main-content

## Über dieses Buch

Integration theory holds a prime position, whether in pure mathematics or in various fields of applied mathematics. It plays a central role in analysis; it is the basis of probability theory and provides an indispensable tool in mathe­ matical physics, in particular in quantum mechanics and statistical mechanics. Therefore, many textbooks devoted to integration theory are already avail­ able. The present book by Michel Simonnet differs from the previous texts in many respects, and, for that reason, it is to be particularly recommended. When dealing with integration theory, some authors choose, as a starting point, the notion of a measure on a family of subsets of a set; this approach is especially well suited to applications in probability theory. Other authors prefer to start with the notion of Radon measure (a continuous linear func­ tional on the space of continuous functions with compact support on a locally compact space) because it plays an important role in analysis and prepares for the study of distribution theory. Starting off with the notion of Daniell measure, Mr. Simonnet provides a unified treatment of these two approaches.

## Inhaltsverzeichnis

### 1. Riesz Spaces

Summary
1.1 This first section introduces the notion of ordered groups and lattices. It might be useful to keep in mind some well known examples, such as Z or R or the space C([a, b]) of real-valued continuous functions on [a, b] with their natural orderings.
1.2 It is devoted to a short study of Riesz spaces, that is, ordered vector spaces which are lattices. The space X of real-valued continuous functions on [a, b], endowed with its usual ordering, is an example of Riesz space.
1.3 In this section we turn our attention to the duals of Riesz spaces. If E is any ordered vector space, there is a natural notion of positive linear form: a linear form I is said to be positive if the image of any positive element of E under I is a positive real number. Notice that if E is the space of real-valued continuous functions on a closed interval [a, b], equipped with its natural ordering, a positive linear form on E is continuous.
1.4 We finally define Daniell measures. A Daniell measure is a linear form L on a space of functions H which has a “finite variation” (Definition 1.4.1) and such that L(f n ) → 0 whenever f n decreases to 0 in H. This last condition should be understood as a continuity condition. Among the examples that the reader may want to keep in mind axe the Riemann integral considered as a positive linear form on the set of continuous functions on [a, b] and also the positive linear forms ε c : Xff(c) where c ∈ [a, b].
Michel Simonnet

### 2. Measures on Semirings

Summary
2.1 Given a nonempty set Ω, the power set of Ω, equipped with symmetric difference and intersection, is a ring. A nonempty subring is called a ring. A σ-ring is a ring which is stable under countable unions. The subset S of semiclosed subintervals [a, b] is not a ring but merely a semiring (Definition 2.1.2). The σ-ring of Halmos sets and the σ-algebra of Borel sets (Definition 2.1.6), in a topological space Ω, are among the most important examples of σ-rings and algebras and will be widely used throughout this book.
2.2 In this section we first define quasi-measures, then measures on semirings. A function μ on a semiring S is a measure if and only if, for any sequence of disjoints sets A i S such that ⋃ A i ⊂ A ∈ S, the series $$\sum {\mu \left( {{A_i}} \right)}$$ converges, and if $$\mu (A) = \sum {\mu \left( {{A_i}} \right)}$$ whenever A = ⋃ A i (Theorem 2.2.3). Any complex measure on a σ-ring is bounded (Proposition 2.2.3). Theorem 2.2.5 gives a basic relationship between abstract measures defined on a semiring S and Daniell measures defined on S-simple functions (“step functions”).
2.3 We introduce Lebesgue measure as a measure defined on the semiring of all semiclosed intervals [α, β] included in some interval I.
Michel Simonnet

### 3. Integrable and Measurable Functions

Summary
3.1 Given a Daniell measure µ, the upper integral of a positive function with respect to Vµ is defined. A positive function is said to be µ-negligible if its upper integral is null. This allows us to define negligible sets and the notion of property true “almost everywhere”. We then prove a few important results such as Beppo Levi’s theorem (Theorem 3.1.1), Fatou’s lemma (Proposition 3.1.2), and the Riesz-Fischer theorem (Theorem 3.1.3) on the completeness of ℒ1 (µ).
3.2 One of the main limitations of Riemann’s integral lies in the fact that it does not yield significant results when it comes to the integral of sequences or series of functions. For example, if a uniformly bounded sequence of Riemann integrable functions converges pointwise on [a, b] to a (necessarily bounded) function f, then f may not be Riemann integrable. The monotone convergence theorem (Theorem 3.2.1), Fatou’s theorem (Proposition 3.2.1), and the Lebesgue dominated convergence theorem (Theorem 3.2.2) will answer some of our questions. For a first lecture, the reader may, without loosing too much, translate “filters” into “sequences”. The next two results, continuity and differentiability of an integral with respect to a parameter (Proposition 3.2.3 and Theorem 3.2.3), are of constant use in analysis.
3.3 In this short section we focus our attention to sets rather than functions and define µ-integrable sets and µ-moderate sets.
3.4 This section is devoted to the definition of σ-measurable spaces, that is, sets endowed with a σ -ring, and the notion of measurable mapping between two such spaces. It is important to notice that this notion of measurability does not depend on a measure.
3.5 The main result of this section is the following form of Egorov’s theorem (Theorem 3.5.1): a sequence of µ-measurable mappings from Ω into a metrizable space which converges locally almost everywhere to f converges uniformly to f on the complement of an arbitrarily small integrable set. Finally, we prove that a µ-measurable function from Ω into a Banach space is integrable if and only if its upper integral with respect to Vµ is finite (Theorem 3.5.3).
3.6 The essential integral of functions is defined, as well as bounded measures.
3.7 First, we define the upper and lower integral of a positive function: f is integrable if and only if its upper and lower integrals are finite and equal. Notice that if f is integrable so is |f|: in this theory, there is no “improper integral”. We then prove Jensen’s inequality (Theorem 3.7.3) which is an important tool both in the theory of probability and in analysis.
3.8 Intuitively, an atom for a measure µ is a µ-integrable set which has no smaller proper subset (smaller and proper both in the sense of measure theory). A measure without atom is said to be diffuse; Lebesgue measure is an example of such a measure. On the other hand, a measure is said to be atomic if each nonnegligible integrable set contains an atom. The counting measure on the semiring of finite subsets of N is an example (cf. Section 6.3). If µ is atomic and f is a function from Ω into a metrizable space, f is measurable if and only if it is constant a.e. on each atom (Theorem 3.8.1).
3.9 A Daniell measure µ defines a measure µ̂ on the ring of integrable sets, called the main prolongation of µ. Similarly, µ defines a measure µ̄, called the essential prolongation of µ, on the ring R̄ of essentially integrable sets. We then study various relationships between these measures.
Michel Simonnet

### 4. Lebesgue Measure on R

Summary
4.1 This is a short review of base-b expansions of a real number. These expansions will be often used throughout the text and the exercises.
4.2 We now define the famous Cantor singular function: continuous, increasing, but its derivative is 0 almost everywhere (for Lebesgue measure).
4.3 We prove the existence of “pathological” sets. In particular, using Cantor singular function we prove the existence of a non-Borelian negligible set.
Michel Simonnet

### 5. L p Spaces

Summary
5.1 In this section we prove several fundamental inequalities. For example, if p, q, and r with 1/p + 1/q = 1/r belong to [0, +∞], and if f, g are two functions on Ω such that N p (f) and N q (g) are finite, then N r (fg)N p (f)N q (g) (Proposition 5.1.2). Theorem 1.1 is a generalization of Minkowski’s inequality: if f and g are two functions on Ω and p ≥ 1, then N p (f + g)N p (f) + N p (g).
5.2 We now generalize to Lp(µ) some of the convergence theorems established in Section 3.2, for example the dominated convergence theorem, and prove the Fischer-Riesz theorem (Proposition 5.2.3). Next, we analyze the duality of L q and L q when p and q are conjugate, that is, when 1/p + 1/q = 1. If F is a Banach space, and p and q are conjugate, then, for every g ∈ L F q (µ), f ↦ ∫ fgdµ is a continuous linear functional on L F p (µ) with norm N q (g) (Theorem 5.2.5). The converse will be dealt with in Chapter 10.
5.3 The notion of convergence in measure is introduced. This section requires some knowledge of uniform spaces.
5.4 This section, which may be omitted, deals with uniformly integrable sets.
Michel Simonnet

### 6. Integrable Functions for Measures on Semirings

Summary
6.1 Let µ be a measure on S semiring in Ω. A ⊂ Ω H is µ-measurable if and only if, for every ES, the set AE is integrable (Proposition 6.1.3). A function from Ω, into some metrizable space is µ-measurable if and only if, for every ES, the restriction of f to E is the limit a.e. of simple mappings (Proposition 6.1.4). Finally, we give two other important necessary and sufficient conditions for f to be µ-measurable.
6.2 Let F be a Banach space with dual F′. If f ∈ ℒ F 1 (µ), then N1(f) = supg∈B |∫ fgdµ| where B is the closed unit ball of St(S, F′) (Theorem 6.2.1).
6.3 This section generalizes the following example. Let S be the semiring of finite subsets of N. Then E ↦ card E, where card E is the cardinality of E, is a measure on S, called the counting measure. It is defined by the unit mass at each point. Such a measure is atomic (cf. Section 3.8).
6.4 In this section we return to the study of prolongations of a measure (cf. Section 3.9).
Michel Simonnet

Summary
7.1 For the convenience of the reader, we review some properties of locally compact HausdorfF spaces, such as Urysohn’s Lemma.
7.2 Let X be a locally compact Hausdorff space. A linear form μ on the space of complex-valued continuous functions with compact support H is a Radon measure if, for every compact KX, the restriction of μ to the space of continuous functions with support in K is continuous. Every positive linear form on H is a Radon measure (Theorem 7.2.1) and every Radon measure is a Daniell measure. Then we define the upper integral with respect to a Radon measure and give results analogous to those obtained in Chapter 3.
7.3 The main result of this section (Theorem 7.3.2) is the following. The upper envelope f of an upwardly directed set, H, of integrable, lower semicontinuous functions such that supg∈H∫gdVμ< ∞ is itself integrable and g converges to f in the mean along the filter of sections of H. Also, a Radon measure is regular in the following sense: if A is measurable and moderate, then, given ε > 0, there is an open set U and a countable union of compact sets F such that Vμ* (U — F) ≤ ε.
7.4 In this section we introduce the notion of Lusin measurable mappings. Intuitively these functions are “almost continuous” on any compact set. A function from X into a metrizable space is Lusin measurable if and only if it is measurable.
7.5 Let xoX. The measure ε x o, defined by ff(xo) for f continuous with compact support, shows that a Radon measure may have some atoms. However, any atom is essentially a point (Proposition 7.5.1). Counting measures provide us with examples of atomic Radon measures.
7.6 One of the goals of measure theory was to improve the Riemann integral. The reader might be happy to know that what we have done so far has substantially improved Riemann’s integral. In this section we prove, in particular, that a bounded function with compact support is Riemann integrable if and only if its set of discontinuity is Lebesgue negligible.
7.7 Various types of convergence, such as vague, weak, and narrow convergence in spaces of measures are studied.
7.8 We continue our investigation of convergence in spaces of measures by looking at tight subsets of M1. Tightness may be viewed as a relative compactness condition.
Michel Simonnet

### 8. Regularity

Summary
8.1 A measure defined on a semiring Φ is said to be strictly regular if its main prolongation is a prolongation of a Radon measure. If each open subset of Ω is a countable union of compact sets, if Φ̃ is the Borel σ-algebra, and if each compact set is μ-integrable, then μ is strictly regular (Theorem 8.1.2).
Michel Simonnet

### 9. Induced Measures and Product Measures

Summary
9.1 Let µ be a measure on a semiring S whose underlying set is X. Let Y be a µ-measurable subset of X. Let T be a semiring whose underlying set is Y. Assume that µ, Y, and T are endowed with adequate properties. We define the measure µ/ T induced by µ on T and see how to integrate with respect to this induced measure (Theorem 9.1.1).
9.2 Let µ′, µ′′ be two measures on semirings S′, S′′ (with Ω′, Ω′′ as their underlying sets, respectively). The measure µ: A′ × A′′ ↦ µ′(A′) µ′′ (A′′) on the semiring S = {A′ × A′′: A′ ∈ S′, A′′ ∈ S′′} (whose underlying set is Ω = Ω′ × Ω′′) is called the product of µ′ and µ′′.V(µ′ ⊗ µ′′) = Vµ′ ⊗ Vµ′′ (Theorem 9.2.1). If f: Ω ↦ [0, +∞] is µ-measurable and µ-moderate, then ∫* f dVµ = ∫* dVµ′(x′) ∫* f(x′,x′′) dVµ′′(x′′) can be computed by means of iterated integrals (Theorem 9.2.5). Likewise, we can compute ∫ f dµ for every µ-integrable mapping from Ω into a Banach space (Theorem 9.2.4, Fubini’s).
9.3 We define Lebesgue measure on an open subset Ω of R k (with k ≥ 1).
Michel Simonnet

Summary
10.1 We define summable families of measures on a same semiring S.
10.2 Let µ be a measure on a semiring S whose underlying set is Ω. A function g: Ω → C is said to be locally µ-integrable whenever gl A a is µ-integrable for every A in S; then the measure gµ: A → ∫ gl A is called the measure with density g relative to µ. V(gµ) = |g|Vµ, (Proposition 10.2.1), and ∫ f dV(gµ) = ∫ f |g| dVµ for all functions f: Ω → [0, +∞] (Theorem 10.2.2). A mapping f from Ω into a Banach space is gµ-measurable (respectively, essentially gµ-integrable) if and only if fg is µ-measurable (respectively, essentially µ-integrable) (Theorem 10.2.1).
When µ is Lebesgue measure on an interval, let us observe that every function continuous on that interval is locally µ-integrable.
10.3 Let µ be a measure on a semiring S. Let ℛ be the ring generated by S. A measure v on S is said to be absolutely continuous with respect to µ if every µ-negligible σ(S)-set is v-negligible. Another equivalent condition is the following condition. For every E in S and for every ɛ > 0, there exists δ > 0 such that for all F in ℛ contained in E and satisfying the inequality |µ|(F) ≤ δ, we have |v|(F) ≤ ɛ (Theorem 10.3.2). This also amounts to saying that v has a density g with respect to µ (Theorem 10.3.1, Radon-Nikodym). Measures µ and v on S are said to be mutually singular whenever inf (|µ|, |v|) = 0. This means that µ and v are concentrated on disjoint sets (Propositions 10.3.3 and 10.3.4). Every measure on S can be written, uniquely, as the sum of a measure gµ absolutely continuous with respect to µ and a measure v such that µ and v are mutually singular (Theorem 10.3.3).
10.4 We combine different operations on the measures.
10.5 We show that L C q (µ) may be regarded as the dual of L C q (µ) (with 1 ≤ p < +∞, q exponent conjugate to p) (Theorems 10.5.1 and 10.5.2). We also characterize those continuous linear functionals on L C (µ) that can be written f ↦ ∫fg dµ for g in L C 1 (µ) (Proposition 10.5.1).
10.6 We describe the dual of L C (µ) (Proposition 10.6.1). A necessary and sufficient condition that this dual be equal to L C 1 (µ) is that Ω be a union of a finite number of atoms and a locally µ-negligible set (Proposition 10.6.2).
Michel Simonnet

### 11. Images of Measures

Summary
11.1 Given a measure μ on S in Ω and a mapping π from Ω into Ω′, we can try to define a measure on a semiring S′ in Ω′ by μ′(A) = μ(π−1(A)). This will define a measure if certain conditions are satisfied, in which case we say that the pair (π, S′) is μ-suited. Then ∫ f dμ′ = ∫(f ○ π)dμ (Theorem 11.1.1).
11.2 In this section, we define compact classes, projective systems of measures, and prove Kolmogorov’s theorem, which gives a sufficient condition for a projective system of measures to define a measure called the projective limit of the system (Theorem 11.2.1). In particular, if for each iI μ i is a positive measure with total mass 1 on S i , and if μ J = ⊗ J μ i for all finite subsets JI, then {μ J } has a projective limit (Theorem 11.2.2).
11.3 We apply the notion of image of a measure to Lebesgue measure on R.
11.4 This is a short introduction to ergodic theory. The main result of this section is Birkhoff’s ergodic theorem: If f: Ω → R is essentially μ-integrable and f k = fu k (where u: Ω → Ω satisfies u(μ) = μ), then (1/n) $$\sum\nolimits_{0 \leqslant k \leqslant n - 1} {{f_k}}$$ converges to some essentially μ-integrable function f* locally μ-a.e. and f* = f*u locally μ-a.e.
Michel Simonnet

### 12. Change of Variables

Summary
12.1 In this section we define derivatives Dμ, Dμ, and D̄μ of a measure μ with respect to Lebesgue measure in V. Next, we analyze the relationships between the Lebesgue decomposition of μ and the properties of its derivatives; for example, if D̄μ is finite everywhere, μ is absolutely continuous with respect to Lebesgue measure.
12.2 First, we study the image of λ under a linear automorphism. The modulus of an automorphism of R k is the absolute value of its determinant (Theorem 12.2.1).
12.3 In this section we prove the change of variables formula: if T is a differentiable homeomorphism from V onto W, then λ W is the image of |J(T)|λ V under T where J(T) is the Jacobian of T (Theorem 12.3.1).
12.4 This section is devoted to polar coordinates in Rn.
Michel Simonnet

### 13. Stieltjes Integral

Summary
13.1 Let I be an interval in R and f a function from I into a metric space (E, d). If J is a subinterval of I, f/J is said to be of finite variation if the set $$\left\{ {\sum\nolimits_\Delta {d\left( {f\left( {{a_{i - 1}}} \right),f\left( {{a_i}} \right)} \right)} } \right\}$$, where ∆ runs over the set of finite partitions of J, is bounded. When E = R, the variation of f is locally bounded if and only if f is the difference of two increasing functions (Proposition 13.1.3).
13.2 Denote by S the natural semiring of I subinterval of R. If μ is any measure on S, every indefinite integral of μ (see Definition 13.2.1) is a function of locally bounded variation (Theorem 13.2.1). The formula for integration by parts holds true for functions of locally bounded variation with no common point of discontinuity (Theorem 13.2.3).
13.3 In this section, which may be omitted, we define line integrals and prove some of their properties.
13.4 We prove the Lebesgue decomposition theorem: If M is a function of locally bounded variation, then M = M1 + M2, where M1 is a singular function and M2 is locally absolutely continuous (Theorem 13.4.4). Also, if M is increasing, then M is differentiable almost everywhere and its derivative is locally integrable (Theorem 13.4.5).
13.5 In this section, which may be omitted. We study the upper and lower derivatives of a function.
Michel Simonnet

### 14. The Fourier Transform in R k

Summary
14.1 In this section, we give some necessary and sufficient conditions for a quasi-measure, defined on the natural semiring of a nonempty open subset of R k , to be a measure (Propositions 14.1.1 to 14.1.3).
14.2 In R k , the distribution function of a probability defined on the Borel σ-algebra is $$F:\left({{x_1}, \cdots {x_k}} \right) \mapsto P\left({\prod\nolimits_{1 \leqslant i \leqslant k} {\left] { - \infty, {x_i}} \right]}} \right)$$. A sequence of probabilities P n with distribution functions F n converges weakly to a probability P with distribution function F if and only if F n (x) converges to F(x) at every point of continuity of F.
14.3 In this section, we define the covariance matrix of a positive measure of order 2 and the normal distribution N(m, σ2).
14.4 Let μ be a bounded measure on the Borel σ-algebra of R k . The Fourier transform of μ is the function ℱμ: x ↦ ∫exp(−2πixy)dμ(y). The function ℱμ is uniformly continuous and bounded (Proposition 14.4.1). Moreover, the Fourier transform is injective (Theorem 14.4.1). We next prove the Riemann—Lebesgue lemma: the Fourier transform of an integrable function vanishes at infinity. Finally, we prove the inversion theorem for bounded measures and two convergence theorems of fundamental importance in the theory of probability.
14.5 This section is devoted to a short study of the normal laws in R k .
Michel Simonnet

### 15. The Strong Law of Large Numbers

Summary
15.1 This section gives some fundamental definitions in the theory of probability, such as the definitions of a probability space and a random variable.
15.2 In this section the fundamental concept of independence is developed.
15.3 We give an example of singular function, which arises naturally from probability theory.
15.4 Given Ω, set Ω n = Ω for all n ∈ N and Ω N = Π n Ω n . The one-sided shift transformation v on ΩN is defined by (x n ) n ≥1 ↦ (x n +i) n ≥i. The probability of a v-invariant event is either 0 or 1 (Proposition 15.4.2) and v is μ-ergodic (Proposition 15.4.3)—see Section 11.4. Then, using Birkhoff’s ergodic theorem, we prove the strong law of large numbers (Theorem 15.4.1).
15.5 A number xm/b n is said to be completely normal if, for every integer k and every k-tuple (u1,…, u k ) of base-b digits, the k-tuple appears in the base-b expansion of x with asymptotic relative frequency 1/b k . Almost every number (for Lebesgue measure) in [0, 1] is completely normal (Proposition 15.5.2).
Michel Simonnet

### 16. The Central Limit Theorem

Summary
16.1 We study some properties of the convergence in law of random variables.
16.2 Let $${\left( {{X_{n,k}}} \right)_{\begin{array}{*{20}{c}} {n \geqslant 1} \\ {1 \leqslant k \leqslant {r_n}} \\ \end{array}}}$$ be a triangular array of independent, centered, and square-integrable random variables. For every n ≥ 1, write $$s_n = \left[ {\sum {_{1 \le k \le r_n } {\rm{ }}Var\left( {X_{n,k} } \right)} } \right]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}$$ and $${s_n} = \sum\nolimits_{1 \leqslant k \leqslant {r_n}} {{X_{n,k}}}$$. The Lindeberg condition is sufficient for S n /s n to converge in law to the normal law (Theorem 16.2.1). Note, incidentally, that this condition is also necessary in the most usual cases (as a consequence of a Feller’s theorem, which is not proved here).
16.3 We prove the central limit theorem (Theorem 16.3.1), as well as some refinements of this theorem.
Michel Simonnet

### 17. Order Statistics

Summary
17.1 Given an n-sample, for almost every ω ∈ Ω, there is a permutation σ = σ ω of {1, …, n} such that Xσ(1) < … < Xσ(n). The coordinates of the random variable defined almost everywhere by X(ω) = (Xσω(1)(ω),…, Xσω(n)(ω)) are called the order statistics of the sample. We compute the distributions of Xσ(k) and X.
17.2 We introduce the notion of median of a sample and give some more convergence theorems.
Michel Simonnet

### 18. Conditional Probability

Summary
18.1 Let (Ω, ℱ, P) be a probability space and let $$\mathcal{D}$$ be a sub-σ-algebra of ℱ. For any random variable Y from (Ω, ℱ) into R k , there exists a unique (up to a.e. equality) random variable Z from (Ω, $$\mathcal{D}$$) into R k such that ∫ A ZdP=∫ A YdP for all A in $$\mathcal{D}$$. Z is called the conditional expected value of Y given 풟. If Y is of order 2, Z is the orthogonal projection of Y on a closed subspace of L2(Ω, P; R k ) (Theorem 18.1.2).
18.2 In this section we prove a measure theoretic converse of the mean value theorem.
18.3 Here, we generalize Jensen’s inequality to conditional expected values: if φ is convex and Y is P-integrable, φ∘ E(Y|풟)≤E(φ(Y) $$\mathcal{D}$$) (Proposition 18.3.1).
18.4 We define the conditional expected value of Y given the random variable X. When the law of X is absolutely continuous with respect to Lebesgue measure, we can compute E(Y|X) (as a limit) outside some P X -negligible set (Proposition 18.4.1).
18.5 This section is devoted to the study of the conditional law of Y given X.
18.6 We compute some conditional laws when P X is defined by masses and when P[X,Y] is absolutely continuous with respect to a product μ⨂ν.
18.7 We prove the existence of conditional laws when Y is an R k valued random variable (Theorem 18.7.1).
Michel Simonnet

### 19. μ-Adequate Family of Measures

Summary
19.1 For Radon measures, the notion of induced measure is quite natural. If X is a locally compact Hausdorff space and Y a locally compact subspace, any Radon measure on X induces a Radon measure on y as follows. If f is continuous on Y with compact support, define g by g(x) = f(x) when xY and g(x) = 0 otherwise. Then f ↦ ∫gdμ, is the measure induced by μ on Y.
19.2 μ-dense families of compact sets, introduced in this section, will be needed later in the text.
19.3 Given a positive Radon measure μ and a μ-dense class D of compact sets, there is a summable family (μα)α∈A of measures such that $$\mu = \sum\nolimits_{\alpha \in A} {{\mu_\alpha}}$$ that supp(μα) belongs to D, and that the sets supp(μα) form a locally countable class (Theorem 19.3.1).
19.4 In this section, we study integration with respect to ∫λ t dμ(t), where t↦λ t At is a μ-adequate family of measures. We prove a result which is similar to Fubini’s theorem (Theorem 19.4.2).
19.5 We specialize the results of Section 19.4 to μ-adapted pairs.
Michel Simonnet

### 20. Radon Measures Defined by Densities

Summary
20.1 Let μ be a Radon measure on a locally compact space X. Let Y be a locally compact subspace of X. Integration with respect to the induced measure μ Y is done in the most natural way (Theorem 20.1.1). Piecing together Radon measures, as done in Theorem 20.1.2, is extremely useful in analysis and differential geometry.
20.2 Let μ be a Radon measure on a locally compact space T. A function g from T into C is said to be locally integrable when gh is μ-integrable for every continuous function h with compact support. In this case gμ: h↦∫ghdμ is a Radon measure. We study how to integrate with respect to gμ (Theorems 20.2.1 and 20.2.2).
20.3 The Radon-Nikodym theorem and Lebesgue’s decomposition theorem still hold for Radon measures.
20.4 From Chapter 10 follow some results on the duality of L C p (μ) spaces, when μ is a Radon measure.
Michel Simonnet

### 21. Images of Radon Measures and Product Measures

Summary
21.1 Let T, X be two locally compact spaces, μ a positive Radon measure on T, and π a µ-measurable mapping from T into X such that g o π is essentially μ-integrable for every g ∈ ℋ(X, C). Denote by ν the image measure, f ↦ ∫(f o π) dμ, of μ under π. Then ∫ f dv = ∫(f o π) dμ for every function fX → [0,+∞]. A mapping f from X into a topological space is ν-measurable when f o π is μ-measurable (Theorem 21.1.1).
21.2 We study the decomposition of a measure in slices. This decomposition will be used in Chapter 23.
21.3 We define the product of two Radon measures.
Michel Simonnet

### 22. Operations on Regular Measures

Summary
22.1 We have seen in Chapter 8 that we can associate a regular measure μ with a Radon measure λ, that is, a continuous linear functional on a space of continuous functions with compact support. We describe herein the functionals associated with the induced measures μ Y , the measures gμ that have a density with respect to μ, and the image measures π(μ).
22.2 We define the σ-ring of Baire sets in a locally compact space.
22.3 We use the results of Section 22.2 to show that the product, μ, of two regular measures, μ′ and μ″, is also a regular measure, provided that the compact sets of the product space are μ-immeasurable (Theorem 22.3.1).
22.4 Section 22.1 allows us to complete the results obtained in Chapter 11 on the change of variable formula.
Michel Simonnet

### 23. Haar Measures

Summary
23.1 In this section we define invariant, relatively invariant, and quasi-invariant measures on a locally compact group.
23.2 On every locally compact group there exists a left-invariant measure. This measure is unique, up to a multiplicative constant (Theorem 23.2.1).
23.3 We define the modular function on a locally compact group.
23.4 Quasi-invariant measures on a locally compact group are all equivalent (Proposition 23.4.3).
23.5 Let G be a locally compact group and H a closed subgroup of G. With every measure λ on the homogeneous space G/H we can associate a measure λ# on G (Theorem 23.5.2) that possesses interesting properties.
23.6 We study integration with respect to λ# (Theorem 23.6.1 and Proposition 23.6.1).
23.7 We reconstitute λ from λ# (Proposition 23.7.4).
23.8 There is only one class of quasi-invariant measures on G/H (Theorems 23.8.1 and 23.8.2).
23.9 In contrast, invariant measures exist on G/H only if the modular functions Δ G and Δ H of G and H coincide on H (Theorem 23.9.2).
23.10 By means of Euler angles we describe the invariant measure on the group SO(n + 1, R) of rotations of R n +1 (Proposition 23.10.3).
23.11 We describe Haar measure on the group SH(n + 1, R) of hyperbolic rotations of R n +1.
In this chapter and the following one, all locally compact spaces will be taken to be Hausdorff, unless otherwise stated.
Michel Simonnet

### 24. Convolution of Measures

Summary
24.1 We define convolution of measures. Two bounded measures are always convolvable (Proposition 24.1.2).
24.2 We supply some necessary and sufficient conditions for a measure and a function to be convolvable (Propositions 24.2.2 and 24.2.3).
24.3 If μ is a measure on G (respectively, a measure with compact support) and if f is a continuous function on G with compact support (respectively, a continuous function), then μ and f are convolvable.
24.4 For every bounded measure μ on G and for every function f in ℒp̄(G) (with 1 ≤ p ≤ +∞ and f are convolvable (Theorems 24.4.1 and 24.4.2).
24.5 For 1 ≤ p ≤ +∞, the endomorphism g ↦ μ̆ * g of A C P (G) is the transpose of f ↦ μ * f (Proposition 24.5.2).
24.6 We study convolution of functions on G.
24.7 We enunciate a few results on regularization of functions (Theorems 24.7.1 and 24.7.2).
24.8 We define Gelfand pairs (important in harmonic analysis). We show that (SO(n + 1, R), SO(n, R)) and (SH(n + 1, R), SO(n, R)) are Gelfand pairs.
In this chapter, all locally compact spaces are supposed to be Hausdorff.
Michel Simonnet

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