Index of terminology

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

Absolute value of a measure : III, 1, 6 and VI, 2, 8. Adapted pair, $\mu^-$ : V, 4, 1. Additive set function : IV, 4, 9. Additivity, complete : IV, 4, 5. Adequate mapping, $\mu^-$ : V, 3, 1 and VI, 1, 1, Footnote. Alien measures : V, 5, 7. Almost everywhere, function defined : IV, 2, 5. Almost everywhere, property true : IV, 2, 3. Atomic measure : III, 1, 3. Band, in a fully lattice-ordered space : II, 1, 5. Barycenter of a measure : IV, 7, 1. Base $\mu$, measure with : V, 5, 2 and VI, 2, 8. Base $\mu$, scalarly of (vectorial measure) : VI, 2, 5. Base $\mu$, vectorial measure with : VI, 2, 4. Belonging to the domain of a diffusion, measure : V, 3, 5. Bishop’s theorem : IV, 7, 5. Boundary, frontier : III, 1, 1, Footnote. Bounded diffusion: V, 3, 5. Bounded in measure, function : IV, 6, 2 and IV, 6, 3. Bounded measure : III, 1, 8 and VI, 2, 9. Carrying a measure, subset : V, 5, 7. Choquet’s theorems : IV, 7, 2 and IV, 7, 6. Clan of subsets of a set : IV, 4, 9. Class, equivalence (of functions), for a measure : IV, 2, 4, IV, 2, 5 and IV, 5, 2. Class, pseudo-image (of a class of measures) : VI, 3, 2. Co-lattice subspace : II, 1, Exer. 3. Compact convergence (in the space of measures) : III, 1, 10. Completely additive : IV, 4, 5. Complex measure : III, 1, 3 and VI, 2, 8. Composed diffusion : V, 3, 6.

Concentrated on a set, measure : V, 5, 7. Conjugate exponents : IV, 6, 4. Conjugate measure : III, 1, 5 and VI, 2, 8. Connected component : V, 6, Exer. 12, Footnote. Conservative mapping : V, 7, Exer. 9. Continuous almost everywhere : IV, 5, Exer. 16. Convergence in mean, convergence in mean of order $p$, convergence in quadratic mean : IV, 3, 3. Convergence in measure : IV, 5, 11. Convergence, almost everywhere : IV, 2, 5. Convergence, compact (in the space of measures) : III, 1, 10. Convergence, strictly compact (in the space of measures) : III, 1, 10. Convergence, vague : III, 1, 9. Convergent filter in a fully lattice-ordered space : II, 1, Exer. 9. Countable at infinity : III, 1, 1, Footnote. Countable convexity, theorem of : IV, 3, 2. Decomposition into slices, of a measure : V, 6, 6 and VI, 3. 1. Decreasing rearrangement of a function : IV, 5, Exer. 29. Defined almost everywhere (function) : IV, 2, 5. Defined locally almost everywhere (function) : IV, 5, 2. Density $g$ (relative to $\theta$), measure with : V, 5, 2 and VI, 2, 8. Density point (of a subset) : V, 6, Exer. 15. Density, of a vectorial measure with respect to a positive measure : VI, 2, 4. Density, with respect to a complex measure : III, 1, 4, V, 5, 2 and VI, 2, 8. Derivates of a function : V, 6, Exer. 12. Diffuse measure : V, 5, 10. Diffusion : V, 3, 5. Diffusion, bounded : V, 3, 5. Diffusion, composed : V, 3, 6. Dirac measure : III, 1, 3. Direct limit, of a direct system of locally convex spaces : III, 1, 1. Direct sum, ordered : II, 1, 4. Discrete measure : III, 1, 3. Disintegration of a measure : V, 6, 6 and VI, 3, 1. Disintegration, of a measure $\mu$ relative to a $\mu$-proper mapping : VI, 3, 1. Disintegration, of a measure $\mu$ relative to a pseudo-image of $\mu$ : VI, 3, 3. Disintegration, of a measure by a measurable equivalence relation : VI, 3, 5. Distribution of a sequence, limit : III, 2, Exer. 5. Domain of a diffusion, measure belonging to : V, 3, 5.

Double integral : III, 4, 1. Dunford–Pettis theorem : VI, 2, 5. Egoroff’s theorem : IV, 5, 4. Equi-integrable set (of functions) : IV, 5, 11. Equidistributed sequence (for a measure) : III, 2, Exer. 5. Equimeasurable functions : IV, 5, Exer. 29. Equivalence class of functions for a measure : IV, 2, 4, IV, 2, 5 and IV, 5, 2. Equivalence relation, Hausdorff : VI, 3, 4. Equivalence relation, measurable : VI, 3, 4. Equivalent measures : V, 5, 6 and VI, 2, 8. Equivalent, $\mu$-equivalent (functions) : IV, 2, 4. Essential upper integral : V, 1, 1. Essentially $\mu$-integrable (set, function) : V, 1, 3. Essentially integrable function : V, 1, 3. Essentially integrable function, $p$-th power : V, 1, 3. Essentially integrable function, for a vectorial measure : VI, 2, 2. Essentially integrable on $A$, function : V, 5, 3. Expansion set (right-, left-) of a function : V, 6, Exer. 12. Extremal point, $\mathscr{H}^-$ : IV, 7, 3. Finite almost everywhere, function : IV, 2, 6. Form, positive linear : II, 2, 1. Form, relatively bounded linear : II, 2, 2. Fully lattice-ordered space : II, 1, 3. Function depending only on a finite number of variables : III, 4, 6. Function, $\mu$-measurable on $A$ : V, 5, 3. Function, $\mu$-moderated : V, 1, 2. Function, $\Phi$-step : IV, 4, 9. Function, $p$-th power essentially integrable : V, 1, 3. Function, $p$-th power integrable : IV, 3, 4. Function, bounded in measure : IV, 6, 2 and IV, 6, 3. Function, defined almost everywhere : IV, 2, 5. Function, defined locally almost everywhere : IV, 5, 2. Function, essentially integrable : V, 1, 3. Function, essentially integrable for a vectorial measure : VI, 2, 2. Function, essentially integrable on $A$ : V, 5, 3. Function, finite almost everywhere : IV, 2, 6. Function, having a property scalarly : VI, 1, 1. Function, integrable, $\mu$-integrable : IV, 4, 1. Function, Lebesgue $n$-th : IV, 6, Exer. 15.

Function, locally integrable : V, 5, 1. Function, locally integrable on A : V, 5, 3. Function, locally negligible, locally $\mu$-negligible : IV, 5, 2. Function, measurable step : IV, 5, 5. Function, measurable, $\mu$-measurable : IV, 5, 1. Function, measurable, defined on a measurable subset of X : IV, 5, 10 and V, 5, 3. Function, negligible, $\mu$-negligible : IV, 2, 1 and IV, 2, 4. Function, numerical : I, 1, 1, Footnote. Function, scalarly essentially integrable : V, 3, 1, VI, 1, 1 and VI, 2, 10. Function, scalarly well integrable : VI, 1, Exer. 19. Function, step, $\Phi$-step : IV, 4, 9. Function, support of : III, 1, 1. Function, universally measurable : V, 3, 4. Function, vaguely $\mu$-measurable : V, 3, 1. Function, vaguely continuous : V, 3, 1. Functions, equi-integrable : IV, 5, 11. Functions, equimeasurable : IV, 5, Exer. 29. Functions, equivalent, $\mu$-equivalent : IV, 2, 4. (GDF) property : VI, 1, 4. Gelfand–Dunford theorem : VI, 1, 4. Gliding hump method : V, 5, Exer. 13. $\mathscr{H}$-extremal point : IV, 7, 3. Hölder’s inequality : I, 2 and IV, 6, 4. Haar orthonormal system : IV, 6, Exer. 17. Hardy’s inequality : IV, 6, Exer. 19. Hausdorff equivalence relation : VI, 3, 4. Image of a measure : V, 6, 1, V, 6, 4 and VI, 2, 10. Imaginary part of a measure : III, 1, 5 and VI, 2, 8. Induced measure, on a locally compact subspace : IV, 5, 7 and VI, 2, 10. Induced measure, on an open subset : III, 2, 1. Inductive limit, of an inductive system of locally convex spaces : see direct limit. Inequality of Hölder : I, 2 and IV, 6, 4. Inequality of Hardy : IV, 6, Exer. 19. Inequality of M. Riesz : IV, 6, Exer. 18. Inequality of Minkowski : I, 2 and IV, 3, 1. Inequality of the mean : IV, 6, 2. Infimum : II, 1, 1, Footnote.

Inner measure : IV, 4, Exer. 7. Integrable function : IV, 4, 1. Integrable function, $p$-th power : IV, 3, 4. Integrable set, $\mu$-integrable set : IV, 4, 5. Integral of a function on $A$ (or extended to $A$) : V, 5, 3. Integral of a scalarly essentially integrable mapping : V, 3, 1, VI, 1, 1 and VI, 2, 10. Integral of an essentially integrable function : V, 1, 3. Integral, double : III, 4, 1. Integral, essential upper : V, 1, 1. Integral, lower : IV, 4, Exer. 5. Integral, multiple, $n$-tuple : III, 4, 4. Integral, of a continuous numerical function with compact support : III, 1, 3. Integral, of a numerical function with respect to a vectorial measure : VI, 2, 2. Integral, of a vector-valued function with respect to a positive measure : IV, 4, 1, V, 1, 3 and VI, 1, 1. Integral, of a vector-valued function with respect to a vectorial measure : VI, 2, 7. Integral, of a weakly continuous function scalarly of compact support : III, 3, 1. Integral, of an integrable function : IV, 4, 1. Integral, upper : IV, 1, 1, IV, 1, 3 and IV, 4, Exer. 5. Integration by parts : V, 8, Exer. 9. Invariant measure, under a homeomorphism : III, 1, 3. Inverse image of a measure under a local homeomorphism : V, 6, 6. Inverse limit, of an inverse system of measures : III, 4, 5. Inverse system (of measures) : III, 4, 5. Isolated subspace, of a Riesz space : II, 1, Exer. 4. Krein’s theorem : IV, 7, Exer. 10. Lattice-ordered vector space : II, 1, 1. Latticial linear mapping : II, 2, Exer. 5. Lebesgue function, $n$-th : IV, 6, Exer. 15. Lebesgue measure : III, 1, 3, III, 4, 1 and III, 4, 4. Lebesgue’s decomposition theorem : V, 5, 7. Lebesgue’s theorem : IV, 3, 7 and IV, 4, 3. Lebesgue–Fubini theorem : V, 8, 4. Lebesgue–Nikodym theorem : V, 5, 5. Lifting property : VI, 2, 5. Limit distribution of a sequence : III, 2, Exer. 5.

Limit superior, limit inferior, in a fully lattice-ordered space : II, 1, Exer. 9. Limit, direct (or inductive) : III, 1, 1. Limit, inverse (or projective) : III, 4, 5. Limit, of a filter in a fully lattice-ordered space : II, 1, Exer. 9. Linear form, positive : II, 2, 1. Linear form, relatively bounded : II, 2, 2. Localization principle, for measurable functions : IV, 5, 2. Localization principle, for measures : III, 2, 1. Locally almost everywhere : IV, 5, 2. Locally countable (set of subsets) : IV, 5, 9. Locally countable family of functions $\geq 0$ : V, 5, 4. Locally integrable function : V, 5, 1. Locally integrable on $A$, function : V, 5, 3. Locally negligible ($\mu$-negligible) function : IV, 5, 2. Locally negligible set, locally $\mu$-negligible set : IV, 5, 2. Lower integral : IV, 4, Exer. 5. $\mu$-adapted pair : V, 4, 1. $\mu$-adequate mapping : V, 3, 1 and VI, 1, 1, Footnote. $\mu$-dense (set of compact subsets) : IV, 5, 8. $\mu$-equivalent functions : IV, 2, 4. $\mu$-integrable function : IV, 4, 1. $\mu$-integrable set : IV, 4, 5. $\mu$-maximum, $\mu$-minimum : IV, 6, 2. $\mu$-measurable (set, function) : IV, 5, 1. $\mu$-measurable equivalence relation : VI, 3, 4. $\mu$-moderated (function, subset) : V, 1, 2. $\mu$-negligible function : IV, 2, 1 and IV, 2, 4. $\mu$-negligible set : IV, 2, 2. $\mu$-pre-adequate mapping : V, 3, 1. $\mu$-proper mapping : V, 6, 1, V, 6, 4 and VI, 2, 10. Majorizable measure : VI, 2, 3. Mapping, $\mu$-adequate : V, 3, 1. Mapping, $\mu$-pre-adequate : V, 3, 1. Mapping, $\mu$-proper (or proper for $\mu$) : V, 6, 1, V, 6, 4 and VI, 2, 10. Mapping, conservative : V, 7, Exer. 9. Mapping, scalarly of compact support : III, 3, 1. Mapping, weakly continuous : III, 3, 1. Mass, total, of a bounded measure : III, 1, 8 and IV, 4, 7. Masses, measure defined by : III, 1, 3.

Maximum in measure, $\mu$-maximum : IV, 6, 2. Mean of a function : III, 1, 8. Mean of order $p$, convergence in : IV, 3. 3. Mean, convergence in : IV, 3, 3. Mean, inequality of the : IV, 6, 2. Measurable (set, function) : IV, 5, 1. Measurable equivalence relation : VI, 3, 4. Measurable function defined on a measurable subset of X : IV, 5, 10 and V, 5, 3. Measurable on A , function : IV, 5, 10 and V, 5, 3. Measurable section : VI, 3, 4. Measurable step function : IV, 5, 5. Measure belonging to the domain of a diffusion : V, 3, 5. Measure concentrated on a set : V, 5, 7. Measure of an integrable set : IV, 4, 5. Measure with base $\mu$ : V, 5, 2. Measure with density $g$ (with respect to $\theta$) : V, 5, 2. Measure, absolute value of : III, 1, 6 and VI, 2, 8. Measure, atomic : III, 1, 3. Measure, bounded : III, 1, 8 and VI, 2, 9. Measure, complex measure : III, 1, 3 and VI, 2, 8. Measure, conjugate : III, 1, 5 and VI, 2, 8. Measure, defined by a unit mass at a point : III, 1, 3. Measure, defined by masses : III, 1, 3. Measure, diffuse : V, 5, 10. Measure, Dirac : III, 1, 3. Measure, discrete : III, 1, 3. Measure, image of : V, 6, 1, V, 6, 4 and VI, 2, 10. Measure, induced : III, 2, 1, IV, 5, 7 and VI, 2, 10. Measure, inner : IV, 4, Exer. 7. Measure, invariant under a homeomorphism : III, 1, 3. Measure, involving no mass in an open set : III, 2, 2. Measure, Lebesgue : III, 1, 3, III, 4, 1 and III, 4, 4. Measure, majorizable, $q$-majorizable : VI, 2, 3. Measure, norm of : III, 1, 8. Measure, outer : IV, 1, 2 and IV, 1, 4. Measure, point : III, 2, 4. Measure, positive : III, 1, 5. Measure, product : III, 4, 1, III, 4, 4, III, 4, 6 and VI, 2, 10.

Measure, pseudo-image : VI, 3, 2. Measure, quotient by an equivalence relation : VI, 3, 5. Measure, real : III, 1, 5 and VI, 2, 1. Measure, real and imaginary parts : III, 1, 5 and VI, 2, 8. Measure, scalar : VI, 2, 1. Measure, support of : III, 2, 2. Measure, vectorial : VI, 2, 1. Measure, with density $g$ with respect to $\mu$ : III, 1, 4, V, 5, 2 and VI, 2, 8. Measures, alien : V, 5, 7. Measures, equivalent : V, 5, 6 and VI, 2, 8. Minimum in measure, $\mu$-minimum : IV, 6, 2. Minkowski’s inequality : I, 2 and IV, 3, 1. Moderated function, $\mu$- : V, 1, 2. Moderated measure : V, 1, 2. Moderated subset, $\mu$- : V, 1, 2. Multiple integral : III, 4, 4. $n$-tuple integral : III, 4, 4. Negligible function : IV, 2, 1 and IV, 2, 4. Negligible set : IV, 2, 2. Norm of a diffusion : V, 3, 5. Norm of a measure : III, 1, 8. Numerical function : I, 1, 1, Footnote. Order $n$, multiple integral of : III, 4, 4. Order $p$, convergence in mean of : IV, 3, 3. Order $p$, equi-integrable of (set of functions) : IV, 5, 11. Ordered direct sum : II, 1, 4. Orthonormal sequence, in $\mathscr{L}^2$ : IV, 6, Exer. 15. Orthonormal system, Haar : IV, 6, Exer. 17. Outer measure : IV, 1, 2 and IV, 1, 4. $p$-th power integrable function : IV, 3, 4. Parts of a measure, real and imaginary : III, 1, 5 and VI, 2, 8. $\Phi$-step function : IV, 4, 9. Point measure : III, 2, 4. Point, $\mathscr{H}$-extremal : IV, 7, 3. Positive linear form : II, 2, 1. Positive measure : III, 1, 5. Pre-adequate mapping, $\mu$- : V, 3, 1. Principle of localization for measurable functions : IV, 5, 2. Principle of localization for measures : III, 2, 1.

Product of a family of positive measures : III, 4, 6. Product of a finite number of measures : III, 4, 4. Product of a measure by a continuous function : III, 1, 4. Product of a measure by a locally integrable function : V, 5, 2. Product of two measures : III, 4, 1 and VI, 2, 10. Projective limit, of a projective system of measures : see inverse limit. Projective system of measures : see inverse system. Proper mapping, $\mu^-$ : V, 6, 1, V, 6, 4 and VI, 2, 10. Property (GDF) : VI, 1, 4. Property true almost everywhere : IV, 2, 3. Property true locally almost everywhere : IV, 5, 2. Property, lifting : VI, 2, 5. Pseudo-image (class, measure) : VI, 3, 2. $q$-majorizable measure : VI, 2, 3. Quadrable set : IV, 5, Exer. 17. Quadratic mean, convergence in : IV, 3, 3. Quasi-integrable function : IV, 5, Exer. 15. Quasi-strong topology : III, 1, Exer. 8. Quotient measure, of a measure by an equivalence relation : VI, 3, 5. Real measure : III, 1, 5 and VI, 2, 1. Real part of a measure : III, 1, 5 and VI, 2, 8. Rearrangement of a function, decreasing : IV, 5, Exer. 29. Recede indefinitely along a filter : III, 2, 2. Relatively bounded linear form : II, 2, 2. Restriction of a measure to a locally compact subset : IV, 5, 7 and VI, 2, 10. Restriction of a measure to an open set : III, 2, 1. Riesz space : II, 1, 1. Riesz, F., theorem of : II, 1, 5. Riesz, M., inequality of : IV, 6, Exer. 18. Scalar measure : VI, 2, 1. Scalarly essentially integrable mapping : V, 3, 1, VI, 1, 1 and VI, 2, 10. Scalarly of base $\mu$, vectorial measure : VI, 2, 5. Scalarly of compact support, mapping : III, 3, 1. Scalarly well integrable function : VI, 1, Exer. 19. Scalarly, function having a property : VI, 1, 1. Section, measurable : VI, 3, 4. Separable : IV, 2, 4, Footnote. Sequence convergent almost everywhere : IV, 2, 5. Sequence equidistributed for a measure : III, 2, Exer. 5.

Series convergent almost everywhere : IV, 2, 5. Set function, additive : IV, 4, 9. Slices, decomposition into (of a measure): V, 6, 6 and VI, 3, 1. Space, fully lattice-ordered : II, 1, 3. Space, Riesz : II, 1, 1. Step function : IV, 4, 9. Step function, $\Phi$ : IV, 4, 9. Step function, measurable : IV, 5, 5. Stieltjes measure : V, 6, Exer. 5. Stone space : II, 1, Exer. 13. Strictly compact convergence (in the space of measures) : III, 1, 10. Strictly compact set, in $\mathscr{K}(X;E)$ : III, 1, 1. Summable family of positive measures : V, 2, 1. Support of a function : III, 1, 1. Support of a measure : III, 2, 2. Support of a vectorial measure : VI, 2, 1. Supremum : II, 1, 1, Footnote. Theorem of Bishop : IV, 7, 5. Theorem of countable convexity : IV, 3, 2. Theorem of Egoroff : IV, 5, 4. Theorem of F. Riesz : II, 1, 5. Theorem of Krein : IV, 7, Exer. 10. Theorem of Lebesgue : IV, 3, 7 and IV, 4, 3. Theorem, Dunford–Pettis : VI, 2, 5. Theorem, Gelfand–Dunford : VI, 1, 4. Theorem, Lebesgue decompositon : V, 5, 7. Theorem, Lebesgue–Fubini : V, 8, 4. Theorem, Lebesgue–Nikodym : V, 5, 5. Theorems of Choquet : IV, 7, 2 and IV, 7, 6. Topology of compact convergence, of strictly compact convergence (on the space of measures) : III, 1, 10. Topology of convergence in mean, in mean of order $p$, in quadratic mean : IV, 3. 3. Topology of convergence in measure : IV, 5, 11. Topology, quasi-strong : III, 1, Exer. 8. Topology, ultrastrong : III, 1, Exer. 15. Topology, vague : III, 1, 9. Total mass, of a bounded measure : III, 1, 8 and IV, 4, 7. Tribe of subsets : IV, 4, 9, Footnote.

True almost everywhere, property : IV, 2, 3. True locally almost everywhere, property : IV, 5, 2. Ultrastrong topology : III, 1, Exer. 15. Uniform structure of convergence in measure : IV, 5, 11. Universally measurable (set, function) : V, 3, 4. Upper integral : IV, 1, 1, IV, 1, 3 and IV, 4, Exer. 5. Upper integral, essential : V, 1, 1. Vague convergence : III, 1, 9. Vague topology : III, 1, 9. Vaguely $\mu$-measurable function : V, 3, 1. Vaguely continuous function : V, 3, 1. Vectorial measure : VI, 2, 1. Vectorial measure of base $\mu$ : VI, 2, 4. Vectorial measure, $q$-majorizable : VI, 2, 3. Vectorial measure, majorizable : VI, 2, 3. Vectorial measure, scalarly with base $\mu$ : VI, 2, 5. Vectorial measure, support of : VI, 2, 1. Weakly continuous mapping : III, 3, 1. Well integrable, scalarly : VI, 1, Exer. 19.