Teoremas ergódicos dos pontos de vista probabilístico e topológico

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(1)Teoremas ergódicos dos pontos de vista probabilístico e topológico Lucas Amorim Vilas Boas. Dissertação apresentada ao Instituto de Matemática e Estatística da Universidade de São Paulo para obtenção do título de Mestre em Ciências Programa: Matemática Aplicada Orientador: Prof. Dr. Fabio Armando Tal. Durante o desenvolvimento deste trabalho o autor recebeu auxílio nanceiro da CNPq. São Paulo, Junho de 2018.

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(3) Ergodic theorems from probabilistic and topological viewpoints Lucas Amorim Vilas Boas. Dissertation presented to the Institute of Mathematics and Statistics of the University of São Paulo to fulfill the requirements of the degree of Master in Science Program: Applied Mathematics Advisor: Prof. Dr. Fabio Armando Tal. During the elaboration of this work the author had the nancial support of CNPq. São Paulo, June 2018.

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(5) Teoremas ergódicos dos pontos de vista probabilístico e topológico. Esta versão da dissertação contém as correções e alterações sugeridas pela Comissão Julgadora durante a defesa da versão original do trabalho, realizada em 17/05/2018. Uma cópia da versão original está disponível no Instituto de Matemática e Estatística da Universidade de São Paulo.. Comissão Julgadora:. . Prof. Dr. Fabio Armando Tal (orientador) - IME-USP. . Prof. Dr. Alejandro Kocsard - IME-UFF. . Prof. Dr. Ricardo dos Santos Freire Junior - IME-USP.

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(7) Ergodic theorems from probabilistic and topological viewpoints. This version of the dissertation incorporates improvements suggested by the Examining Board in the event of the defense of the original version of this work, in 05/17/2018. A copy of the original version is available at the Institute of Mathematics and Statistics of the University of São Paulo.. Examining Board:. . Prof. Fabio Armando Tal, PhD (advisor) - IME-USP. . Prof. Alejandro Kocsard, PhD - IME-UFF. . Prof. Ricardo dos Santos Freire Junior, PhD - IME-USP.

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(9) Acknowledgments First of all, I would like to thank my advisor Prof. Fabio Armando Tal, for this opportunity and for his kindness, guidance and shared knowledge. I would also like to thank Prof. Jorge Sotomayor and Prof. Salvador Zanata, who also introduced me to the quest of dynamical systems. In fact, the structural role of the Brazilian dynamical systems community is to be acknowledged. Nonetheless, I would like to thank the support of my family and friends and the lasting care and patience of my girlfriend Brenda Poubel during this journey.. i.

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(11) Resumo AMORIM, L. V. B.. Teoremas ergódicos dos pontos de vista probabilístico e topológico.. 2018. 87 f. Dissertação (Mestrado) - Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, 2018.. Teoremas ergódicos são resultados medida-teóricos clássicos em sistemas dinâmicos ou, mais precisamente, teoria ergódica. Eles armam que a convergência das médias de Birkho é típica, em um sentido de medida. Este trabalho objetiva explicar como esses resultados podem ser reinterpretados à luz da topologia e da teoria das probabilidades. A primeira relação é apresentada por meio de um análogo em categoria de Baire de uma versão habitual do teorema ergódico de Birkho (assumindo ergodicidade). Ao invés de convergência das médias de Birkho, o comportamento topologicamente típico será oposto: as médias não convergirão de modo dramático. A segunda relação é apresentada examinando como a lei dos grandes números interage com o teorema ergódico de Birkho (assumindo ergodicidade). A lei dos grandes números pode ser obtida como corolário do teorema ergódico de Birkho. Entretanto, ela permite um novo ponto de vista, pois mantém as conclusões do teorema ergódico de Birkho (assumindo ergodicidade) mesmo no caso não-ergódico, ao custo de que haja um certo tipo de independência.. Palavras-chave: teorema ergódico, análogo topológico, categoria de Baire, lei dos grandes números.. iii.

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(13) Abstract AMORIM, L. V. B.. Ergodic theorems from probabilistic and topological viewpoints. 2018.. 87 p. Dissertation (Master) - Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, 2018.. Ergodic theorems are classic measure theoretical results in dynamical systems or, more precisely, ergodic theory. They state that the convergence of Birkho averages is typical, in a measuretheoretical sense. This work aims to explain how these results can be re-ìnterpreted in light of topology and probability theory. The rst relationship is presented through a Baire category analogue of a standard version of Birkho 's ergodic (assuming ergodicity). Instead of convergence of Birkho averages, the topological typical behavior will be the opposite: averages do not converge in a dramatic way. The second relationship is presented by examining how the law of large numbers interacts with Birkho 's ergodic theorem (assuming ergodicity). The law of large numbers can be obtained as corollary of Birkho 's ergodic theorem. However, the law provides a new point of view, as it guarantees the conclusions of Birkho 's ergodic theorem (assuming ergodicity) will hold even in the non-ergodic case, at the cost of requiring some sort of independence.. Keywords: ergodic theorem, topological analogue, Baire category, law of large numbers.. v.

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(15) Contents List of abbreviations. ix. List of symbols. xi. 1 Introduction. 1. 2 Prerequisites. 3. 2.1. Measure Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.2. Functional analysis and the space of measures. . . . . . . . . . . . . . . . . . . . . . . .. 20. 2.3. Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 24. 2.4. Dynamical Systems and Ergodic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . .. 33. 2.5. Topological Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 38. 3 Birkho's Ergodic Theorems: Measure Theoretical and Topological Analogues. 3. 45. 3.1. Birkho Ergodic Theorem (Measure Theoretical) . . . . . . . . . . . . . . . . . . . . . .. 45. 3.2. Birkho Ergodic Theorem Topological Analogue . . . . . . . . . . . . . . . . . . . . . .. 61. 4 Law of Large Numbers: From Probability to Ergodic Theory 4.1. Unifying Notation and Framework. 4.2. Law of Large Numbers. 69. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 69. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 74. 5 Final Remarks. 81. 5.1. Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 81. 5.2. Future studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 82. Bibliography. 83. vii.

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(17) List of abbreviations LCH. Locally compact Hausdor space. SM. Separable metric space. LCH+SM. Locally compact Hausdor separable metric space. BET. Birkho 's ergodic theorem. IID. Independent and identically distributed. LLN. Law of large numbers. a.e.. Almost everywhere. a.a.. Almost all. ix.

(18) x. LIST OF ABBREVIATIONS.

(19) List of symbols The set of non-negative natural numbers. N ∪˙ X. {0, 1, . . .}. Union of disjoint sets. n. XN X∞. = ∏nj=0 X = {s ∶ {0, . . . , n} → X}, n ∈ N. = ∏j∈N X = ∏∞ j=0 X = {s ∶ N → X} N = X , this notation will be used when. referring to. xi. X n , n ∈ N,. and. XN. in a row:. X e , e ∈ N ∪ {∞}.

(20) xii. LIST OF SYMBOLS.

(21) Chapter 1. Introduction Interactions between measurable and topological dynamics, as well as between probability theory and ergodic theory, are known to be fruitful. Many theorems in ergodic theory can be enriched by tracing analogues to topological dynamics. A paradigmatic case is an analogy between Poincare's recurrence and Birkho 's topological recurrence. On the other hand, many denitions and results in ergodic theory are very naturally understood in probabilistic terms, due to the very nature of objects making up ergodic theory. The idea is to strengthen these connections, making them work for ergodic theorems. For our concerns, Birkho 's ergodic theorem is the major one. Later, this theorem will be presented indepth. For now, it can be summarized as stating measure-theoretical typical convergence of the so-called Birkho averages. One objective of this dissertation is to present a topological (or a topological dynamics) analogue to Birkho 's ergodic theorem. The major issue is that measure-theoretical and topological structures do not always translate perfectly. The kind of topological dynamics in which we are looking for analogues is the abstract one, whose topological structure and dynamical system acting on it are quite general. Distributional properties of orbits in those systems, to be understood roughly as properties of the set of measures accumulated by averages of Diracs supported along orbits, will play a key role. These properties are to be understood as topological in the sense that they do not require invariant measures. Also, we will often try to verify whether those and other properties hold typically in the sense of Baire category. Each of these subjects will be addressed during this text. See sections 2.5 and 3.2. Another objective of this dissertation is to present a probability theory analogue to Birkho 's ergodic theorem. In this case, the analogue is well-known, namely, the law of large numbers. Our aim is to scrutinize how these theorems relate. The major issue is that, in probability theory, no system exists at all. It is necessary to create a conceptual bridge between ergodic and probability theories. Our inclination will be that of a dynamicist, i.e., using the bridge to adapt content from probability theory to ergodic theory. In particular, the law of large number will be adapted and this adaptation compared to Birkho 's ergodic theorem. To create this bridge, a lot is discussed in terms of approaching a dynamical system as a stochastic process and, conversely, approaching a stochastic process as modeled by a dynamical system. Another central feature is to interpret independence (and identical distribution) in the context of dynamics. Again, each of these subjects will be addressed during this text. See sections 2.3 and 4.1. To give a glimpse of what is coming, the topological analogue of Birkho 's ergodic theorem (assuming ergodicity), to be presented later in the text, shows that the typical behavior of Birkho averages is, perhaps surprisingly, a dramatic non-convergence. This discussion is carried in section 3.2 and is due to [Win10]. Also, we show that the law of large numbers follows from Birkho 's ergodic theorem. However, the law provides a new point of view, as it guarantees the conclusions of Birkho 's ergodic theorem. 1.

(22) 2. INTRODUCTION. 1.0. (assuming ergodicity) will hold even in the non-ergodic case, at the cost of requiring some sort of independence. This discussion is carried in section 4.2 and its proof relies heavily on the previous conceptual bridge, both of which are presented by the author in an original way. Even though no reference is used, these subjects some how standard implicit knowledge for the mathematician working on this interface..

(23) Chapter 2. Prerequisites In this chapter, during sections 2.1 and 2.2, it is assumed the reader is familiar with measure theory and functional analysis as in [Fol07]. During section 2.3, it is assumed the reader is familiar with probability theory as in the rst three chapters of [Çn11]. During section 2.4, it is assumed the reader is familiar with dynamical systems and ergodic theory as in the rst ve chapters of [VO16]. Unless locally mentioned, these texts are references for each respective section. The last section, 2.5, about topological dynamical systems, is a collection of results from many sources, according to the taste of the author. Those proofs are quite standard, but some of them, about topological dynamics per se , were adapted. The exposition in each section summarizes many aspects of the texts mentioned above but do not prove every proposition in detail, especially in sections 2.1, 2.2 and 2.3, where technical proofs are omitted and only important proofs are sketched or indicated. However, proofs start to show up complete in sections 2.4 and 2.5, as we move into the core subject of this dissertation. The experienced reader can skip this chapter and consult it only when necessary. The reader lacking any prerequisite can fulll it during this chapter, at least to the extent necessary to make sense of the rest of the dissertation. Sections 2.4 and 2.5, however, are pretty much complete and oer a decent basis for the results they are presenting.. 2.1 Measure Theory During this section, we review some measure theory in order to align notation and the statement of major results. Proofs of purely technical content are omitted, while proofs of instructive or important results are briey sketched or indicated.. ¯ ∶= {−∞} ∪ R ∪ {∞}. The order [−∞, ∞] = R in [−∞, ∞] is given by −∞ < x < ∞, ∀x ∈ R. Therefore every set in [−∞, ∞] is bounded and has inmum and supremum, hence every sequence in [−∞, ∞] has lim supn = inf k≥1 supn≥k and lim inf n = supk≥1 inf n≥k . The topology in [−∞, ∞] is that generated by the subbasis consisting of the open intervals, (a, b) = {x ∈ [−∞, ∞] ∶ a < x < b}, (a, ∞] = {x ∈ [−∞, ∞] ∶ x > a}, [−∞, b) = {x ∈ [−∞, ∞] ∶ x < b} and [−∞, ∞], with a, b ∈ [−∞, ∞]. Closed and semi-closed intervals in [−∞, ∞] are dened naturally. A sequence in this topology converges if, and only if, lim inf and lim sup are coincident. The arithmetic operations + ∶ [−∞, ∞] × [−∞, ∞] ∖ {(−∞, ∞), (∞, −∞)} → [−∞, ∞] and ⋅ ∶ [−∞, ∞] × [−∞, ∞] → [−∞, ∞] are dened (extending the real operations) as: x + ±∞ = ±∞ = ±∞ + x, ∀x ∈ R; ±∞ + ±∞ = ±∞; x ⋅ ±∞ = ±∞ = ±∞ ⋅ x, ∀x > 0, x ⋅ ±∞ = ∓∞ = ±∞ ⋅ x, ∀x < 0 and 0 ⋅ ±∞ = 0 = ±∞ ⋅ 0. We then dene σ -algebras, collections of subsets of a space X (whose elements are called measurable sets) on which we can conveniently prescribe specic functions assigning values in [0, ∞], For our convenience, we dene the extended reals. to be interpreted as generalized areas or volumes.. Denition 2.1.1. σ -algebra).. A ⊂ P(X) (the set of c parts of X ). A collection A is an algebra (a σ -algebra) on X if: (i) ∅ ∈ A, (ii) A ∈ A ⇒ A ∈ A, k ∞ (iii) if An ∈ A, for n ∈ {0, . . . , k} (or n ≥ 0), then ∪n=0 An ∈ A (or ∪n=0 An ∈ A). (Algebra and. Let. X. 3. be a nonempty set and.

(24) PREREQUISITES. 4. 2.1. It is easy to note that P(X) and {∅, X} are σ -algebras and that the arbitrary intersection σ -algebras on X is a σ -algebra on X . It is then possible to dene the σ -algebra generated by Y ⊂ P(X), as the intersection of all σ -algebras containing Y , σ(Y) ∶= ⋂ A. It is well dened of. (as. P(X). containing. A (Y).. Y⊂A. always makes the intersection non-trivial), unique and understood as smallest. Y.. The denition of the algebra generated by. Y ⊂ P(X). σ -algebra. is analogous and denoted by. If (X, τ ) is a topological space, the σ -algebra σ(τ ) is called the Borel σ -algebra of X , denoted BX . Its elements are called Borelians sets. When dealing with a topological space, we will implicitly assume it is equipped with a Borel σ -algebra, unless otherwise stated. A pair (X, A), where X is a nonempty set and A is a σ -algebra on X , is called a measurable space.. Denition 2.1.2 (Measure).. (X, A) be a measurable space. A measure µ on (X, A) (or X , or A, if clear from the context) is a function µ ∶ A → [0, ∞] such that: (i) µ(∅) = 0 and (ii) if (An )n≥0 is ˙ a union of disjoint sets, µ(∪˙ n≥0 An ) = ∑n≥0 µ(An ). a sequence of disjoint sets in A, then, denoting ∪ Let. (X, A) equipped with a measure µ on (X, A), (X, A, µ), is called a measure X is a topological space, a measure on (X, BX ) is called Borelian measure on X . Let (X, A, µ) be a measure space. We say µ is: (i) a probability if µ(X) = 1, (ii) nite if µ(X) < ∞, (iii) σ -nite if ∃(An )n≥0 a sequence in A such that X = ∪n An and µ(An ) < ∞, ∀n ≥ 0 and (iv) seminite if for each A ∈ A such that µ(A) = ∞, ∃E ∈ A, E ⊂ A, such that 0 < µ(E) < ∞. A measurable space. space. If. The following proposition summarizes the basic working properties of measure spaces.. Proposition 2.1.3 (Properties of measures).. Let (X, A, µ) be a measure space. Then: A, B ∈ A, A ⊂ B , µ(A) ≤ µ(B) (ii) (Subadditivity) for any sequence (An )n≥0 in A, µ(∪n An ) ≤ ∑n µ(An ) (iii) (Continuity from below) for any increasing sequence (An )n≥0 in A (with An ⊂ An+1 , n ≥ 0), µ(∪n An ) = limn µ(An ) (iv) (Continuity from above) for any decreasing sequence (An )n≥0 in A, (with An ⊃ An+1 , n ≥ 0) and such that µ(Aj ) < ∞ for some j ≥ 0, µ(∩n An ) = limn µ(An ). (i) (Monotonicity) for any. all all. for for. Dealing with a nite measure space, we can count on a very general approximation property, using elements from a generating algebra.. Proposition 2.1.4 (Approximating with a generating algebra). X generating A. ˙ E △ D = (E ∖ D)∪(D ∖ E).. space and where. A. an algebra on. Then. Let (X, A, µ) be a nite measure ∀E ∈ A, > 0, ∃D ∈ A such that µ(E △ D) < ,. On a measure space (X, A, µ), if a property P holds for every x ∈ X except possibly on a set E with µ(E) = 0 (a null set), we say P is true µ-almost everywhere (abbreviated µ-a.e.), or for µ-almost every x ∈ X (abbreviated µ-a.e.x ∈ X ). A measure space (X, A, µ) such that every subset of a null set is measurable (hence a null set, by monotonicity) is called complete.. Proposition 2.1.5 (Completing a measure space).. Let (X, A, µ) be a measure space, N = {N ⊂ X ∶ A¯ = {B ⊂ X ∶ ∃A ∈ A, N ∈ N , M ⊂ N , such that B = A ∪ M }. Then A¯ is a σ -algebra (called the completion of A with respect to µ) and there is a unique extension µ ¯ of µ (called the ¯ completion of µ), that makes (X, A, µ ¯) a complete measure space, namely µ ¯(B) = µ ¯(A∪M ) = µ(A).. µ(N ) = 0}. and. We now focus on the rst tool used to construct measures, the so-called Caratheodory construction (the second is Riesz-Markov representation theorem). In special, we address the construction of Lesbegue-Stieltjes measures on. Denition 2.1.6. R.. .. ∗ be a set. An outer measure µ on X is a set function ∗ ∗ ∗ ∗ on P(X), µ ∶ P(X) → [0, ∞] such that (i) µ (∅) = 0, (ii) C ⊂ D ⊂ X ⇒ µ (C) ≤ µ (D) and (iii) ∗ ∗ if (Cn )n≥0 is a sequence of subsets of X , then µ (∪n≥0 Cn ) ≤ ∑n≥0 µ (Cn ). (Outer measure). Let. X.

(25) MEASURE THEORY. 2.1. Proposition 2.1.7 P(X). .. (Creating outer measures from pre-outer measures). be such that (i). ∅∈S. and (ii). ∃(Sn )n≥0. a sequence of elements in. Let. S. X. 5. S ⊂ ∪n Sn = X (S. be a set and. such that. is to be a family of simple" sets that we know how to measure, e.g, a set of intervals/squares/cubes) 0 0 0 and µ ∶ S → [0, ∞] be such that µ (∅) = 0 (µ is called a pre-outer measure and simply assign a prototypical measure).. ∗ 0 For each C ⊂ X , dene µ (C) = inf{∑n µ (Sn ) ∶ (Sn )n≥0 is a sequence in ∗ 0 Then µ is an outer measure, called the outer measure induced by µ .. S. such that. C ⊂ ∪n Sn }.. We are going to induce a measure from an outer measure (or ultimately from a pre-outer measure).. A ⊂ X to be µ∗ -measurable if splits well every other set" with respect to µ , i.e., if µ (E) = µ (E ∩ A) + µ∗ (E ∩ Ac ), ∀E ⊂ X . Since the inequality (≤) is immediate, it is ∗ ∗ ∗ c ∗ enough to check that µ (E) ≥ µ (E ∩ A) + µ (E ∩ A ), ∀E ⊂ X such that µ (E) < ∞ (since, when ∗ µ (E) = ∞, no checking is required). First we dene a set. ∗. ∗. ∗. Theorem 2.1.8 ∗. (Caratheodory's theorem - creating measures from outer measures). .. Let. X. be a. µ an outer measure on X . Then {A ⊂ X ∶ A is µ∗ -measurable} is a σ -algebra on X , denoted σ(µ∗ ). ∗ ∗ ∗ (ii) µ ∣σ(µ∗ ) is a measure on (X, σ(µ )), called the measure induced by µ ∗ ∗ ∗ ∗ (iii) for each C ⊂ X such that µ (C) = 0, it holds that C ∈ σ(µ ); therefore (X, σ(µ ), µ ∣σ(µ∗ ) ) ∗ complete. Denote µ ∣σ(µ∗ ) by µ ˆ.. set and (i). is. The preceding theorem will be applied to a specic case, which will be useful for constructing Lesbegue-Stieltjes measures on. σ -algebra. R:. extending pre-measures (on an algebra) to measures (on the. generated by that algebra).. Denition 2.1.9. (Pre-measures). Let X a set and A be an algebra on X . A pre-measure µ0 (X, A ) (or X , or A , if clear from the context) is a function µ0 ∶ A → [0, ∞] such that: (i) µ0 (∅) = 0 and (ii) if (An )n≥0 is a sequence of disjoint sets in A such that ∪˙ n≥0 An ∈ A , then µ0 (∪˙ n≥0 An ) = ∑n≥0 µ0 (An ).. on. Those denitions of probability, nite,. σ -nite. and seminite measures extend naturally to pre-. measures.. Theorem 2.1.10 (Caratheodory's extension theorem - creating measures from pre-measures). X. be a set,. A. be an algebra on. X. and. µ0 ∶ A → [0, ∞]. a pre-measure on. Let. (X, A ).. Use proposition 2.1.7, facing A as the family of simple" sets, S ∶= A , and µ0 as a pre-outer 0 ∗ 0 measure, µ ∶= µ0 , to obtain the outer measure µ ∶ P(X) → [0, ∞] on X induced by µ = µ0 (in ∗ 0 general, proposition 2.1.7 will not guarantee µ extends µ ). Use Caratheodory's theorem, to obtain a complete measure µ ˆ ∶= µ∗ ∣σ(µ∗ ) in σ(µ∗ ) (obviously µ ˆ ∗ coincides with µ where it is dened). The present theorem adds that, when starting from a pre-measure, as herein assumed: [`eoe' and `uoe' for existence and uniqueness of extension] ∗ ∗ 0 (eoe, i) µ ∣A = µ0 , i.e., µ extends µ = µ0 . ∗ (eoe, ii) A ⊂ σ(µ ), therefore µ ˆ is a measure dened on a σ -algebra containing. µ0 ,. i.e.,. µ ˆ(A) = µ0 (A), ∀A ∈ A . σ(A ) ⊂ σ(µ∗ ),. (eoe, iii) naturally. therefore. µ ∶= µ ˆ∣σ(A ) = µ∗ ∣σ(A ). A. which extends. is a measure dened on the. A which extends µ0 . ν ∶ σ(A ) → [0, ∞] is another measure extending µ0 , then ν ≤ µ, i.e., ν(A) ≤ µ(A), ∀A ∈ σ(A ). The equality holds for those A ∈ σ(A ) such that µ(A) = µ∗ (A) < ∞. (uoe, v) if µ0 is σ -nite and ν ∶ σ(A ) → [0, ∞] is another measure extending µ0 , then ν = µ. ∗ (vi) if µ0 is σ -nite, then (X, σ(µ ), µ ˆ) equals (X, σ(A ), µ ¯), the completion of (X, σ(A ), µ). σ -algebra. generated by. (uoe, iv) if. We are now ready to dene the Lesbegue-Stieltjes measures on measures on. R. R,. a major class of Borelian. (in fact, every Borelian measure which is nite on bounded measurable sets is a. Lesbegue-Stieltjes measure and vice-versa)..

(26) PREREQUISITES. 6. 2.1. As a motivation, start with a Borelian measure on and dene its centralized distribution function. R which is nite on bounded measurable sets F ∶ R → R given by. ⎧ µ((0, x]) x>0 ⎪ ⎪ ⎪ ⎪ F (x) = ⎨0 x=0 ⎪ ⎪ ⎪ ⎪ ⎩−µ((x, 0]) x < 0. Using proposition 2.1.3, it follows that F is non-decreasing, right-continuous and satises (a) µ((x, y] ∩ R) = F (y) − F (x), for x, y ∈ [−∞, ∞], x ≤ y , (b) F (∞) ∶= limx→∞ F (x) ∈ [0, ∞] and (c) F (−∞) ∶= limx→−∞ F (x) ∈ [−∞, 0]. Next, starting with a non-decreasing right-continuous function F , we construct a measure µ whose centered distribution is F . The family of simple" sets to which we will assign a prototypical measure is the family S = {(a, b] ∩ R ∶ a, b ∈ [−∞, ∞], a ≤ b} of h-intervals. It has the following convenient property.. Proposition 2.1.11. above is an elementary family, i.e., (i) nite union of disjoint elements of Moreover, if elements in. E. E. .. S of h-intervals introduced S, T ∈ S ⇒ S ∩ T ∈ S and (iii) S ∈ S ⇒ S c is a. (Properties of the set of h-intervals). S. ∅ ∈ S,. (ii). The family. (in this case, two elements).. is an elementary family, then the collection. A (E) = D (E). σ(S) = BR , σ(D (S)) = BR .. D (E). of nite disjoint unions of. is an algebra. In particular,. Finally, since. Is is now possible to dene a Lesbegue-Stieltjes pre-measure.. Proposition 2.1.12 (Lesbegue-Stieltjes pre-measure).. Let F ∶ R → R be non-decreasing and rightF (∞) ∶= limx→∞ F (x) ∈ [0, ∞] and F (−∞) ∶= limx→−∞ F (x) ∈ [−∞, 0]. Dene on D (S) ˙ nj=0 (aj , bj ] ∩ R) = ∑nj=0 [F (bj ) − F (aj )]. Then µ0 is a pre-measure on function µ0 , assigning µ0 (∪ algebra D (S) = A (S).. continuous, the the. Now Caratheodory's extension theorem is used to obtain the Lesbegue-Stieltjes measure.. Theorem 2.1.13 continuous, with. .. (Lesbegue-Stieltjes measure). F (∞) ∶= limx→∞ F (x) ∈ [0, ∞]. F ∶ R → R be non-decreasing F (−∞) ∶= limx→−∞ F (x) ∈ [−∞, 0].. Let. and. and right-. Then, using Caratheodory's extension theorem and its notation, there exists a unique (µ0 is ∗ Borelian measure µF on (R, BR ), and an associated completion µF on (R, σ(µ )), whose. σ -nite). centered distribution is. µF. F.. In particular,. µF ((a, b] ∩ R) = F (a) − F (b),. for. a, b ∈ [−∞, ∞], a ≤ b. (so. is nite on bounded measurable sets and hence on compact sets). Let. G∶R→R. be another non-decreasing and right-continuous function: then. µG = µ F ⇔ F − G. is constant.. µ on (R, BR ) is nite on bounded measurable sets (or on compact sets), then there F ∶ R → R non-decreasing and right-continuous (namely, take F as µ's centered distribution) that µ = µF .. Reciprocally, if exists such. µF or µ¯F , which we may, by abuse of notation, jointly denote by µF . When µ is a nite measure on R, its cumulative distribution, G ∶ R → R given by G(x) = µ((−∞, x]), is non-decreasing, right-continuous and satises (a) µ((x, y] ∩ R) = F (y) − F (x), for x, y ∈ [−∞, ∞], x ≤ y , (b) G(∞) ∶= limx→∞ G(x) ∈ [0, ∞] and (c) G(−∞) ∶= 0. In this case, by the previous theorem, G and F dier by a constant, namely µ((−∞, 0]). We call the Lesbegue-Stieltjes measure associated to. F. either. Our last assertion about Lesbegue-Stieltjes measures is a collection of their regularity properties.. Proposition 2.1.14. (Regularity of Lesbegue-Stieltjes measures). Let µ be a complete LesbegueR associated with F ∶ R → R be non-decreasing and right-continuous, with F (∞) ∶= limx→∞ F (x) ∈ [0, ∞] and F (−∞) ∶= limx→−∞ F (x) ∈ [−∞, 0]. Let Mµ be the domain of µ. Then, ∀E ∈ Mµ : ∞ ∞ (i) µ(E) = inf{∑j=0 µ((aj , bj ] ∩ R) ∶ E ⊂ ∪j=0 (aj , bj ] ∩ R, aj , bj ∈ [−∞, ∞], aj ≤ bj } (immediate ∗ from the denition of µ induced from the pre (outer) measure dened in proposition 2.1.12).. Stieltjes measure on.

(27) MEASURE THEORY. 2.1. 7. ∞ µ(E) = inf{∑∞ j=0 [F (bj ) − F (aj )] ∶ E ⊂ ∪j=0 (aj , bj ] ∩ R, aj , bj ∈ [−∞, ∞], aj ≤ bj } (immediate from µ being the Lesbegue-Stieltjes associated to F ). ∞ ∞ (iii) µ(E) = inf{∑j=0 µ((aj , bj ) ∩ R) ∶ E ⊂ ∪j=0 (aj , bj ) ∩ R, aj , bj ∈ [−∞, ∞], aj ≤ bj } (a version (ii). of outer regularity). µ(E) = inf{µ(U ) ∶ U open, E ⊂ U ⊂ R} (outer regularity) (v) µ(E) = sup{µ(K) ∶ K compact, K ⊂ E ⊂ R} (inner regularity) (vi) E = V ∖ N1 , where V ∈ Gδ (R) (the set of countable intersections of open sets of R) µ(N1 ) = 0. (vii) E = H ∪ N2 , where H ∈ Fσ (R) (the set of countable unions of closed sets of R) µ(N2 ) = 0. (viii) such that µ(E) < ∞, it holds that ∀ > 0, ∃A ⊂ R a nite union of open intervals such µ(E △ A) < (a version of proposition 2.1.4). (iv). and and that. We then focus on the most special Lesbegue-Stieltjes measure. It is the so-called Lesbegue. m, associated with F = idR . As in proposition 2.1.13, by abuse of notation, m refers to the measure of both (R, BR , µid ) and its completion (R, BR , µid ). The former σ -algebra, BR , characterized in propositions 2.1.13 and 2.1.10, is by denoted L. In fact, BR ⊊ L ⊊ P(X), but measure, denoted by. we will not give a hint why. The next statement shows that the Lesbegue measure is algebraically well-behaved.. Proposition 2.1.15. (Lesbegue measure behaves well with translations and dilations). If E ∈ L, s, r ∈ R, s + E = {s + x ∶ x ∈ E} ∈ L, rE = {rx ∶ x ∈ E} ∈ L, m(s + E) = m(E) and m(rE) = ∣r∣m(E).. then, for. We now start to shift our review towards the integration theory based on such measures. Later, we will generalize the concept of measures by broadening its codomain.. (X, A) and (Y, B) (or on (X, A) f −1 (B) ∈ A, ∀B ∈ B . In practice, this. By denition, a measurable function between measurable spaces if the codomain is implicit),. f ∶ (X, A) → (Y, B),. satises. property can be veried as follows.. Proposition 2.1.16 (Checking measurability in a generating algebra). measurable spaces. A, ∀B ∈ G ,. then. f. (X, A). and. (Y, B),. with. B. Let. f. be a function between −1 generated by an algebra G , i.e., σ(G) = B . If f (B) ∈. is measurable.. H = {B ⊂ Y ∶ f −1 (B) ∈ A}. Note that H is itself a σ -algebra and G ⊂ H, hence B = σ(G) ⊂ H. As a consequence, f is measurable. By denition, a measure preserving function between measure spaces (X, A, µ) and (Y, B, ν), f ∶ (X, A) → (Y, B), is a measurable function such that ν(B) = µ(f −1 (B)), ∀B ∈ B . In practice, this The argument behind this proposition is pretty standard. Let. property can be veried as follows.. Proposition 2.1.17. .. (Checking measure preservation in a generating algebra). surable function between measure spaces −1 i.e., σ(G) = B . If µ(f (B)) = ν(B), ∀B. (X, A, µ) and (Y, B, ν), with B generated ∈ G , then f is measure preserving.. Let. f. be a mea-. by an algebra. G,. H = {B ⊂ Y ∶ µ(f −1 (B)) = B = σ(G) ⊂ H. As a consequence, f is. The argument behind the proposition is precisely like the last one. Let. ν(B)}.. Note that. H. is itself a. σ -algebra. and. G ⊂ H,. hence. measure preserving. Two measurable spaces bijection. h∶X →Y,. (X, A) and (Y, B) are said to be isomorphic if there exists a measurable. admitting a measurable inverse.. (X, A, µ) and (Y, B, ν) are said to be strictly isomorphic if there exists a h ∶ X → Y , admitting a measurable inverse (which will be measure almost isomorphic if, in each, there are subsets X0 ⊂ X and Y0 ⊂ Y of. Two measure spaces. measure-preserving bijection preserving). They are. full measure so that the corresponding restricted measure spaces are strictly isomorphic. Given a measure space. (X, A, µ). A∣X0 = {A ∩ X0 ∶ A ∈ A}. and and. X0 ⊂ X , µ∣ A∣. X0. the restricted measure space is. is a standard function restriction.. (X0 , A∣X0 , µ∣ A∣. X0. ),. where.

(28) PREREQUISITES. 8. 2.1. Obviously, the composition of measurable (measure preserving) functions is a measurable (measure preserving) function.. Proposition 2.1.18 (Measurable functions under elementary and limiting operations are measurable). If f, g ∶ X → K , K ∈ {C, [−∞, ∞]}, are measurable, then f +g , f g and f /g (when dened) are. hj ∶ X → [−∞, ∞], j ≥ 0, are measurable, then a(x) = supj hj (x), b(x) = inf j hj (x), c(x) = lim supj hj (x), d(x) = lim inf j hj (x) are measurable. In particular, max{h1 , h2 }, min{h1 , h2 }, ∣h1 ∣ and limj hj (if dened for all x ∈ X ) are measurable. measurable. Additionally, if. σ -algebra. Also, continuous functions are measurable functions with respect to the Borelian. on. the domain and codomain, since we can check the preimage property for the generating collection of open sets. A simple function on. (X, A), φ ∶ X → C. (or. R),. is a nite linear combination of characteristic. functions of measurable sets using complex (or real) coecients. It can be equivalently expressed by dierent linear combinations, even though we x a standard representation: since the image of a simple function is nite (while a measurable function with nite image is, conversely, simple), we let. Ej = f −1 ({zj }),. where. Img(f ) = {z0 , . . . , zn },. and. φ = ∑nj=0 zj XEj .. As a sum of measurable. functions, a simple function is also measurable.. (X, A) (X, A) in. φ = ≥ 0, j ≥ 0), be a simple function on standard representation and let µ n be a measure on X . We dene the integral of φ with respect to µ by ∫ φdµ ∶= ∑j=0 aj µ(Ej ) ∈ [0, ∞] (remember 0⋅∞ = 0). This integral is homogeneous of degree one (taking scalars c ∈ [0, ∞)), additive and monotonic. We also write ∫ φdµ = ∫ φ = ∫ φ(x)dµ(x) = ∫ φ(x)µ(dx), ∫A φdµ = ∫ φXA dµ and ∫ = ∫X . This notation can be extended naturally to all integrals we will dene later and will not be First, we assign integrals to simple functions on. ∑nj=0 aj XEj (aj. taking values in. [0, ∞].. Let. repeated.. (X, A) taking values in [0, ∞] (these L+ (X) and µ be a measure on X . We dene ∫ f dµ ∶= sup{∫ φdµ ∶ φ simple function on (X, A), 0 ≤ φ ≤ f }. By the previous monotonicity, this denition extends the former. If f has nite integral we call it (µ-)integrable, while if it has innite integral we call it (µ-)quasi-integrable. Second, we assign integrals to measurable functions on. L+ (X)).. functions comprise the set. Let. f. be in. Proposition 2.1.19 (Integration is indierent to changes in sets of measure zero, v1). +. L (X). and. µ. a measure on. X.. ∫ f dµ = 0. Then. if, and only if,. Let. f. be in. f = 0 µ-a.e.. Additionally, this integral is homogeneous of degree one (taking scalars. c ∈ [0, ∞)). and mono-. tonic. It is also additive, as a consequence of the next two results.. Proposition 2.1.20. (Measurable Functions are approximated by simple functions). .. Let. (X, A). f ∶ X → [0, ∞] (or C). Then there exists a sequence (φn )n≥0 of positive (X, A) such that 0 ≤ ∣φ1 ∣ ≤ . . . ≤ ∣φn ∣ ≤ . . . ≤ ∣f ∣, φn → f on any set on which f is bounded.. be a measurable space and. (complex) valued simple functions on pointwise and uniformly. The following theorem is the rst major convergence theorem. Together with the last proposition and monotonicity, it also permits to write the integral of functions increasingly converging to. Theorem 2.1.21. f. f ∈ L+ (X). as a limit of integrals of simple. instead of the original supremum.. (fn )n≥0 is a sequence lim f dµ = limn ∫ fn dµ. n n ∫ space. If. in. +. L (X). such that. Hence, under the same hypothesis, it follows that. .. (X, A, µ) fn (x) ↗ f (x), µ-a.e.x ∈ X ,. (Monotone Convergence Theorem - a.e. version). Let. be a measure then. ∫ f dµ =. ∫ ∑n fn dµ = ∑n ∫ fn dµ.. The second major converge theorem is the following.. Theorem 2.1.22 a sequence in. . Let (X, A, µ) lim inf f dµ ≤ lim inf n n n ∫ fn dµ. ∫. (Fatou`s Lemma - a.e. version). L+ (X),. then. be a measure space. If. (fn )n≥0. is.

(29) MEASURE THEORY. 2.1. 9. We will now extend the domain of functions to which we (try to) assign integrals to those taking. [−∞, ∞] and C. (X, A, µ) be a measure space, f ∶ X → [−∞, ∞] be a measurable function and f + , f − ∶ X → [−∞, ∞] be given, respectively, by f + (x) = max{f (x), 0} and f − (x) = max{−f (x), 0}, so + − + − + − that f = f − f and ∣f ∣ = f + f . When ∫ f dµ and ∫ f dµ are not both innite, we dene the + − integral of f with respect to µ to be ∫ f dµ ∶= ∫ f dµ − ∫ f dµ ∈ [−∞, ∞]. If f has nite integral we call it (µ-)integrable, while if it has innite integral we call it (µ-)quasi-integrable. Now, not every values in Let. measurable function will be one of them, i.e., not every measurable function will be assigned an integral, as when. + ∫ f dµ. and. − ∫ f dµ. are both innite. Clearly,. f. is integrable if, and only if,. ∣f ∣. is. integrable.. (X, A, µ) be a measure space and f ∶ X → C be a measurable function. When ∫ Re(f )dµ ∫ Im(f )dµ are both nite (Re(f ) and Im(f ) are measurable), we dene the integral of f with respect to µ to be ∫ f dµ ∶= ∫ Re(f )dµ + i ∫ Im(f )dµ ∈ C. Here (µ-)integrable functions are exactly those measurable functions to which we assign integrals. The concept of (µ-)quasi-integrable Let. and. functions is not applicable anymore, as now the integral is complex-valued. We highlight the following properties.. Proposition 2.1.23 (The set of integrable functions is a vector space).. The set of real (complex). valued integrable functions is a real (complex) vector space and the integral is a linear functional on it.. Proposition 2.1.24. (A triangular inequality, v1). )integrable (taking values in. [0, ∞], [−∞, ∞]. or. .. C),. (X, A, µ) be a measure then ∣ ∫ f dµ∣ ≤ ∫ ∣f ∣dµ. Let. space. If. f. is (µ-. Proposition 2.1.25 (Integration is indierent to changes in sets of measure zero, v2). Let (X, A, µ) f and g are (µ-)integrable (taking values in [0, ∞], [−∞, ∞] f dµ = gdµ, ∀E ∈ B if, and only if, ∫ ∣f − g∣dµ = 0 if, and only if, f = g , µ-a.e.. ∫E ∫E. be a measure space. If. or. C),. then. Finally, we state the last major convergence theorem.. Theorem 2.1.26 (Dominated convergence theorem).. (X, A, µ) be a measure space. If (fn )n≥0 [0, ∞], [−∞, ∞] or C) such that fn → f µ-a.e. and there exists an integrable function g (taking values in [0, ∞]), with ∣fn ∣ ≤ g µ-a.e. ∀n ≥ 0, then f is integrable and ∫ f dµ = ∫ limn fn dµ = limn ∫ fn dµ. Hence, considering (fn )n≥0 as before and g = ∑n ∣fn ∣. If ∫ ∑n ∣fn ∣dµ = ∑n ∫ ∣fn ∣dµ (by monotone convergence theorem) is nite, then ∑n fn converges µ-a.e. to an integrable function f and ∫ f dµ = ∫ ∑n fn dµ = ∑n ∫ fn dµ. Let. is a sequence of (µ-)integrable functions (taking values in. We now shift our review towards the theory of. Lp. spaces. During the exposition of this subject,. (X, A, µ). 1 C-valued measurable function. For p ∈ (0, ∞), dene ∥f ∥p ∶= (∫ ∣f ∣p dµ) /p ∈ [0, ∞]; for p = ∞, dene ∥f ∥∞ = inf{C ∈ [0, ∞] ∶ ∣f ∣ ≤ C µ-a.e.} ∈ [0, ∞] (which adapts the uniform norm to the p present setting). Now, for p ∈ (0, ∞], dene L (µ, C) ∶= {f ∶ X → C; f is measurable and ∥f ∥p < ∞}. 1 Note that L (µ, C) is simply the set of C-valued integrable functions. 1 With pointwise addition and multiplication by scalars, the set L (µ, C) is a complex linear 1 subspace of {f ∶ X → C} and ∫ ⋅dµ ∶ L (µ, C) → C is a linear functional. We could want to consider measurable functions taking values in [−∞, ∞] or [0, ∞]. The problem consider a xed measure space Let. f. be a. is that sums and dierences of functions in those sets are not well dened. Nonetheless, every denition in the next-to-last paragraph, not involving sums, can be directly adapted, by changing C to [−∞, ∞] or [0, ∞].. K ∈ {[−∞, ∞], [0, ∞]}, every K -valued measurable measure function f such that f ({−∞}) and f ({∞}) have null measure, in particular for every f ∈ Lp (µ, K) (in the aforementioned adapted sense), we can consider the K -valued measurable measure function fˆ = f Xf −1 (R) , whose image is in R and such that fˆ = f , µ-a.e.. Every following result can be extended to −1. Moreover, for. −1.

(30) PREREQUISITES. 10. K ∈ {[−∞, ∞], [0, ∞]}. 2.1. in the sense of considering. fˆ. instead of. ndings. As such, every following result is presented for A pair. p, q ∈ [1, ∞]. f. K = C.. is said to be a pair of conjugate exponents if. and collecting the associated. p−1 + q −1 = 1. (here. 1. and. ∞. are. conjugate exponents).. Proposition 2.1.27. .. (Ho ¨lder inequality) Let f, g ∶ X → C, be measurable functions and p, q, r ∈ 1 1 [1, ∞] be such that /p + /q = 1/r. Then ∥f g∥r ≤ ∥f ∥p ∥g∥q . Note we are not stating, a priori, that f , g or f g are in Lp (µ, C), Lq (µ, C) or Lr (µ, C).. Proposition 2.1.28. (Minkowski inequality). [1, ∞). Then ∥f + g∥p ≤ ∥f ∥p + ∥g∥p . Lp (µ, C).. .. f, g ∶ X → C,. Let. be measurable functions and. Note we are not stating, a priori, that. f, g. or. f +g. p∈. are in. p ∈ [1, ∞], Lp (µ, C) is a vector space and ∥ ⋅ ∥p is an associated seminorm (which can assume value 0 at not identically 0 functions f , as long as f = 0 µ-a.e.). p For each p ∈ [1, ∞], we dene an equivalence relation on L (µ, C), by stating f ∼ g whenever p ∥f − g∥p = 0, i.e., f = g µ-a.e.. So we obtain N ∶= {f ∈ L (µ, C) ∶ ∥f ∥p = 0}, a complex linear subspace p p p p of L (µ, C). Now, we dene the set of equivalence classes on L (µ, C), L (µ, C) ∶= L (µ, C)/N = {[f ] ∶ f ∈ Lp (µ, C)}, where [f ] ∶= {g ∈ Lp (µ, C) ∶ f ∼ g , i.e., f − g ∈ N } is called the equivalence class of f . p Adapting vector addition, multiplication by scalars and ∥ ⋅ ∥p to L (µ, C), by using representaFrom Minkowski inequality, for. tives of each equivalence class and performing the previously dened operations (these denitions will be well-dened), we have:. Proposition 2.1.29 (L p is a Banach space).. For. p ∈ [1, ∞], (L p (µ, C), ∥ ⋅ ∥p ). is a Banach space. (complete normed vector space).. ([fn ])n≥0 and [f ], all of ∥[fn ] − [f ]∥p → 0. This same. As a normed space, it has an associated notion of convergence: given them in. L p (µ, C), ([fn ])n≥0. is said to converge to. f. in. Lp. whenever. notion of convergence allows the following result.. Proposition 2.1.30 (Simple functions are dense in L p ).. S ∶= {[φ] ∈ L p (µ, C) ∶ φ = ∑kn=0 an XEn p dense in L (µ, C).. is simple with. For. an ∈ C. p ∈ [1, ∞], the set of simple functions and µ(En ) < ∞, for n = 1, . . . , k} is. L p (µ, C), given (fn )n≥0 Lp whenever ∥fn − f ∥p → 0.. Borrowing the norm just dened on. (fn )n≥0. is said to converge to. f. in. and. f,. all of them in. From now on, an abuse of notation will be allowed by denoting both by. L (µ, C), p. Lp (µ, C). relying on the context to make the meaning clear. In particular,. Lp or L p convergences. p of L (µ, C), by taking its. convergence in order to refer to either. From an element. reader can always obtain an element. equivalence class.. We conclude the review on. Lp. Lp (µ, C),. L p (µ, C) p we may use L p of L (µ, C), the and. spaces with the following three propositions. The rst of them is. a direct application of dominated convergence theorem.. Proposition 2.1.31. Lp convergence). Let p ∈ [1, ∞), fn , ∃g ∈ Lp (µ, C) such that ∣fn ∣ ≤ g , µ-a.e., ∀n ≥ 0,. (Almost everywhere convergence and. n ≥ 0, and f be C-valued measurable functions. and fn → f , µ-a.e., then ∥fn − f ∥p → 0.. If. Proposition 2.1.32 (A function in Lp is a sum of one from a higher exponent space and another q p r from a lower). If 0 < p < q < r ≤ ∞, then L (µ, C) ⊂ L (µ, C) + L (µ, C). Proposition 2.1.33 (The Lp inclusion). and. ∥f ∥p ≤ ∥f ∥q µ(X). 1/p−1/q. If. µ(X) < ∞. and. then. Lq (µ, C) ⊂ Lp (µ, C). .. In the above proposition, if we relax the niteness of ample:. 0 < p < q ≤ ∞,. X = [1, ∞], µ = m, ∫. 1/xdµ. = ∞,. but. ∫. 1/x2 dµ. µ,. we get the following simple counterex-. < ∞.. We now shift our review towards some results about product measures..

(31) MEASURE THEORY. 2.1. 11. Let ((Xα , Aα ))α∈Λ be an arbitrary family of measurable spaces, X be a set and (fα ∶ X → Xα )α∈Λ be a family of functions. Then there exists the smallest σ -algebra A on X that makes every fα measurable, called the σ -algebra on X induced by the family (fα )α∈Λ : A = σ(T ), where T = {V ⊂ X ∶ ∃α ∈ Λ and A ∈ Bα such that V = fα −1 (A)}. A consequence of the σ -algebra induced by the family (fα )α∈Λ to the measurability of functions is not only that each fα gets measurable, but also that:. Proposition 2.1.34. (Measurability of the composition with the inducing family). Let X be a (Xα , Aα )α∈Λ be a family of measurable spaces, (fα ∶ X → Xα )α∈Λ be a family of functions, A be the σ -algebra on X induced by the family (fα )α∈Λ , (Y, B) be another measure space and g ∶ (Y, B) → (X, A) be a function. Then g is measurable ⇔ fα ○ g ∶ (Y, B) → (Xα , Aα ) is measurable, ∀α ∈ Λ (the ⇒ implication is. set,. trivial, since the composition of measurable functions is measurable).. ((Xα , Bα ))α∈Λ be an arbitrary family of measurable spaces, X be their cartesian product, X = ∏α∈Λ Xα , and (prα ∶ X → Xα )α∈Λ be the family of canonical projections. The σ -algebra on X induced by the family (prα )α∈Λ is called the product σ -algebra (on the product space X ) and denoted by ⊗α∈Λ Aα . One should ask whether ⊗α∈Λ Aα equals the σ -algebra generated by the set of rectangles whose α-components are measurable sets in Aα ", i.e., whether it equals σ(R), where R = {∏α∈Λ Aα ⊂ X ∶ Aα ∈ Aα , α ∈ Λ}. It will be the case when Λ is countable. This is because (i) in general, T ⊂ R, since V ∈ T ⇒ V = prα∗ −1 (A) for some α∗ ∈ Λ and A ∈ Aα∗ ⇒ V = ∏α∈Λ Uα , where Uα = A if α = α∗ and Uα = Xα otherwise ⇒ V ∈ R and (ii) when Λ is countable, R ⊂ T , since V ∈ R ⇒ V = ∏α∈Λ Aα = ∩α∈Λ prα −1 (Aα ) ∈ S , once each prα −1 (Aα ) ∈ ⊗α∈Λ Aα and the intersection is countable. Moreover, if, for each α ∈ Λ, Aα is generated by a set Gα , a stronger version will hold: ⊗α∈Λ Aα = σ(R′ ), where R′ = {∏α∈Λ Gα ⊂ X ∶ Gα ∈ Gα , α ∈ Λ}. In the present context, the last proposition has a direct corollary: a function g taking values in Let. the product space is measurable if (and only if ) its coordinate functions are also measurable. The following result explores, when each algebra. BX .. BXα. and. X = ∏α∈Λ Xα. Xα. is a topological space equipped with its Borel. is equipped with the product topology, how. ⊗α∈Λ BXα. σ-. compares to. Proposition 2.1.35. (Product of Borel σ -algebras and Borel σ -algebra of the product topology). ((Xα , τα , BXα ))α∈Λ be a family of topological spaces each equipped with its Borel σ -algebra and X = ∏α∈Λ Xα equipped with the product topology τ . Then: (i) ⊗α∈Λ BXα ⊂ BX (ii) ⊗α∈Λ BXα = BX in the following cases: (a) ∀α ∈ Λ, τα has a countable basis (separable metric spaces are a special case) and all but a countable quantity of the Xα are singletons or, equivalently, (b) τ has a countable basis. e−1 ¯ or C instead of R As a corollary, BRe = ⊗j=0 BR , e ∈ N ∪ {∞}, analogously with R. Let. ((Xj , Aj , µj ))nj=0 , we wish to construct a n n product measure on the product measurable space (∏j=0 Xj , ⊗j=0 Aj ). With no loss of generality, Now, starting with nite family of measure spaces. we restrict for the case when. n = 2,. (X, A, µ) and (Y, B, ν). For such, R = {A × B ⊂ X × Y ∶ A ∈ A, B ∈ B}. and use the measure spaces. we assign a suitable prototypical (product) measure in the set. of rectangles, adapt it to an algebra and leverage on Caratheodory's extension theorem. In order to do so, we rephrase proposition 2.1.11 to the set of rectangles.. Proposition 2.1.36. . The family R of rectangles introduced ∅ = ∅×∅ ∈ R, (ii) (A×B), (C ×D) ∈ R ⇒ (A×B)∩(C ×D) = (A ∩ C) × (B ∩ D) ∈ R and (iii) (A × B) ∈ S ⇒ (A × B)c = (X × B c ) ∪ (Ac × B) = (Ac × X) ∪ (A × B c ) is a nite union of disjoint elements of R (in this case, two elements). Moreover, if E is an elementary family, then the collection D (E) of nite disjoint union of elements in E is an algebra. In particular A (E) = D (E). Finally, since σ(R) = A ⊗ B , σ(A (R)) = A ⊗ B . (Properties of the set of rectangles). above is an elementary family, i.e., (i).

(32) PREREQUISITES. 12. 2.1. It is now possible to dene a (product) pre-measure.. Proposition 2.1.37. .. (Product pre-measure) Let (X, A, µ) and (Y, B, ν) be two measure spaces, n E = ∪˙ j=0 Aj × Bj be a disjoint union of rectangles in R and (µ × ν)0 ∶ D (R) → [0, ∞] be the set ˙ nj=0 Aj × Bj ↦ (µ × ν)0 (∪˙ nj=0 Aj × Bj ) = ∑nj=0 µ(Aj )ν(Bj ). In special, if A × B function assigning E = ∪ ˙ nj=0 Aj ×Bj ) = ∑nj=0 (µ×ν)0 (Aj ×Bj ). is one such rectangle, (µ×ν)0 (A×B) = µ(A)ν(B) and (µ×ν)0 (∪ Then (µ×ν)0 (is well dened, independent of the representation chosen for E and) is a pre-measure. on the algebra. D (R) = A (R).. Now Caratheodory's extension theorem is used to obtain the product measure.. Theorem 2.1.38 (Product measure).. Let. (X, A, µ). and. (Y, B, ν). be two measure spaces.. Then, by Caratehordory's extension theorem, there exists a measure (µ × ν)0 , and an associated completion µ × ν on (X × Y, σ((µ × ν)∗ )). Also, if. µ. and. ν. σ -nite,. are. µ×ν. on. A⊗B. extending. (µ × ν)0 is σ -nite pre-measure. Therefore, the extension η on A ⊗ B other than µ × ν is such that η(A × B) = will have to coincide with (µ × ν)0 on D (R) = A (R)).. then. is unique. In particular, no other measure. µ(A)ν(B), ∀A ∈ A, B ∈ B. (because it. µ1 ×. . .×µn . But one can check that A1 ⊗ . . . ⊗ An = µ1 × . . . × µn = (µ1 × . . . × µn−1 ) × µn , so, with minor adaptations, it suces to understand the case when n = 2. The product measure implies certain relations will hold between µ × ν integration in X × Y , µ integration on X and ν integration on Y . The following theorem summarizes these relations. An analogous construction can be done to dene. (A1 ⊗ . . . ⊗ An−1 ) ⊗ An. and. Theorem 2.1.39 (Fubini-Tonelli theorem).. Let (X, A), (Y, B) and (X × Y, A ⊗ B) be measurable µ, ν (and, consequently, µ × ν ) be σ -nite measures on those spaces, respectively. y (a.1) if E ∈ A ⊗ B , then Ex ∶= {y ∈ Y ∶ (x, y) ∈ E} ∈ B, ∀x ∈ X and E ∶= {x ∈ X ∶ (x, y) ∈ E} ∈ A, ∀y ∈ Y . Also, if, f ∶ (X × Y, A ⊗ B) → (Z, C) is measurable, then fx ∶ y ∈ (Y, B) ↦ f (x, y) ∈ (Z, C) y is measurable , ∀x ∈ X , and f ∶ x ∈ (X, A) ↦ f (x, y) ∈ (Z, C) is measurable, ∀y ∈ Y . y (a.2) if E ∈ A ⊗ B , then the functions x ∈ X ↦ ν(Ex ) ∈ [0, ∞] and y ∈ Y ↦ µ(E ) ∈ [0, ∞] are y measurable and µ × ν(E) = ∫ ν(Ex )dµ(x) = ∫ µ(E )dν(y) + (b) (Tonelli) if f is in L (X × Y ), then (b.i) g ∶ x ∈ X ↦ ∫ fx (y)dν(y) ∈ [0, ∞] and h ∶ y ∈ Y ↦ y + + ∫ f (x)dµ(x) ∈ [0, ∞] (well dened, by (a.1)) are in L (X) and L (Y ), satisfying: (b.ii) [∗] ∫ f d(µ × ν) = ∫ g(x)dµ(x) = ∫ [∫ fx (y)dν(y)]dµ(x) = ∫ [∫ f (x, y)dν(y)]dµ(x), and aaaaaaa ∫ f d(µ × ν) = ∫ h(y)dν(y) = ∫ [∫ fy (x)dµ(x)]dν(y) = ∫ [∫ f (x, y)dµ(x)]dν(y). 1 1 y 1 (c) (Fubini) if f is in L (µ × ν), then (c.i) fx is in L (ν), for µ-a.e.x ∈ X , and f is in L (µ), y for ν -a.e.y ∈ Y , (c.ii) g ∶ x ∈ X ↦ ∫ fx (y)dν(y) ∈ [0, ∞] and h ∶ y ∈ Y ↦ ∫ f (x)dµ(x) ∈ [0, ∞] are, 1 1 respectively, µ-a.e. dened and ν -a.e. dened and, respectively, in L (µ) and L (ν), satisfying: (c.iii) the formula [∗], as above. spaces and. Acquainted to the product measure, using proposition 2.1.38, we can dene the Lesbegue mea-. mn on Rn , n ∈ N, as m × . . . × m (n times), being each m the Lesbegue measure on (R, BR ) n n (or (R, L), it is equivalent). It's domain will be denoted L , with BRn ⊂ L . Again, by an abuse of n n notation, we may refer to m as both the standard Lesbegue measure on L or it's restriction to BRn . More, if the context is clear, we may refer to mn simply by m.. sure. The following proposition about regularity is a direct consequence of proposition 2.1.14 applied to. m.. Proposition 2.1.40 (Regularity of Lesbegue measure on Rn ).. For all. E ∈ Ln :. m (E) = inf{m (U ) ∶ U open, E ⊂ U ⊂ R } (outer regularity) n n n (ii) m (E) = sup{m (K) ∶ K compact, K ⊂ E ⊂ R } (inner regularity) n n (iii) E = V ∖ N1 , where V ∈ Gδ (R ) (the set of countable intersections of open sets of R ) and mn (N1 ) = 0. n n (iv) E = H ∪ N2 , where H ∈ Fσ (R ) (the set of countable unions of closed sets of R ) and n m (N2 ) = 0. n n (v) such that m (E) < ∞, it holds that ∀ > 0, ∃(Rj )j=0 a nite collection of disjoint rectangles n ˙ j=0 Rj) < . whose sides are intervals such that µ(E △ ∪ (i). n. n. n.

(33) MEASURE THEORY. 2.1. The. n-dimensional. 13. Lesbegue measure also inherits a good behavior with respect to translation,. and linear transformations (hence dilations and rotations). We will not state these results explicitly. During the study of dynamical systems, we will be interested in considering, not only nite product spaces, but also countable product spaces. The discussion of this topic is mostly inuenced by the appendix of [VO16]. Here, we throw in a little bit of topology, as we did before when discussing nite product of measure spaces. In this discussion, it is very convenient to restrict ourselves to probabilities spaces. We adopt this restriction. Let. (Xj , Aj , µj )j∈N. be a countable family of probability spaces and. (∏j∈N Xj , ⊗j∈N Aj ). be the. associated product measurable space, dened after proposition 2.1.34. Once we are dealing with a countable product, the product. R. j -components. of rectangles whose. are in. Aj. σ -algebra is generated by the family. (see discussion after proposition 2.1.34).. C [m; Bm , . . . , Bm+k ] = {x = (x0 , x1 , . . .) ∈ ∏j∈N Xj ∶ xm ∈ Bm , . . . , xm+k ∈ Bm+k }, m ≥ 0, k ≥ 0, Bj ∈ Aj for m ≤ j ≤ m + k . Therefore, the family of (measurable) cylinders C also generates the product σ -algebra, i.e., σ(C) = ⊗j∈N Aj . Using the same notation and attitude of propositions 2.1.11 and 2.1.36, the family C is also an elementary family. So A (C) = D (C) and σ(D (C)) = σ(C) = ⊗j∈N Aj . If each Xj is equipped with a topology τj (in which case we assume Aj = BXj ), the product space ∏j∈N Xj is equipped with the product topology ⊗j∈N τj , the coarest topology on the product space making all the projections continuous (as we know, B∏ coincides with ⊗j∈N BXj whenever each j∈N Xj τj has a countable basis, see proposition 2.1.35). Additionally, if each (Xj , τj ) is compact, so will be (∏j∈N Xj , ∏j∈N τj ), by the Tychono 's theorem from topology. These rectangles are, in turn, obtained as countable intersections of elements in the family. of (measurable) cylinders. Also, it is a well known result in topology that, when the product is more than nite, the product topology will not generally coincide with the box topology (that one having as basis rectangles whose. j -components. are in. τj ).. Moreover, a countable product of open sets. the product topology if, and only if,. Uj = Xj. is in. j 's. These sets are called [m; Bm , . . . , Bm+k ], where, now,. for all but a nitely many. (topological) cylinders and denote them pretty much as before:. Bj ∈ τj .. ∏∞ j=0 Uj , Uj ∈ τj ,. They form a basis for the product topology. Accordingly, here, a countable intersection of. them, consists of a rectangle, but generally not of an open set (as countable intersections of open sets are not necessarily open). Now, is easy to assign prototypical measures to the family of cylinders. These assignments will. σ -algebra generated by them, ⊗j∈N µj . We will not go explicitly into. be extended, using the Caratheodory's construction, to the the product. σ -algebra,. dening the product measure. i.e., to details. of this constructions, once the major ideas were already provided.. Proposition 2.1.41. . Let (Xj , Aj , µj )j∈N be a countable family (∏j∈N Xj , ⊗j∈N Aj ) be the associated product measurable space. Then there exists a unique probability ⊗j∈N µj on (∏j∈N Xj , ⊗j∈N Aj ) such that ⊗j∈N µj ([m; Bm , . . . , Bm+k ]) = µm (Bm ) ⋅ . . . ⋅ µm+k (Bm+k ), for all (measurable) cylinder [m; Bm , . . . , Bm+k ]. In particular, a given rectangle ∏n∈N An , An ∈ An , is such that ∏j∈N An = ∩n∈N Cn , where Cn = [0; A0 , . . . An ], hence, by continuity from below, ⊗j∈N µj (∏j∈N An ) = limn ⊗j∈N µj (Cn ). (Kolmogorov extension theorem). of probability spaces and. (∏j∈N Xj , ⊗j∈N Aj , ⊗j∈N µj ) is called the product space of ((Xj , Aj , µj ))j∈N . (Xj , Aj , µj )'s are all equal to a same probability space (X, A, µ), the product space (∏j∈N X, ⊗j∈N A, ⊗j∈N µ) can be denoted (X N , AN , µN ), case in which µN is called a Bernoulli measure derived from µ. In the above case, suppose X is nite, with n elements, X = {s0 , . . . , sn−1 }, and equipped with X X the discrete σ -algebra and the (compact) discrete topology A = 2 , τ = 2 . As such, the measure µ is completely described by knowing µ({s0 }) = p0 , . . . , µ({sn−1 }) = pn−1 . The probability space When. In this situation, every (measurable or topological) cylinder can be written as a nite union of the. [m; bm , . . . , bm+k ] = {x = (x0 , x1 , . . .) ∈ ∏j∈N X ∶ xm = bm , . . . , xm+k = m ≤ j ≤ m + k . As a consequence, the family E of elementary cylinders also generates the product σ -algebra and serve as a basis for the product topology. Case in. so called elementary cylinders:. bm+k }, m ≥ 0, k ≥ 0, bj ∈ X. for.

(34) PREREQUISITES. 14. 2.1. which proposition 2.1.41 could extend only a prototypical measure assigned to elementary cylinders:. ⊗j∈N µj ([m; bm , . . . , bm+k ]) = pm ⋅ . . . ⋅ pm+k , for all elementary cylinders [m; bm , . . . , bm+k ]. To conclude this topic, also when X is nite, the product topology is not only compact but also N metrizable, for example, by the distance d((xn )n≥0 , (yn )n≥0 ) = θ , where θ ∈ (0, 1) is a xed number and N = N ((xn )n≥0 , (yn )n≥0 ) is the largest N ≥ 0 such that xi = yi , ∀∣i∣ ≤ N . When X = {s0 , s1 }, or, ∣xj −yj ∣ easier, X = {0, 1}, an equivalent distance for metrizing the product topology is ∑j∈N . 2j This machinery for countable products can be developed analogously, with minor modications, for. Z. instead of. N.. Here we conclude the rst part of this review. In the second part, we generalize the concept of measure, by broadening its codomain from. [0, ∞]. to. ¯ R. or. C,. then we introduce the concept of. Radon-Nikodym derivative and nally present Riesz-Markov representation theorem.. Denition 2.1.42. (Signed measure). Let (X, A) be a measurable space. A signed measure ν on ¯ such that: (i) ν(∅) = 0, (ii) X , or A, if clear from the context) is a function ν ∶ A → R the image of ν do not contain −∞ and ∞ at the same time (might contain one), (iii) if (An )n≥0 is a ˙ n≥0 An ) = ∑n≥0 ν(An )+ − ∑n≥0 ν(An )− (one of the summands sequence of disjoint sets in A, then ν(∪ + − must be nite), where ν(A) = max{ν(A), 0}, ν(A) = max{−ν(A), 0}.. (X, A). (or. From now on we will call measures by positive measures. First is instructive to present natural examples of signed measures that later will be known to characterize them in all generality: if is nite, then a. ν = µ 1 − µ2. µ-quasi-integrable. µ1. and. µ2. (X, A), where one of them ¯ is (X, A) and f ∶ X → R by ν(A) = ∫A f dµ is a signed. are positive measures on. is a signed measure; if. µ. is a positive measure on. function, then the set function. ¯ ν∶A→R. dened. measure. The following basic properties of signed measures can be compared to those of positive measures.. Proposition 2.1.43. (Properties of signed measures). .. Let. (X, A, ν). be a signed measure space.. Then: (i) (No more monotonicity) Try taking a measurable set with negative (signed) measure, and note that. ∅. is a subset with greater measure.. (ii) (No more subadditivity) Try taking two measurable sets with negative measures and whose intersection has also negative measure, and note that the sum of their measures is smaller than the measure of the union. (iii) (Continuity from below) (iv) (Continuity from above). j ≥ 0 ⇒ ν(∩n An ) = limn ν(An ).. (An )n≥0 an increasing sequence in A ⇒ ν(∪n An ) = limn ν(An ). (An )n≥0 a decreasing sequence in A such that ν(Aj ) ∈ R for some ν(A) = ∫A f dµ let us motivate some defA, ν must assign positive values to every signs over A, we may nd measurable subsets. The example where a signed measure is of the form initions: when. f. takes only one sign, say positive, over. measurable subset of where. f. A,. but if. is negative, to which. ν. f. assume dierent. assigns negative values.. ν a signed measure on (X, A), we dene A ∈ A to be positive (negative, null) ′ ′ if ν(A ) ≥ 0 (≤ 0, = 0), ∀A ⊂ A, A ∈ A (in the previous motivation, that happens exactly when f ≥ 0(≤ 0, = 0) µ-a.e. in A). Naturally, a subset of a positive (negative, null) set is also positive Hence, with. ′. (negative, null), as well as a countable union of positive (negative, null) sets. The following two theorems characterize a signed measure. ν. as the dierence of two positive. measures.. Theorem 2.1.44 (Hahn decomposition).. Let. ν. be a signed measure on. (X, A).. Then there exists. such that P ∪N = X and P ∩N = ∅. This (Hahn) decomposition ′ ′ is essentially unique, since any other pair P , N with the same property has symmetric dierences ′ ′ P △ P = N △ N with ν measure 0, i.e., null sets can be allocated freely between what is to be taken. positive set. P. and a negative one. N. as positive and negative parts of the decomposition..

(35) MEASURE THEORY. 2.1. 15. The full proof of the theorem is intricate, so we give not a sketch but a starting hint that. ν does not assume the value ∞ and A ranges over the positive sets (it will always exits, since ∅ is one such). That gives is a sequence (Pn )n≥0 of positive sets growing in terms of measure, i.e., ν(Pn ) ↗ m. Dene P = ∪n Pn . It is positive since it is the union of positive sets, and ν(P ) = m, by continuity from below. Dene N = X ∖ P . What is left, the intricate part, is proving that N is negative.. illuminates what goes on. Suppose, with no loss of generality, dene. m. as the. sup. of. ν(A). as. Now we dene a concept that will be useful in the following theorem and later on. After the following theorem we dene another related concept. Two signed measures ν1 and ν2 on (X, A) are mutually singular, denoted ν1 ⊥ ν2 , if there exists A1 , A2 ∈ A such that A1 ∪ A2 = X, A1 ∩ A2 = ∅, ν1 (A2 ) = 0, ν2 (A1 ) = 0. Intuitively, this says that ν1 and ν2 assign mass in disjoint regions of the space. The previous theorem divided the space X in two parts on which the measure ν is sign compatible". In the next theorem we want to plug, in each part, positive measures µ1 and µ2 whose dierence equals ν , and it is reasonable to expect that µ1 and µ2 are mutually singular.. Theorem 2.1.45 (Jordan decomposition). unique positive measures. ν+. and. ν−. on. Let. (X, A). ν. be a signed measure on ν = ν + − ν − and. such that. (X, A). Then ν + ⊥ ν −.. there exists. ν + , and ν − , are natural. That is, use Hahn − decomposition for ν to nd P and N , dene ν (A) = ν(A ∩ P ), and ν (A) = −ν(A ∩ N ). As + − + − dened, ν and ν are positive measures such that ν = ν − ν . Uniqueness follow from the essential As a sketch proof, note that the candidates for. +. uniqueness of Hahn decomposition. With the Jordan decomposition of variation of. +. ν , ∣ν∣ = ν + ν. −. ν = ν + − ν −,. we can dene a positive measure called the total. .. A denition that will come up later side by side with mutual singularity is absolute continuity.. ν on (X, A) is absolutely continuous with respect to a positive ν ≪ µ, if µ(E) = 0 ⇒ ν(E) = 0. Clearly ν ≪ µ ⇔ ∣ν∣ ≪ µ ⇔ ν + ≪ µ. µ on ν ≪ µ.. A signed measure. measure. (X, A),. and. denoted. −. We should interpret mutual singularity and absolute continuity as antithetical, in the sense that if. ν⊥µ. and. ν ≪ µ,. then. ν ≡ 0.. The following proposition characterizes absolute continuity under niteness and explains why we call it like that.. Proposition 2.1.46 signed measure and. ∣ν∣(A) < . whenever. . Let ν be a nite ν ≪ µ ⇔ ∀ > 0, ∃δ > 0 such that. (Characterization of absolute continuity under niteness). µ a positive µ(A) < δ .. measure on. (X, A).. Then. A direct and important use of the concept of total variation, introduced above, is to write ν f dµ for some positive measure µ and µ-quasi-integrable f . For such, use Hahn's decomposition X = P ∪ N and Jordan's decomposition ν = ν + − ν − , and let be µ = ∣ν∣ and f = XP − XN . Note f dµ(A) = ∫A XP − XN d(ν + + ν − ) = ∫A XP dν + − ∫A XN dν + + ∫A XP dν − − ∫A XN dν − = ν + (A ∩ P ) − ν + (A ∩ N ) + ν − (A ∩ P ) − ν − (A ∩ N ) = ν + (A) − ν − (A) = ν(A). Therefore ν = f dµ. We can easily dene integrals with respect to signed measures. Let ν be a signed measure on ¯ ), such (X, A) and f be a measurable function on (X, A) (taking values in K = C, [0, ∞] or R 1 + 1 − + − 1 that f ∈ L (ν ) or f ∈ L (ν ). Now dene ∫ f dν ∶= ∫ f dν − ∫ f dν ∈ K . Here L (ν, K) is the 1 set of K -valued functions whose integral is dened and nite. It is no surprise that L (ν, K) = 1 + 1 − L (ν , K) ∩ L (ν , K) (when K = C this is less important, since every function to which an integral 1 ∫ ⋅dη is assigned is in L (η, C)). Note that we can still assign (innite) integrals to some functions 1 ¯ ). outside L (ν, K) (when considering K = [0, ∞] or R as. Proposition 2.1.47. .. (Properties of the total variation of signed measures). Let. (X, A). Then: ∣ν(A)∣ ≤ ∣ν∣(A), ∀A ∈ A. 1 ¯ = L1 (∣ν∣, R) ¯ and f ∈ L1 (ν, R) ¯ ⇒ ∣ ∫ f dν∣ ≤ ∫ ∣f ∣d∣ν∣. (ii) L (ν, R) (iii) ∣ν∣(A) = sup{∣ ∫A f dν∣ ∶ ∣f ∣ ≤ 1}. (iv) if ν1 and ν2 both omit an innity of the same sign, ∣ν1 + ν2 ∣ ≤ ∣ν1 ∣ + ∣ν2 ∣.. measures on (i). ν, ν1 , ν2. be signed.