g be $\mu$ *-measurable in* X *(§5, No. 3, Example)*.

This follows from Prop. 3 of §4.

Corollary

— For a subset B of X to be $\mu_X$ *-measurable, it is necessary and sufficient that* B be $\mu$ *-measurable*.

Theorem 1

— Let g be a function defined on X, with values in $\overline{\mathbf{R}}$ or in a Banach space. For g to be essentially $\mu_X$ *-integrable, it is necessary* and sufficient that $g$ be essentially $\mu$ -integrable in $X$ (§ 5, No. 3, Example), in which case

\int g \, d\mu_X = \int_X g \, d\mu.

This follows from Th. 2 of § 4.

Corollary 1

For a subset B of X to be essentially $\mu_X$ -integrable, it is necessary and sufficient that it be essentially $\mu$ -integrable, in which case $\mu_X(B) = \mu(B)$ .

Corollary 2

Let g be a complex function defined on T and locally $\mu$ -integrable; the restriction $g_X$ of g to X is then locally $\mu_X$ -integrable, and

(g \cdot \mu)_X = g_X \cdot \mu_X.

This follows at once from Th. 1, applied to the functions $fg_X$ ( $f \in \mathscr{K}(X; \mathbf{C})$ ), and the definition of the measure induced on X by a complex measure (Ch. IV, § 5, No. 7).

Corollary 3

Let $\theta$ be a complex measure on T; then

|\theta|_X = |\theta_X|.

Set $|\theta| = \mu$ and apply Cor. 2 on taking $g$ to be a complex function of absolute value 1 such that $\theta = g \cdot \mu$ (§ 5, No. 5, Cor. 3 of Th. 2); then $\theta_X = g_X \cdot \mu_X$ ; but $g_X$ is a function of absolute value 1, and the formula (6) follows from Prop. 2 of § 5, No. 2.

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 corollaries 1,2 extend at once to a complex measure.

Scholium

— For every function $f$ (resp. $g$ ) defined on $X$ (resp. $T$ ) with values in the Banach space $F$ or in $\overline{\mathbf{R}}$ , let us denote by $\zeta(f)$ (resp. $\rho(g)$ ) the extension by 0 of $f$ to $T$ (resp. the restriction of $g$ to $X$ ). Then $\zeta(\rho(g)) = \varphi_X \cdot g$ , $\rho(\zeta(f)) = f$ . We denote by $\mu'$ the measure $\varphi_X \cdot \mu$ on $T$ . For every $p \in [1, +\infty]$ , Props. 1 and 2 imply that $\zeta$ maps $\mathscr{L}_F^p(X, \mu_X)$ into $\mathscr{L}_F^p(T, \mu')$ , and that $\rho$ maps $\mathscr{L}_F^p(T, \mu')$ onto $\mathscr{L}_F^p(X, \mu_X)$ , with preservation of norm in both cases, as well as of the integral when $p = 1$ (Th. 1); passing to the associated Hausdorff spaces, we obtain two isomorphisms inverse to each other. Similarly, if $\zeta$ and $\rho$ are applied to positive numerical functions, the essential upper integral is preserved (Prop. 1). Thus, if we agree to identify a function on $X$ with a function on T that is zero on X − T, and the measure $\mu_X$ with the measure $\mu'$ , problems concerning induced measures are reduced to problems concerning measures defined by densities, treated in §5. This sort of reasoning is applicable to complex measures as well, by Cor. 3 of Th. 1.

2. Properties of induced measures

Proposition 3

— Let X be a locally compact subspace of T, and $\lambda$ a complex measure on X. The following properties are equivalent: a) the canonical injection $i : X \to T$ is $\lambda$ -proper; b) for every compact subset K of T, $K \cap X$ is essentially $\lambda$ -integrable; c) every point $t \in T$ admits a neighborhood V such that $V \cap X$ is essentially $\lambda$ -integrable; d) there exists a measure $\theta$ on T such that $\theta_X = \lambda$ . If these equivalent conditions are satisfied, we have, with notations as in d),

(i(\lambda))_X = \lambda \quad \text{and} \quad i(\lambda) = i(\theta_X) = \varphi_X \cdot \theta .

The injection $i$ being continuous, the equivalence of the properties a), b) and c) follows from Prop. 1 of §6, and the remark that follows it, applied to the positive measure $|\lambda|$ . If $\lambda$ is induced on X by a measure $\theta$ on T, then $|\lambda| = |\theta|_X$ (formula (6)), consequently $|\lambda|(K \cap X) = |\theta|(K \cap X) \leq |\theta|(K) < +\infty$ (Prop. 1) for every compact subset K of T, so that d) implies b). Suppose that a) is satisfied, and let us show that $(i(\lambda))_X = \lambda$ , which will imply d). Let g be an element of $\mathscr{H}(X; \mathbf{C})$ ; denoting by $g'$ the extension by 0 of g to T we have, by the definition of induced measure and then by Prop. 7 of §6, No. 4,

\int g \, d(i(\lambda))_X = \int g' \, d(i(\lambda)) = \int (g' \circ i) \, d\lambda = \int g \, d\lambda .

This completes the proof of the equivalence of the four properties. If $\lambda = \theta_X$ and $g \in \mathscr{H}(T; \mathbf{C})$ , then

\int g \, d(i(\theta_X)) = \int (g \circ i) \, d(\theta_X) = \int g \varphi_X \, d\theta ,

because $g \varphi_X$ is the extension by 0 of $g \circ i$ to T. This proves the second formula of (7).

Corollary 1

— If X is closed, then every complex measure $\lambda$ on X is induced by a measure on T.

For, if K is a compact set in T then K ∩ X is compact, hence λ-integrable.

Corollary 2

— Let θ be a complex measure on T, π a θ-proper mapping of T into a locally compact space Y, and π_X its restriction to X. Then π_X is θ_X-proper, and π_X(θ_X) = π(φ_X · θ).

For, π_X = π ∘ i, where i is the canonical injection X → T. When θ is positive the corollary may therefore be deduced from Prop. 3 and the transitivity of image measures (§6, No. 3, Prop. 4). The case of a non-positive complex measure then follows by linearity.

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 (θ_X)_Y induced by θ_X on Y is equal to θ_Y ('transitivity of induced measures').

It suffices to observe that if g is an element of $\mathscr{H}(Y; \mathbf{C})$ , then the extension by 0 of g to T may be obtained by extending by 0 the extension by 0 of g to X, or again, making use of the identifications of the Scholium, that $\varphi_Y · \theta = \varphi_Y(\varphi_X · \theta)$ (§5, No. 4, Prop. 8).

Proposition 5

— Let $(\lambda_\alpha)_{\alpha \in A}$ 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 $\lambda_\alpha|X$ is then bounded above in $\mathscr{M}(X)$ , and

\sup_{\alpha \in A} (\lambda_\alpha|X) = \lambda|X.

In view of the identifications in the Scholium, this proposition is a special case of Prop. 5 of §5, No. 4.

Corollary

— Let $(\mu_i)_{i \in I}$ be a summable family of positive measures on T, with sum μ. The family of induced measures $\mu_i|X$ is then summable, and

\sum_{i \in I} (\mu_i|X) = \mu|X.

Proposition 6

— Let $\Lambda : t \mapsto \lambda_t$ be a μ-adequate mapping of T into $\mathscr{M}_+(X)$ , where X is a locally compact space that is countable at infinity, and let Y be a locally compact subspace of X. Set $\int \lambda_t d\mu(t) = \nu$ . The mapping $t \mapsto \lambda_t|Y$ of T into $\mathscr{M}_+(Y)$ is then μ-adequate, and

\int (\lambda_t|Y) d\mu(t) = \nu|Y.

Taking into account the identifications in the Scholium, this proposition is a special case of Prop. 7 of §5, No. 4.

Exercises

See the exercises for § 7.