Index of terminology

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

Abstract measure, integral: IX, 3, Exer. 4. Additive set function: IX, 3, 2. Additive, countably (set function): IX, 3, 2. Adjoint of a linear mapping: IX, Annex, 2. Algebra, triangular (upper, lower): VII, 3, 3. Almost everywhere: IX, 1, 9. Associated promeasure, to a measure: IX, 6, 1. Atomic gauge: IX, 3, Exer. 9. Base $\mu$, measure with: IX, 2, 2. Bochner’s theorem: IX, 6, 12. Bounded encumbrance: IX, 1, 1. Bounded measure: IX, 1, 2. Bounded set function: IX, 3, 2. Brunn–Minkowski inequality: VII, 1, Exer. 25. Cardinal, strongly inaccessible: IX, 3, Exer. 13, footnote. Cardinal, ulamian: IX, 3, Exer. 11. Character of a monoid: IX, 5, 7. Concentrated (encumbrance) on a subset: IX, 1, 1. Concentrated (measure) on a subset: IX, 1, 4. Continuous linear representation: VIII, 2, 1. Contragredient (linear representation): VIII, 2, 2. Convolution product of a finite sequence of measures: VIII, 1, 1. Convolution product of functions: VIII, 4, 5. Convolvable functions: VIII, 4, 5. Convolvable measure and function: VIII, 4, 1. Convolvable, $\varphi$-convolvable (finite sequence of measures): VIII, 1, 1. Countably additive set function: IX, 3, 2. Covariance matrix of a Gaussian measure: IX, 6, 6. Covariance of a Gaussian promeasure on $\mathbf{R}^T$: IX, 6, 6. Cramped set of bounded measures: VIII, 3, Exer. 10.

Crushing: IX, 1, 8. Decomposition, Iwasawa’s, of $\mathbf{GL}(n, K)$: VII, 3, 3. Density of a measure with respect to another: IX, 2, 2. Diffuse gauge: IX, 3, Exer. 9. Disintegration of a measure: IX, 2, 7. Domain, fundamental: VII, 2, 10. Encumbrance: IX, 1, 1. Encumbrance, bounded: IX, 1, 1. Encumbrance, concentrated on a subset: IX, 1, 1. Encumbrance, image: IX, 1, 1. Encumbrance, induced: IX, 1, 1. Encumbrance, locally bounded: IX, 1, 1. Equicontinuous linear representation: VIII, 2, 1. Essential upper integral: IX, 1, 2. Essentially bounded function: IX, 1, 10. Essentially integrable function: IX, 1, 10. Finite sequence, $\varphi$-convolvable, of measures: VIII, 1, 1. Form, nuclear quadratic: IX, Annex, 1. Full submonoid: IX, 5, 7. Function of positive type: IX, 6, 12. Function, essentially bounded: IX, 1, 10. Function, essentially integrable: IX, 1, 10. Function, generating (of a sequence): IX, 5, 7. Function, integrable: IX, 1, 10. Function, locally integrable: IX, 2, 2. Function, locally negligible: IX, 1, 4. Function, measurable: IX, 1, 5. Function, moderated: IX, 1, 9. Function, modular (of a locally compact group): VII, 1, 3. Function, negligible: IX, 1, 9. Function, universally measurable: IX, 2, 7. Fundamental domain: VII, 2, 10. G-covering: VII, 1, Exer. 27. G-filling: VII, 1, Exer. 27. G-paving: VII, 1, Exer. 27. Gauge on a set X: IX, 3, Exer. 9. Gaussian measure, promeasure: IX, 6, 5. Gaussian measure with covariance matrix $C$: IX, 6, 6. Gaussian promeasure with covariance K: IX, 6, 6.

Gaussian promeasure, canonical, on a real Hilbert space: IX, 6, 6. General linear group: VII, 3, 3. Generating function of a sequence: IX, 5, 7. Group, large triangular (upper, lower): VII, 3, 3. Group, special triangular (upper, lower): VII, 3, 3. Group, strict triangular (upper, lower): VII, 3, 3. Group, unimodular: VII, 1, 3. Haar measure (left, right): VII, 1, 2. Hilbert–Schmidt mapping: IX, Annex, 2. Image of a measure: IX, 2, 3. Image of a promeasure: IX, 6. 2. Image of an encumbrance: IX, 1, 1. Induced encumbrance: IX, 1, 1. Induced measure: IX, 2, 1. Inequality, Brunn–Minkowski: VII, 1, Exer. 25. Inner regular set function: IX, 3, 2. Integers, $p$-adic: VII, 1, 6. Integrable function: IX, 1, 10. Integrable set: IX, 1, 9. Integral, abstract: IX, 3, Exer. 4. Integral, essential upper: IX, 1, 2. Integral, upper: IX, 1, 9. Integral of a function: IX, 1, 10. Invariant measure (left-, right-) on a group: VII, 1, 1. Invariant measure, under a group of operators: VII, 1, 1. Inverse (or projective) limit of measures on an inverse (or projective) limit of locally compact groups: VII, 1, 6. Inverse (or projective) limit of measures: IX, 4, 2. Inverse (or projective) system of measures: IX, 4, 2. Isometric linear representation: VIII, 2, 1. Iwasawa decomposition of $\mathbf{GL}(n, K)$: VII, 3, 3. Kernel of positive type: IX, 6, 6. Lagrange’s theorem: VII, 1, Exer. 29. Laplace transformation: IX, 5, 7. Large triangular group (upper, lower): VII, 3, 3. Left Haar measure: VII, 1, 2. Left multiplier, of a relatively invariant measure on a locally compact group: VII, 1, 8. Left-invariant measure on a group: VII, 1, 1.

Levy, P., theorem of: IX, 5, Exer. 13. Lifting of measures: IX, 2, 4. Limit, inverse (or projective), of measures on an inverse (or projective) limit of locally compact groups: VII, 1, 6. Limit, inverse (or projective), of measures: IX, 4, 2. Linear representation, transposed, contragredient of a linear representation: VIII, 2, 2. Locally almost everywhere: IX, 1, 4. Locally bounded encumbrance: IX, 1, 1. Locally bounded set function: IX, 3, 2. Locally integrable function: IX, 2, 2. Locally negligible function: IX, 1, 4. Locally negligible set: IX, 1, 4. Mapping, Hilbert–Schmidt: IX, Annex, 2. Mapping, $\mu$-proper: IX, 2, 3. Margins of a measure on a function space: IX, 4, 3. Mass, total (of a promeasure): IX, 6, 1. Mean, orbital: VII, 2, 2. Measurable function: IX, 1, 5. Measurable set: IX, 1, 5. Measure: IX, 1, 2. Measure, abstract: IX, 3, Exer. 4. Measure on $\mathbf{Q}_p$, normalized Haar: VII, 1, 6. Measure, bounded: IX, 1, 2. Measure, Gaussian with covariance matrix $C$: IX, 6, 6. Measure, Gaussian with variance $Q$: IX, 6, 5. Measure, Haar (left, right), on a locally compact group: VII, 1, 2. Measure, image: IX, 2, 3. Measure, induced: IX, 2, 1. Measure, invariant (relatively invariant, quasi-invariant) under a group of operators: VII, 1, 1. Measure, left-invariant (right-invariant), on a locally compact group: VII, 1, 1. Measure, lifting of: IX, 2, 4. Measure, moderated: IX, 1, 9. Measure, normalized Haar, on a compact group, on a discrete group: VII, 1, 3. Measure, outer (of a set): IX, 1, 9. Measure, product: IX, 2, 5. Measure, quasi-invariant, on a locally compact group: VII, 1, 9. Measure, relatively invariant, on a locally compact group: VII, 1, 8.

Measure, Wiener: IX, 6, 7. Measure, with base $\mu$: IX, 2, 2. Measure, with density $f$ with respect to a measure $\mu$: IX, 2, 2. Measured space: IX, 6, 7, Footnote (2). Minkowski’s theorem: VII, 1, Exer. 27. Minlos’s theorem: IX, 6, 10. Moderated function: IX, 1, 9. Moderated measure: IX, 1, 9. Moderated set: IX, 1, 9. Modular function of a locally compact group: VII, 1, 3. Modulus of a locally compact group: VII, 1, 3. Modulus of an automorphism: VII, 1, 4. Multiplier (left, right), of a relatively invariant measure on a locally compact group: VII, 1, 8. Multiplier on a product $G \times X$ of a group $G$ and a set $X$ on which $G$ operates: VIII, 2, 3. Multiplier, of a measure relatively invariant under a group of operators: VII, 1, 1. Negligible function: IX, 1, 9. Negligible set: IX, 1, 9. Normalized Haar measure on a compact group, on a discrete group: VII, 1, 3. Normalized Haar measure on $Q_p$: VII, 1, 6. Nuclear quadratic form: IX, Annex, 1. Nuclear space: IX, 6, 10. Orbital mean: VII, 2, 2. Outer measure: IX, 1, 9. $p$-adic integers: VII, 1, 6. Positive premeasure, measure: IX, 1, 2. Positive type, function of: IX, 6, 12. Positive type, kernel of: IX, 6, 6. Premeasure: IX, 1, 2. Premeasure, positive: IX, 1, 2. Premeasure, real: IX, 1, 2. Product of a family of measures: IX, 4, 3. Product of two measures: IX, 2, 5. Product, convolution (of a measure and a function): VIII, 4, 1. Product, convolution (of functions): VIII, 4, 5. Product, convolution (of measures), with respect to a mapping: VIII, 1, 1. Projective (or inverse) limit of measures: IX, 4, 2.

Projective (or inverse) system of measures: IX, 4, 2. Prokhorov’s conditions: IX, 4, 2 and IX, 5, 5. Promeasure: IX, 6, 1. Promeasure associated with a measure: IX, 6, 1. Promeasure, canonical Gaussian, on a real Hilbert space: IX, 6, 6. Promeasure, Fourier transform of: IX, 6, 3. Promeasure, Gaussian with covariance K, on $\mathbf{R}^T$: IX, 6, 6. Promeasure, Gaussian with variance Q: IX, 6, 5. Promeasure, image: IX, 6, 2. Promeasure, total mass of: IX, 6, 1. Proper, $\mu$- (mapping): IX, 2, 3. Quadratic form, nuclear: IX, Annex, 1. Quasi-invariant measure on a locally compact group: VII, 1, 9. Quasi-invariant measure, under a group of operators: VII, 1, 1. Quotient of a measure on a locally compact space X by a Haar measure of a group operating on X: VII, 2, 2. Radon space: IX, 3, 3. Real measure, premeasure: IX, 1, 2. Regular representation (left, right): VIII, 2, 5. Regularization: VIII, 4, 7. Relatively invariant measure, on a locally compact group: VII, 1, 8. Relatively invariant measure, under a group of operators: VII, 1, 1. Representation, continuous (separately continuous, equicontinuous, isometric): VIII, 2, 1. Representation, regular (left, right): VIII, 2, 5. Representation, unitary: VIII, 2, Exer. 4. Right Haar measure: VII, 1, 2. Right multiplier, of a relatively invariant measure on a locally compact group: VII, 1, 8. Right-invariant measure on a group: VII, 1, 1. Sazonov topology: IX, 6, 10. Separately continuous (linear representation): VIII, 2, 1. Set function, additive: IX, 3, 2. Set function, bounded: IX, 3, 2. Set function, countably additive: IX, 3, 2. Set function, inner regular: IX, 3, 2. Set function, locally bounded: IX, 3, 2. Space, nuclear: IX, 6, 10. Space, Radon: IX, 3, 3.

Space, strongly Radon: IX, 3, 3. Special triangular group (upper, lower): VII, 3, 3. Strict triangular group (upper, lower): VII, 3, 3. Strongly inaccessible cardinal: IX, 3, Exer. 13, footnote. Strongly Radon space: IX, 3, 3. Sub-inverse (or sub-projective) system of measures: IX, 4, 2. Submonoid, full: IX, 5, 7. Sum of a family of measures: IX, 1, 7. Summable family of measures: IX, 1, 7. Support of a measure: IX, 1, 6. System, inverse (or projective), of measures: IX, 4, 2. System, sub-inverse (or sub-projective), of measures: IX, 4, 2. Theorem, Bochner’s: IX, 6, 12. Theorem, Lagrange’s: VII, 1, Exer. 29. Theorem, Minkowski’s: VII, 1, Exer. 27. Theorem, Minlos’s: IX, 6, 10. Theorem, P. Lévy’s: IX, 5, Exer. 13. Theorem, Thue’s: VII, 1, Exer. 28. Thue’s theorem: VII, 1, Exer. 28. Tight convergence, topology of: IX, 5, 3. Tight topology: IX, 5, 3. Topology, Sazonov’s: IX, 6, 10. Topology, tight: IX, 5, 3. Total mass of a promeasure: IX, 6, 1. Trace of a quadratic form with respect to another: IX, Annex, 1. Transform, Fourier (of a measure, of a promeasure): IX, 6, 3. Transform, Laplace (of a measure): IX, 5, 7. Transposed linear representation: VIII, 2, 2. Triangular algebra: VII, 3, 3. Triangular group (large, strict, special): VII, 3, 3. Ulam cardinal: IX, 3, Exer. 11. Ulam ultrafilter: IX, 3, Exer. 11. Unimodular group: VII, 1, 3. Unitary representation: VIII, 2, Exer. 4. Universally measurable function: IX, 2, 7. Universally measurable set: IX, 3, 3. Upper integral: IX, 1, 9. Variance of a measure: IX, 6, 5. Wiener measure: IX, 6, 7.