Index of notations

Reference numbers indicate, in order, the chapter, section and subsection (or, exceptionally, exercise).

Chapter III :

$\mathscr{C}(X;E), \mathscr{C}(X), \mathscr{C}(X,A;E), \mathscr{K}(X;E), \mathscr{K}(X), \mathscr{K}(X,A;E), \mathscr{K}(X,A), \mathscr{K}_+(X)$ (X a locally compact space, E a topological vector space) : III, 1, 1. Supp(f) (f a function with values in a vector space or in $\overline{\mathbf{R}}$) : III, 1, 1. $\mathscr{C}^b(X;E), \mathscr{C}^0(X;E)$ : III, 1, 2. $\|f\|$ (f a function with values in a normed space) : III, 1, 2. $\mu(f), \langle f, \mu \rangle, \int f d\mu, \int f \mu, \int f(x) d\mu(x), \int f(x)\mu(x)$ (f a function in $\mathscr{K}(X;\mathbf{C})$, $\mu$ a (complex) measure) : III, 1, 3. $\mathscr{M}(X;\mathbf{C}), \mathscr{M}(X), \mathscr{M}_\sigma(X;\mathbf{C}), \mathscr{M}_\sigma(X)$ : III, 1, 3. $\varepsilon_a$ : III, 1, 3. g · $\mu$ (g a function in $\mathscr{C}(X;\mathbf{C})$) : III, 1, 4. $\overline{\mu}, \Re \mu, \Im \mu$ : III, 1, 5. $\mathscr{M}(X;\mathbf{R}), \mathscr{M}(X), \mathscr{M}_+(X)$ : III, 1, 5. $\mu \leq \nu$ ($\mu, \nu$ real measures) : III, 1, 5. $\mu^+, \mu^-, |\mu|$ ($\mu$ a real measure) : III, 1, 5. $|\mu|$ ($\mu$ a complex measure) : III, 1, 6. $\|\mu\|$ ($\mu$ a measure) : III, 1, 8. $\mathscr{M}^1(X,\mathbf{R}), \mathscr{M}^1(X)$ : III, 1, 8. $\mu|Y$ ($\mu$ a measure on X, Y an open subspace of X) : III, 2, 1. Supp($\mu$) ($\mu$ a measure) : III, 2, 2. $\langle f, z' \rangle$ : III, 3, 1. $\widetilde{\mathscr{K}}(X;E)$ : III, 3, 1. $\int f d\mu, \int f \mu, \int f(x) d\mu(x), \int f(x)\mu(x)$ (f a function in $\widetilde{\mathscr{K}}(X;E)$) : III, 3, 1. $\int d\mu(y) \int f(x,y) d\lambda(x)$ : III, 4, 1. $\iint f d\lambda d\mu, \iint f d\mu d\lambda, \iint f \lambda \mu, \iint f \mu \lambda, \iint f(x,y) d\lambda(x) d\mu(y), \iint f(x,y) d\mu(y) d\lambda(x), \iint f(x,y) \lambda(x) \mu(y), \iint f(x,y) \mu(y) \lambda(x)$ : III, 4, 1.

λ ⊗ μ (λ, μ measures) : III, 4, 2. μ₁ ⊗ μ₂ ⊗ ⋯ ⊗ μₙ, $\bigotimes_{i=1}^{n} \mu_i$ : III, 4, 4. $\int f\ d\mu_1\ d\mu_2\ldots\ d\mu_n$, $\iint \ldots \int f\ d\mu_1\ d\mu_2\ldots\ d\mu_n$, $\int f(\mu_1 \otimes \mu_2 \otimes \cdots \otimes \mu_n)$, $\iint \ldots \int f(x_1, x_2, \ldots, x_n)\ d\mu_1(x_1)\ d\mu_2(x_2)\ldots\ d\mu_n(x_n)$, $\iint \ldots \int f(x_1, x_2, \ldots, x_n)\mu_1(x_1)\mu_2(x_2)\cdots\mu_n(x_n)$ : III, 4, 4. $\bigotimes_{\lambda \in L} \mu_\lambda$ : III, 4, 6.

Chapter IV :

φ_A : IV, 1, 1. $\mathscr{H}$₊, $\mathscr{J}$₊(X), $\mathscr{J}$₊ : IV, 1, 1. μ*(f) (μ a positive measure) : IV, 1, 1, IV, 1, 3 and IV, 4, Exer. 5. ∫* f dμ, ∫* fμ, ∫* f(x) dμ(x), ∫* f(x)μ(x) (f a function ≥ 0, μ a positive measure) : IV, 1, 3. μ*(A) (A a subset of X, μ a positive measure) : IV, 1, 2 and IV, 1, 4. f̃ : IV, 2, 4 and IV, 2, 5. φ((f̃ₙ)), f̃ + g̃, αf̃, g̃f̃ : IV, 2, 4. f̃ ≤ g̃, supₙ f̃ₙ, infₙ f̃ₙ, lim supₙ→∞ f̃ₙ, lim infₙ→∞ f̃ₙ (f, g, fₙ numerical functions) : IV, 2, 6. |z| (z a point of a normed space) : IV, 3, 2. |f| (f a function with values in a normed space) : IV, 3, 2. N_p(f, μ), N_p(f), N_p(f̃) (1 ≤ p < +∞) : IV, 3, 2. $\mathscr{G}_F(X), \mathscr{G}_F$ : IV, 3, 3. $\mathscr{G}_F^p(X, \mu), \mathscr{G}_F^p(\mu), \mathscr{G}_F^p$ : IV, 3, 3. $\mathscr{G}_F$ : IV, 3, 3. $\mathscr{G}_F, \mathscr{D}_F^p(X, \mu), \mathscr{D}_F^p(\mu), \mathscr{D}_F^p, L_F^p(X, \mu), L_F^p(\mu), L_F^p, \mathscr{D}^p, L^p (1 \leq p < +\infty)$ : IV, 3, 4. ‖f̃‖_p (1 ≤ p < +∞) : IV, 3, 4. μ(f), ∫ f dμ, ∫ f(x) dμ(x), ∫ fμ, ∫ f(x)μ(x), μ(f̃) (f a μ-integrable function with values in a Banach space) : IV, 4, 1. μ(A) (A a μ-integrable set) : IV, 4, 5. $\mathscr{Q}'(X; C)$ : IV, 4, 8. $\mathscr{Q}$(Φ), $\mathscr{Q}_F$(Φ) (Φ a clan of sets) : IV, 4, 9. μ_*(f) (f a function) : IV, 4, Exer. 5.

$\mu_*(A)$ (A a set) : IV, 4, Exer. 7. $\int_A f d\mu, \int_A f\mu, \int_A^* f d\mu, \int_A^* f\mu$ : IV, 5, 6. $\mathscr{S}(A, \mu; F), \mathscr{S}_F(A, \mu), \mathscr{S}_F(\mu), \mathscr{S}_F$ : IV, 5, 11. $W(V, B, \delta)$ : IV, 5, 11. $S(A, \mu; F), S_F(A, \mu), S_F(\mu), S_F$ : IV, 5, 11. $f^*$ (the decreasing rearrangement of $f$) : IV, 5, Exer. 29. $M_\infty(f), m_\infty(f)$ : IV, 6, 2. $N_\infty(f)$ : IV, 6, 3. $\mathscr{L}_F^\infty(X, \mu), \mathscr{L}_F^\infty(\mu), \mathscr{L}_F^\infty, \mathscr{N}_F^\infty$ : IV, 6, 3. $\|\dot{f}\|_\infty, N_\infty(\dot{f})$ : IV, 6, 3. $L_F^\infty(X, \mu), L_F^\infty(\mu), L_F^\infty, \mathscr{L}^\infty, L^\infty$ : IV, 6, 3. $b_\mu$ : IV, 7, 1. $\mathrm{Ch}_\mathscr{H}(X), \mathrm{Ch}(X), \check{S}_\mathscr{H}(X), \check{S}(X)$ : IV, 7, 3.

Chapter V :

$\mathscr{F}_+(E), \mathscr{F}_+$ (E a set) : V, Preliminary conventions. $\mu^\bullet(f), \mu^\bullet(A), \int^\bullet f d\mu, \int^\bullet f(t) d\mu(t), \int^\bullet f\mu$ : V, 1, 1. $\overline{\mathscr{F}}_F^p(T, \mu), \overline{\mathscr{F}}_F^p(\mu), \overline{\mathscr{F}}_F^p$ : V, 1, 3. $\overline{N}_p(f), \overline{\mathscr{L}}_F^p(T, \mu), \overline{\mathscr{L}}_F^p(\mu), \overline{\mathscr{L}}_F^p$ : V, 1, 3. $\overline{\mathscr{L}}_F^p(T, \theta)$ ($\theta$ a complex measure) : V, 1, 3. $\int \lambda_t d\mu(t)$ ($t \mapsto \lambda_t$ a family of positive measures) : V, 3, 1. $\int d\mu(t) \int f(x) d\lambda_t(x)$ : V, 3, 1. $\|\Lambda\|$ ($\Lambda$ a diffusion) : V, 3, 5. $\langle \eta, h \rangle$ : V, 3, 5. $\Lambda f, \mu \Lambda$ : V, 3, 5. $\Lambda H$ : V, 3, 6. $\mathscr{L}_{\text{loc}}^1(T, \mu; F), L_{\text{loc}}^1(T, \mu; F)$ : V, 5, 1. $u \cdot \theta$ ($u$ a complex function, $\theta$ a complex measure) : V, 5, 2. $\int_A^\bullet f d\mu$ : V, 5, 3. $u(\mu_1, \ldots, \mu_n)$ ($u$ a positively homogeneous numerical function) : V, 5, 9. $\pi(\mu)$ ($\pi$ a $\mu$-proper mapping): V, 6, 1. $\pi(\theta)$ ($\theta$ a complex measure, $\pi$ an $|\theta|$-proper mapping) : V, 6, 4. $\pi^{-1}(\mu)$ ($\pi$ a local homeomorphism): V, 6, 6. $\iint^* f(t, t') d\mu(t) d\mu'(t'), \iint^\bullet f(t, t') d\mu(t) d\mu'(t'), \iint f(t, t') d\mu(t) d\mu'(t')$ : V, 8, 1

Chapter VI:

F', F'', {F'}^*, F_\sigma (F a Hausdorff locally convex space) : VI, Introduction. \mathscr{K}(T), \mathscr{K}_R(T), \mathscr{K}_C(T), \mathscr{K}(T,A), \mathscr{K}_C(T,A) : VI, Introduction. \langle f, z' \rangle, \langle z', f \rangle : VI, 1. \int f d\mu, \int f(t) d\mu(t) (f a vector-valued function, \mu a positive measure) : VI, 1, 1. gf, fg (f a vector-valued function, g a scalar function) : VI, 1, 1. \mathscr{C}'(T) : VI, 1, 6. \int f dm, \int f(t) dm(t) (f a numerical function, m a vectorial measure) : VI, 2, 1 and VI, 2, 2. g \cdot m (g a numerical function, m a vectorial measure) : VI, 2, 1. $\mathscr{L}(m)$ : VI, 2, 2. q(m), |m| (q a semi-norm, m a vectorial measure) : VI, 2, 3. f \cdot \mu (f a vector-valued function, \mu a positive measure) : VI, 2, 4. $\mathscr{L}_{F_s'}^\infty, L_{F_s'}^\infty$ : VI, 2, 5. \langle f, g \rangle (f, g vector-valued functions) : VI, 2, 6. I_{\Phi,m}, \int f dm (f a vector-valued function, m a vectorial measure) : VI, 2, 7. |m|, \int f dm (m a complex measure) : VI, 2, 8, III, 1, 6. \mathscr{L}_F^p(T,m), \overline{\mathscr{L}}_F^p(T,m), L_F^p(T,m) (m a complex measure) : VI, 2, 8, V, 1, 3. h \cdot m (m a complex measure) : VI, 2, 8. \overline{m} (m a complex measure) : VI, 2, 8, III, 1, 5. \|m\| (m a complex measure) : VI, 2, 9, III, 1, 8. \pi(m), m_Y, m \otimes m' (m, m' complex measures) : VI, 2, 10. \mathscr{B}(F_1, F_2), r\Phi, l\Phi : VI, App., 1. E_\sigma, F_\sigma, E'_s, F'_s, \mathscr{B}(E,F) : VI, App., 1. \Lambda_{F'}^p(T,\mu), M_p, M'_p : VI, 1, Exer. 16.