Measure and Integration

Transcription

1 Wayne State University Mathematics Faculty Research Publications Mathematics Measure and Integration Jose L. Menaldi Wayne State University, Recommended Citation Menaldi, Jose L., "Measure and Integration" (2016). Mathematics Faculty Research Publications. Paper This Book is brought to you for free and open access by the Mathematics at It has been accepted for inclusion in Mathematics Faculty Research Publications by an authorized administrator of

2 Measure and Integration 1 Jose-Luis Menaldi 2 Current Version: 11 November First Version: c Copyright No part of this book may be reproduced by any process without prior written permission from the author. 2 Wayne State University, Department of Mathematics, Detroit, MI 48202, USA ( menaldi@wayne.edu). 3 Long Title. Measure and Integration: Theory and Exercises 4 This book is being progressively updated and expanded. If you discover any errors or you have any improvements to suggest, please the author.

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4 Contents Preface Introduction v vii 0 Background Cardinality Frequent Axioms Metrizable Spaces Some Basic Lemmas Measurable Spaces Classes of Sets Borel Sets and Topology Measurable Functions Some Examples Various Tools Measure Theory Abstract Measures Caratheodory s Arguments Inner Approach Geometric Construction Lebesgue Measures Measures and Topology Borel Measures On Metric Spaces On Locally Compact Spaces Product Measures Integration Theory Definition and Properties Cartesian Products Convergence in Measure Almost Measurable Functions iii

5 iv Contents 4.5 Typical Function Spaces Integrals on Euclidean Spaces Multidimensional Riemann Integrals Riemann-Stieltjes Integrals Diadic Riemann Integrals Lebesgue Measure on Manifolds Hausdorff Measure Area and Co-area Formulae Measures and Integrals Signed Measures Essential Supremum Orthogonal Projection Uniform Integrability Representation Theorems Elements of Real Analysis Differentiation and Approximation Partition of the Unity Lebesgue Points Functions of one variable Lebesgue Spaces Trigonometric Series Some Complements Appendix - Solutions to Exercises Measurable Spaces Measure Theory Measures and Topology Integration Theory Integrals on Euclidean Spaces Measures and Integrals Elements of Real Analysis Notation 391 Bibliography 395 Index 403

6 Preface This project has several parts, of which this book is the first one. The second part deals with basic function spaces, particularly the theory of distributions, while part three is dedicated to elementary probability (after measure theory). In part four, stochastic integrals are studied in some details, and in part five, stochastic ordinary differential equations are discussed, with a clear emphasis on estimates. Each part was designed independent (as possible) of the others, but it makes a lot of sense to consider all five parts as a sequence. The last two parts are derived from a previous course to supplement stochastic optimal control theory, while the first three parts of these lectures begun when preparing and teaching a short course in Elementary Probability with a first introduction to measure theory at the University of Parma (Italy) during the Winter Semester of Later preparing to teach our regular Real Analysis series at Wayne State University during the academic year 2007, and after reviewing many books with suitable material, I decided to enlarge and to complete my notes instead of adopting one of books commonly used here and there. Clearly, there are many excellent books from which a (two semesters) Real Analysis course can be taught, but perhaps each instructor has a unique opinion and a particular selection of topics. Nevertheless, most of instructors will agree (in principle) with a selection of sections included in this book. As mentioned earlier, this course grew out of an interest in Probability, but without rushing throughout the measure and integration (theory), what in most cases is the difference between students in analysis with a pure interest versus a more applied orientation. Thus, the reader will note a subtle insistence in the extension of measures form a semi-ring, in some general properties of measures on topological spaces (a chapter that can be skipped during the first reading) and in various spaces of measures and measurable functions. In a way, the approach is to cover first the essential and to deal later with the complements. Most of the style is formal (propositions, theorems, remarks), but there are instances where a more narrative presentation is used, the purpose being to force the student to pause and fill-in the details. Practically, there are no specific section of exercises, giving to the instructor the freedom of choosing problems from various sources (and according to a particular interest of subjects) and reinforcing the desired orientation. There is no intention to diminish the difficulty of the material to put students at ease, on the contrary, all points are presented as blunt as possible, even sometimes v

7 vi Preface shorten some proofs, but with appropriate references. For instance, we assume that most of Chapter 0 (the background) is somehow known, even if not all will be actually needed, but it is preferable to warn earlier the reader of the deep analysis ahead. The first three chapters (1, 2 and 4) give the basic stuff about abstract measure and integration, while Chapter 5 and 6 complement the material in two opposite directions. Chapter 3 is a little more demanding. Finally, in Chapter 7, we are ready to see the results of the theory. This book is written for the instructor rather than for the student in a sense that the instructor (familiar with the material) has to fill-in some (small) details and selects exercises to give a personal direction to the course. It should be taken more as Lecture Notes, addressed indirectly (via an instructor) to the student. In a way, the student seeing this material for the first time may be overwhelmed, but with time and dedication the reader can check most of the points indicated in the references to complete some hard details. Perhaps the expression of a guided tour could be used here. In the appendix, all exercises are re-listed by section, but now, most of them have a (possible) solution. Certainly, this appendix is not for the first reading, i.e., this part is meant to be read after having struggled (a little) with the exercises. Sometimes, there are many ways of solving problems, and depending of what was developed in the theory, solving the exercises could have alternative ways. The instructor will find that some exercises are trivial while other are not simple. It is clear that what we may call Exercises in one textbook could be called Propositions in others. This part one has a large number of exercises, but as the material get more complicated (i.e., in several chapters in parts two and three), a few or not at all exercises are given. The combination of parts I, II, and III is neither a comprehensive course in measure and integration (but a certain number of generalizations suitable for probability are included), nor a basic course in probability (but most of language used in probability is discussed), nor a functional analysis course (but function spaces and the three essential principles are addressed), nor a course in theory of distribution (but most of the key component are there). One of the objectives of these first three books is to show the reader a large open door to probability (and partial differential equations), without committing oneself to probability (or partial differential equations) and without ignoring hard parts in measure and integration theory. Michigan (USA), Jose-Luis Menaldi, June 2015

8 Introduction As indicated by the title, these lecture notes concern measure and integration theory. The objective is the d-dimensional Lebesgue integral, but in going there, some general properties valid for measures in metric spaces are developed. Instead of taking a direct way to reach our goal (after a background chapter), we prefer a more systematic approach in which family of sets and measurable functions are presented first, without a direct motivation. Chapter 2 (abstract measures) begins with the classic Caratheodory construction (with the typical example of the Lebesgue measure), followed by the inner measure approach and a little of the geometric construction. Chapter 3 (can be skipped in the first reading) continues with Borel measures (which can also be used to establish more specific properties on the Lebesgue measure). Basic on integrals is developed in Chapter 4, where the convergence theorem are obtained, and in Chapters 5 and 6, we give some complement on measure and integration theory (including Riemann-Stieltjes integrals and signed measures). Finally, in Chapter 7, we consider most of the useful results applicable to Euclidean d-dimensional spaces. As much as possible, each section is kept independent of other sections in the same chapter. For instance, to have a quick historical evolution of ideas within the integrals (from Cauchy to Lebesgue), the interested reader may take a look at the book Burk [23, Chapter I] or Chae [25, Chapter I and II]. Perhaps checking most of Gordon [54], the reader may find a detailed discussion on various type of integrals. Now, before going into more details and as a preview of what is to come, a quick discussion on discrete measures (including the typical convergence theorems) is in order. Essentially based on property of the sup, recall that for any series of nonnegative real numbers i=1 a i, with a i 0, the sum i=1 a i is the sup{ n i=1 a i : n 1} and therefore: (1) if ι is a bijective function from the positive integers into themselves then i=1 a i = i=1 a ι(i); (2) if I 1, I 2,... is a partition (finite or non) of the positive integers then i=1 a i = n i I n a i. It is convenient to use a discrete example as a motivation for our discussion. Let be a non empty set and denote by 2 the family of all the subsets of, and then, choose a finite or at most countable subset I of and a sequence of strictly positive real numbers {a i : i I}. Consider m : 2 [0, ] defined by m(a) = i I a i1 A (i), where 1 A (i) = 1 {i A} is equal to 1 only if i A and zero otherwise. Note that initially, all a i are nonnegative real numbers, but we vii

9 viii Preface can include the symbol + preserving the linear order and the meaning of the series, i.e., all series have nonnegative terms, a converging series has a finite value and a non convergence (divergent) series has the symbol + as its value. Therefore we have by definition (1) m( ) = 0 and the following property (so-called σ-additivity), (2) if A = i=1 A i with A i A j = for any i j, then m(a) = i=1 m(a i). This function (defined on sets) m is called a discrete measure, the set I is the set of atoms and a i is the measure (or weight) of the atom i. Clearly, to define m we need only to know the values m({i}) for any i in the finite or countable set I. An element N of 2 is called negligible with respect to m if m(n) = 0. In the case of discrete measures, any subset of N of I is negligible. If m() = 1 then we say that m is a discrete probability measure. A function f : R is called integrable with respect to m if the series i I f(i) m({i}) = i I f(i) a i converges, and in this case the integral with respect to m is defined as the following real number: fdm = fdm = f(i) m({i}) = f(i) a i. i I i I Even if the series diverges, if f is nonnegative then we can define the integral as above (a nonnegative number when the series converges or the symbol + otherwise). Next, a function f is called quasi-integrable with respect to m if either the positive part f + or the negative part f is integrable. The class of integrable functions is denoted by L or L(, 2, m) if necessary. A couple of properties are immediately proved for the integral: (1) if c R and f, g L then cf + g L and (cf + g)dm = c fdm + gdm; (2) if f g, quasi-integrable then fdm gdm; (3) f L if and only if f L and in this case fdm f dm; (4) if f 0 and fdm = 0 then f = 0 except in a negligible set. There are three main ways of taking limit inside the integral for a sequence f n of functions: (a) Beppo Levi s monotone convergence: if 0 f n f n+1 and f(x) = lim n f n (x) for any x then fdm = lim n fn dm; (b) Fatou s lemma: suppose that f n 0 and f(x) = lim inf n f n (x) then fdm lim infn fn dm; (c) Lebesgue s dominated convergence: if f n g with g L and f(x) = lim n f n (x) exists for any x then fdm = lim n fn dm. Essentially, any one of the three theorems can be deduced from any of the others. For instance, use (a) on g n (x) = inf k n f k (x) to get (b) and use (b) on g ± f n to deduce (c). To prove (a), for any number C < fdm there exists a finite set J I of atoms such that i J f(i) m({i}) > C. Since J is finite, for every ε > 0 there exists N = N(ε, C, J) such that i J f n(i) m({i}) > C ε for every n N. Because f n 0 and C, ε are arbitrary we deduce fdm lim n fn dm, and the equality follows. For instance, if = {1, 2,...} is the set of strictly positive integers then m({i}) = 2 i defines a probability measure. Consider the sequence of functions

10 Preface ix {f n } given by f n (x) = 2 n 1 {n=x}. We can verify that each f n is integrable, and that the sequence {f n } converges to the function identically zero but f n dm = 1 for every n, i.e., we cannot use (a) or (c) and the inequality in (b) is strict. Negligible sets (or sets of measure zero) do not play an essential role for discrete measures since there is a largest set of measure zero, namely I, i.e., for a given discrete measure m with atoms I we may ignore the complement of I. However, in general, negligible sets are a fundamental part, i.e., almost everything happens except a set of measure zero, referred to as almost everywhere (a.e.) or almost surely (a.s.) when a probability is used. The generalization of these arguments is the basis of measure and integration theory. However, before being able to reach the classic example of the Lebesgue- Borel measure and the extension of the Riemann-Stieltjes integral, several points should be discussed. Certainly, there are many (text)books on measure theory and integration at various level of difficulties (too many to make a non exhaustive reasonable list), but let me mention that the classic textbooks Halmos [57] and Natanson [86] (among others) were available (besides lecture notes) for me (when studying this subject for my first time). Now, for the (student) reader that just want to take a (serious) panoramic view, (among other) the textbooks Bass [8] and Pollard [90] are an excellent option; and in Richardson [93, Chapter 1, pp. 1 9], a quick history on the subject is given. Perhaps the reader may be interested in a concrete view (and less traditional) of the material, e.g. Swartz [113, Chapters 1 4, pp ]. In several parts of this lecture notes, there are precise references to (text)books (with pages) that the reader may check to enlarge details of proofs or viewpoints (and discussions) on the various aspects of measure (and integration) theory covered in this manuscript.

11 x Preface

12 Chapter 0 Background Before going into some details and without really discussion set theory (e.g. see Halmos [58]) we have to consider a couple of issues. On the other hand, we need to recall some basic topology, e.g. most of what is presented in the preliminaries section of Yosida [122, Chapter 0, pp. 1 22]. Alternatively, the reader may check part of the material in Royden [98, Chapters 1,2,7,8 and 9] or Dshalalow [36, Part I, pp ]. Certainly, being familiar with most of the material developed in basic mathematical analysis books (e.g., Apostol [4] or Hoffman [63] or Rudin [99]) yields a comfortable background, while being familiar with the material covered in elementary mathematical analysis books (e.g., Kirkwood [70] or Lewin [76] or Trench [116]) provides an almost sufficient background. It not the intention to write a comprehensive treatment of Measure Theory (as e.g., the five volumes of Fremlin [46]), but the reader may find a little more than expected. Reading what follows for the first time could be very dense, so that the reader should have some acquaintance with most of the concepts discussed in this preliminary chapter. Clearly, not every aspect (of this preliminary chapter) is needed later, but it is preferred to face these possible difficulties now and not later, when the actual focus of interest is revelled. 0.1 Cardinality We want to count the number of element of any set. If a set is finite, then the number of elements (also called the cardinal of the set) is a natural or nonnegative integer number (zero if the set is empty). However, if the set is infinite then some consideration should be made. To define the cardinal of a set in general, we say that two sets have the same cardinal or are equipotent if there exits a bijection between them. Since equipotent is a reflexive, symmetric and transitive relation, we have equivalence classes of sets with the same cardinal. Also we say that the cardinal of a set A is not greater than the cardinal of another set B if there is a injective function from A into B, usually denoted by card(a) card(b), and we can show that card(a) card(b) and card(b) 1

13 2 Chapter 0. Background card(a) imply that A and B have the same cardinal. Similarly, if A and B have the same cardinal then we write card(a) = card(b), and also card(a) < card(b) with obvious meaning. We use the symbol ℵ 0 (aleph-nought) for the cardinal of N the set of natural (or nonnegative integer) numbers, and we show that ℵ 0 is first nonfinite cardinal. Any set in the class ℵ 0 is called countable infinite or denumerable, while countable sets, may be finite or infinite. With time, the two names countable and denumerable are used indistinctly and if necessary, we have to specify finite or infinite for countable sets. It can be shown that the integer numbers Z and the rational numbers Q are both countable sets, moreover, the union or the finite (Cartesian) product does not change the cardinality of infinite sets, e.g., if {A i : i 1} is a sequence of countable sets A i then i=1 A i and n i=1 A i are also countable, for any positive integer n. However, we can also show that the cardinal of 2 A, the set of the parts of a nonempty set A (i.e., the set of all subsets of A, which can be identified with the product {0, 1} A ) has cardinal strictly greater than card(a). Nevertheless, if A is an infinite set then the set composed by all subsets of A having a finite number of element (called the finite-parts of A) have the same cardinal as A. Indeed, if A = {1, 2,...} then the set 2 A of the F finite-parts of A can be represented (omitting the empty set) as finite sequences a = {a 1,..., a n } of elements a i in A. Thus, if {2, 3, 5, 7,..., p i,...} is the sequence of all prime numbers then for each a there is a unique positive integer m = 2 a1 3 a2 5 a3 7 a4 p an n, and because the factorization in term of the prime numbers is unique, the mapping a m is one-to-one, i.e., 2 A is countable. F Representing real numbers in binary form, we observe that 2 {0,1,...} has the same cardinal as the real numbers R, which is strictly greater than ℵ 0. The cardinality of R is called cardinality of continuum and denoted by 2 ℵ0. However, we do not know whether or not there exists a set A with cardinal ℵ such that ℵ 0 < ℵ and ℵ < 2 ℵ0. Anyway, it is customary to use the notation ℵ 1 = 2 ℵ0, ℵ 2 = 2 ℵ1, and so on. The (generalized) continuum hypothesis states that for any infinite set A there is no set with cardinal strictly between the cardinal of A and the cardinal of 2 A. In particular for ℵ 0 and 2 ℵ0, this assumption (so-called continuum hypothesis) has an equivalent formulation as follows: the set R can be well-ordered in such a way that each element of R is preceded by only countably many elements, i.e., there is a relation satisfies (a) for any two real numbers x and y we have x y or y x or x = y (linear order), (b) for every real number x we have x x (reflexive), and (c) every nonempty subset of real numbers A has a first number, i.e., there exist a 0 in A such that a 0 a for any a in A (well-ordered), and the extra condition (d) for every real number x the set of real numbers y x is a countable set. It can also be proved that the cardinal number of R (which is called the continuum cardinal) is indeed the cardinal of 2 N = ℵ 1 (i.e., the part of N). Furthermore, the continuum hypothesis (i.e., there is not cardinal number between ℵ 0 and the continuum) is independent of the axioms of set theory (i.e., it cannot be proved or disproved using those axioms). For instance, the reader may check the books Ciesielski [26], Cohen [27], Gol-

14 0.2. Frequent Axioms 3 drei [53], Moschovakis [84] or Smullyan and Fitting [107], for a comprehensive treatment in cardinality and set theory axioms. Also Pugh [91, Chapter 1, pp. 1 50] or Strichartz [111, Chapter 1, pp. 1 24] is a suitable initial reading. 0.2 Frequent Axioms Sometimes, we need to differentiate sets with the same cardinal based on other characteristics, e.g., a natural order of numbers or the natural inclusion for collection or family of sets. A partially ordered set (X, ) is a set X (family of sets) and a relation, which is transitive (a b and b c imply a c) and antisymmetric (a b and b a imply a = b). An order (on a set X) is called (1) linear (or total, and the set X is linearly ordered) if for every a and b in X we have either a b or b a, and (2) well-order (and the set X is well-ordered) if (1) holds and any nonempty subset A of X has a minimum element, i.e., there is a in A such that a a, for every a in A. Certainly (several) well-order can be given to a finite set, and any subset of integer numbers with a finite infimum inhered a well-order from the integer numbers. A typical situation is to partially order a collection of sets with the inclusion. Note that the natural order of the real numbers R is a linear order, but not a well-order. Also, the set R I of all real-valued functions on a set I (of more than one element), with the natural partial order (x i ) (y i ) if x i y i for every i in I, is not a linear order. In a partially ordered set not all elements are comparable, i.e., we may have two elements a and b such that neither a b nor b a. Thus, given a partially ordered set (X, ) and a subset A X, we say that x in X is an upper bound of A if a x for every a in A, and if x belongs to A then x is the maximum element of A. Maximum has little use for partially ordered set, instead, we say that m in A is a maximal element, chain of A if for any a in A such that m a we have a = m (i.e., m is larger or equal to any other element a in A that can be compared with m). A chain in X is a subset C X such that becomes a linear order on C. There are several equivalent ways of expressing the well-ordering principle (or axioms), e.g., Hausdorff Maximal Principle: Every partially ordered set (X, ) has a maximal chain, i.e., a subset C of X such that (C, ) is a linearly ordered set and (C, ) is not a linearly ordered set, for any subset C strictly containing C. Zorn s Lemma: Every nonempty partially ordered set has a maximal element if any chain has an upper bound. Zermelo s Axiom: Every set can be well-ordered, i.e., if X is a set, then there is some well-order on X, i.e., is a linear order (all elements in X are comparable) and every nonempty subset of X has a first element. A typical use is when a construction of sets satisfying some properties is

15 4 Chapter 0. Background partially ordered (e.g., by the inclusion), and we deduce the existence of a maximal set satisfying those properties. Related to the above assumptions, but independent from other axioms of the set theory, is the so-called Axiom of Choice (AoC), which can also be expressed in various equivalent ways, e.g., AoC Form (a): The Cartesian product of any nonempty family of nonempty sets must be a nonempty set, i.e., if {A i : i I} is a family of sets such that I and A i, for any i I then there exits at least one choice a i A i, for any i I. AoC Form (b): If {A i : i I} is a family of arbitrary nonempty disjoint sets indexed by a set I, then there exists a set consisting of exactly one element form each A i, with i I. All these axioms come into play when dealing with uncountable sets. Beside using cardinality to classify sets (mainly sets involving numbers), we may push further and classify well-ordered sets. Thus, similarly to the cardinality, we say that two well-ordered sets (X, ) and (Y, ) have the same ordinal if there is a bijection between them preserving the order. Thus, to each well-ordered set we associate an ordinal (an equivalence class). It clear that finite ordinals are the sets of natural numbers {1, 2,..., n}, n = 1, 2,..., with the natural order, and the first infinite ordinal is the set of natural numbers N = {1, 2,...} (or equivalently any infinite subset of integer number with a finite infimum). Each ordinal has a next ordinal, i.e., given an ordinal (X, ) we define (X + 1, ) to be X + 1 = X { }, with X and x, for every x in X. However, each ordinal not necessarily has a previous (or precedent) ordinal, e.g., there is not an ordinal X such that X + 1 = N. Similarly to cardinals, we say that the ordinal of a well-ordered set (X, ) is not greater than the ordinal of another well-ordered set (Y, ) if there is a injective function from X into Y preserving the linear order, usually denoted by ord(x) ord(y ). Thus, we can show that if ord(x) ord(y ) and ord(y ) ord(x) then (X, ) and (Y, ) have the same ordinal. Moreover, there are many properties satisfied by the ordinal (e.g., see Brown and Pearcy [21, Chapter 5, pp ], Kelley [67, Appendix, ]), we state for future reference Ordinal Order: The set (actually class or family of sets) of all ordinals is wellordered (the above order denoted by and the strict order by <, i.e., the and the ), namely, every nonempty subset (subfamily) of ordinals has a first ordinal. Moreover, given an ordinal x, the set {y < x} of all ordinal strictly precedent to x has the same ordinal as x. For instance, we may call ω 0 the first uncountable ordinal, i.e., any ordinal ω < ω 0 is countable (finite or infinite). It is clear that there are plainly of ordinals between N and ω 0. Thus, transfinite induction and recursion can be used with ordinals, i.e., first for any element a of a well-ordered set (X, ) define the initial segment of X determined by a, i.e., I(a) = {x X : x a, x a}, to have the

16 0.3. Metrizable Spaces 5 Transfinite Induction Principle: If a subset A of a well-ordered set X satisfies (for every a in X) the condition I(a) A a A then A is indeed the whole set, i.e., A = X. It is clear that if a is the minimum element in X A then by definition I(a) A and therefore a belongs to A. A neat case is when the well-ordered set X is actually the natural number, i.e., the ordinary mathematical induction. Similarly to the ordinary recursion argument, where a function f on the natural numbers can be defined by specifying f(0) and then defining f(n) in terms of f(0),..., f(n 1), we have the recursion principle for well-ordered set, e.g., see Dudley [37, Section I.3, pp ]. Alternatively, the reader may check Folland [45, Chapter 0, pp. 1 17], for a short and clean discussion on the above points. Also see Berberian [12, Chapter 1, pp. 1 85]. 0.3 Metrizable Spaces Recall that a topology τ on a nonempty set X is a collection (or class or family) of open subsets of X, such that (1) X and are open sets, (2) any union of open sets is an open set, (3) any finite intersection of open sets is an open set. It is not necessary to describe the whole family of open sets τ, usually it suffices to give a base for the topology τ, i.e., a subfamily β τ of open sets such that for any open set O τ and any point x in O there is a member B β in the base satisfying x B O. Also, if τ 1 and τ 2 and two topology on X then τ 1 is stronger (or finer) than τ 2, or equivalently, τ 2 is weaker (coarser) than τ 1 if τ 1 τ 2, i.e., every open set in (X, τ 1 ) is an open set in (X, τ 2 ). Thus, closed sets are complements of open sets, and the (sequential) convergence (also cluster, interior, boundary, compact, connect, dense, etc) is then defined. Clearly, an abstract space with a topology is called a topological space. Actually, remark that we use only with topological spaces where points are separated closed sets, i.e., Hausdorff spaces, so that these properties are implicitly assumed everywhere (even if it is seldom restated) in the text, unless explicitly mentioned otherwise. A metric on a set X is a function d : X X R satisfying for every x and y in X the following conditions: (a) d(x, y) 0 and d(x, y) = 0 if and only if x = y, (b) d(x, y) = d(y, x) and (c) d(x, y) d(x, z) + d(z, y) for every z in X. The couple (X, d) is called a metric space, which becomes a topological space with the open sets defined by means of the (base of) open balls B(x, r) = {y X : d(x, y) < r}, for any x in X and any r > 0. On the other hand, a metrizable space is a topological space X in which a metric can be defined (but not really used) so that (X, d) has a topology equivalent to the initial one given on X (where the topology has a simple characterization). In a metric space, for every x in X the countable family of balls B(x, 1/n), n = 1, 2,..., forms a neighborhood-base at x and so, the (X, d) topology is first-countable or it satisfies the first axiom of countability. In a first-countable topology (in particular in a metric space), we can use only convergent sequences to define its topology, i.e., a subset A of X is closed for the topology induced by the metric d if and only if for every sequence {a n : n 1} of points in A such

17 6 Chapter 0. Background that d(a n, a) 0 as n, for some a in X, we have that the limit point a belongs to A. On the other hand, a topological space X is called separable if there exists a countable dense subset Q X, i.e., Q is countable and its closure Q is the whole space X. Hence, a separable metric space (X, d) is second-countable or it satisfies the second axiom of countability, i.e., its contains a countable base, namely, the family of balls B(q, 1/n) with q in a countable dense set Q and n 1. Actually the converse is also true, i.e., a metric space (X, d) is secondcountable if and only if it is separable, and similarly, a topological space with a countable base is separable and metrizable. Moreover any locally compact (or vector topological) space with a countable base is metrizable, but certainly, the converse is false. For instance, the reader may want to take a quick look at the book Kelley [67, Chapter 4, ]. Recall that a topological space is called sequentially compact if every sequence admits a convergent subsequence. Any sequentially compact metric space is separable, and for a sequential space (i.e., where convergent sequences are used to define its topology), compactness and sequentially compactness are equivalent. Given a family {(X i, d i ) : i I} of metric spaces the product space X = i I X i is a topological space with the product topology which may not be metrizable. For a countable family I = {1, 2,...}, we may define the metric d(x, y) = i=1 2 i d i (x i, y i ) 1 + d i (x i, y y ), x = (x i), y = (y i ), which induces an equivalent topology on X, i.e., a countable product space X = i=1 X i is indeed metrizable with the above metric d. We may consider uniform continuity and Cauchy sequences in a metric space (X, d). Thus, (X, d) is complete if any Cauchy sequence has a limit, i.e., if d(x n, x m ) 0 as n, m then there exists x such that d(x n, x) 0 as n. If a space (X, d) is not complete then we can completed it in the same way as we pass from the rational number to the real numbers. However, the concept of completeness is not a topological property, i.e., on a given space we may have two metrics yielding equivalent topologies but only one of them is complete. Anyway, every compact metric space is complete. A complete separable metrizable (or separable completely metrizable) space is called a Polish space, i.e., a separable topological space X with a metric yielding a complete metric space (X, d). For instance, the space C(R d ) of all real-valued continuous functions is a Polish space with the metric d(f, g) = 2 n [ f g ] n, [ f ] n = sup f(x) }. 1 + [ f g ] n x n n=1 In probability, the sample spaces are Polish spaces, most of the times, we use the space of continuous functions from R into R d or the space of all cad-lag functions from R into R d, i.e., functions continuous from the right and having limits from the left.

18 0.3. Metrizable Spaces 7 A subset K of a metric space (X, d) is called totally bounded if for every ε > 0 there exists a finite number of points x 1,..., x n in K such that K n i=1 B(x i, ε), i.e., any x in K is within a distance ε from the set {x 1,..., x n }. It is very instructive (but no simple) to show that a subset K (of a complete metric space) is totally bounded if and only if the closure of K is compact, e.g., Yosida [122, Section 0.2, pp ]. A vector topological space has a topology compatible with the vector structure, i.e., such that the addition and the scalar multiplication (of vectors) are continuous operations (e.g., check the books Kothe [73] or Schaefer [103]). An example is the so-called locally convex spaces, and better, a normed space X, which is vector space with a norm, i.e., a nonnegative function defined on X such that (a) λx = λ x, for every x in X and λ in R, (b) x+y x + y, for every x and y in X, and (c) x 0 for every x in X and x = 0 only if x = 0. Given a norm, we define a metric d(x, y) = x y (but not any metric comes from a norm), which yields the topology. In a normed space, any set that can be covered by a ball is called a bounded set. But, only on finite dimensional normed spaces, any closed and bounded set is necessarily compacts. A complete normed space is called a Banach space. The space C b (X) of all real-valued (or complex-valued) bounded continuous functions on a Hausdorff topological space X, with the sup-norm f = sup { f(x) : x X }, is a typical example of an infinite dimensional Banach space. A topological space X is said to be locally compact if every point has a compact neighborhood, i.e., an open set with compact closure. This implies that for every point x in X and any open set U containing x there is another open set V containing x such that V U and the closure V is compact. A locally compact Banach space is necessarily a space of finite dimension (i.e., homeomorphic to some R d, d 1). Again, better than a norm is an inner or scalar product, i.e., a bilinear maps (, ) from X X into R satisfying (a) (λx+y, z) = λ(x, z)+(y, z), for every x, y in X and λ in R, (b) (x, y) = (y, x), for every x, y in X, and (c) (x, x) 0 for every x in X and (x, x) = 0 only if x = 0. From an inner product we can define a norm x = (x, x), indeed, by considering the discriminant of the positive quadratic form λ (x + λy, x + λ) we obtain the Cauchy inequality (x, y) x y, for every x and y, which yields the triangular inequality (b) for the norm. Certainly, not any norm comes from an inner product, indeed, a norm is derived form an inner product if and only if the parallelogram law is satisfied, i.e., x+y 2 + x y 2 = 2 x 2 +2 y 2, for every x and y in X, in which case the inner product is defined by the polarization identity x+y 2 = x 2 + y 2 +2R(x, y). A complete normed space where the norm comes from an inner product is called a Hilbert space. A typical infinite dimensional Hilbert space is l 2, the space of real-valued sequences a = {a n } satisfying n a2 n <, with the inner product (a, b) = n a nb n. A more elaborate example is space L 2 (K), with K a compact subset

19 8 Chapter 0. Background of R d, which is the completion of C b (K) = C(K), space of continuous functions, with the inner product (f, g) = f(x) g(x) dx. K By means of the theory of the integral we are able to study in great detail spaces similar to this one. Moreover, we may have complex Banach and Hilbert spaces, i.e., the vector space is on the complex field C, and λ denotes the modulus (instead of the absolute value) when λ belongs to C for the condition (a) of norm. In the case of a inner product, the condition (b) becomes (x, y) = (y, x), where the over-line means complex conjugate, i.e., the application (, ) is sesquilinear (instead or bilinear) with complex values. As seen later, general discussions of good measures are focus on a separable complete metrizable space (i.e., a Polish space), while general discussions of (nonlinear) functions are considered on locally compact spaces with a countable bases. As expected, the typical oversimplified example is R n. For instance, the reader may take a look at DiBenedetto[32, Chapters 1 and 2, pp. 1 64] or Hewitt and Stromberg [62, Chapter 2, pp ] or Royden [98, Chapters 1 and 2, pp. 1 53] or Rudin [101, Chapter 1, pp. 3 40], for a quick review on topology and continuous functions. Also most of Brown and Pearcy [21], and Pugh [91, Chapter 2, pp ] are a suitable initial reading. 0.4 Some Basic Lemmas There are a couple of very useful results, but for our treatment this may be considered on the side, such as Lemma 0.1 (Urysohn). Let A and B be two nonempty, disjoint and closed sets in a metric space (X, d). If a and b are two distinct real numbers then the function x g(x) = a + (b a)d(x, A) d(x, A) + d(x, B) is continuous and g(x) = a for any x in A and g(x) = b for any x in B. Proof. The distance from a point to a set is defined as d(x, A) = inf{d(x, y) : y A} and the triangular inequality yields that d(x, A) d(y, A) d(x, y), which proves that the above function g is continuous. Proposition 0.2 (Tietze s Extension). Let f be a bounded real-valued function defined on a closed subset C of a metric space (X, d). Then there exists a continuous extension g of f to the entire space X.

20 0.4. Some Basic Lemmas 9 Proof. A proof of Tietze s Extension is based on the following construction, which shows that an extension g is uniform limit in X of the series k g k of continuous functions. First, with a = sup C f(x) define A = {x C : f(x) a/3} and B = {x C : f(x) a/3} to find a continuous function g 1 : X [ a/3, a/3] satisfying f(x) g 1 (x) 2a/3, for every x in C. Next with f 1 = f g 1, a 1 = 2a/3 define A 1 = {x C : f 1 (x) a 1 /3} and B 1 = {x C : f 1 (x) a 1 /3} to find a continuous function g 2 : X [ a 1 /3, a 1 /3] satisfying f 1 (x) g 2 (x) 2a 1 /3, for every x in C. Finally, by induction it follows that f(x) n k=1 g k(x) 2 n a3 n, for every x in C and that g n (x) 2 n 1 a3 n, for any x. Statements in Lemma 0.1 and Proposition 0.2 are also valid for more general topological spaces, e.g., Hausdorff locally compact spaces and normal spaces. Another key result frequently used, is Weierstrauss approximation theorem, Theorem 0.3 (Weierstrauss). If f is a real-valued continuous function on the compact interval [a, b] then there exits a sequence of polynomials {p n } such that p n (x) f(x) uniformly on [a, b]. Proof. For a given real-valued continuous function f on [a, b] the change of variable y = (x a)/(b a) reduces the discussion to the case where the compact interval [a, b] is [0, 1]. Moreover, for f on [0, 1] the change of function g(x) = [f(x) f(0)] x[f(1) f(0)] yields g(0) = g(1) = 1. Briefly, it should be proven that for any real-valued continuous function on the real line R which vanishes outside the interval [0, 1] there exists a sequence of polynomials {p n } such that p n (x) f(x) uniformly on [0, 1]. Note that f is uniformly continuous on R. Consider the polynomials q n (x) = c n (1 x 2 ) n, n = 1, 2,..., where the coefficients c n are chosen so that 1 c n (1 x 2 ) n dx = 1, n 1. 1 Remark that the function (1 x 2 ) n 1+nx 2 vanishes at x = 0 and has derivative positive on the open interval (0, 1) to deduce that (1 x 2 ) n 1 nx 2, x [0, 1]. Hence, the calculation 1 1 (1 x 2 ) n dx = (1 x 2 ) n dx 2 2 1/ n 0 1/ n 0 (1 x 2 ) n dx (1 nx 2 )dx > 1 n implies that 0 c n 1/ n, for every n 1. Therefore, for any δ > 0, the following estimate 0 q n (x) n(1 δ 2 ) n, x [ 1, 1], with x δ,

21 10 Chapter 0. Background holds true. Next, define the polynomial p n (x) = 1 1 f(x + y)q n (y)dy = 1 0 f(z)q n (z x)dy, x [ 1, 1]. By continuity, for every ε > 0 there exists δ > 0 such that x y < δ implies f(x) f(y) < ε and also, f(x) C < for every x n [0, 1]. Thus p n (x) f(x) 1 2C f(x + y) f(x) q n (y)dy 1 δ 1 δ q n (y)dy + ε q n (y)dy + 2C δ 4C n(1 δ 2 ) n + ε, which shows that p n (x) f(x) uniformly on [0, 1]. 1 δ q n (y) Revising the above proof, it should be clear that the same argument is valid when the function f takes complex values and then the coefficients of the polynomials p n are complex. Remark 0.4. A vector space F of real-valued functions defined a nonempty set X, is called an algebra if for any f and g in F the product function fg belongs also to F. A family F of real-valued functions is said to separate points in X if for every x y in X there exists a function f in F such that f(x) f(y). If X is a Hausdorff topological space then C(X) the space of all continuous real-valued functions defined on X is an algebra that separate points. The so-called Stone- Weierstrauss Theorem states that if K is a compact Hausdorff space and F is an algebra of functions in C(K) which separate points and contains constants functions, then for every ε > 0 and any g in C(K) there exists f in F such that sup { f(x) g(x) : x X } = f g < ε, i.e., F is dense in C(K) for the sup-norm. Moreover, K is stable under the complex-conjugate operation then the same result is valid for complex-valued functions, e.g, check the books DiBenedetto [32, Sections IV.16 18, pp ], Rudin [99, Chapter 7, ] or Yosida [122, Section 0.2, pp. 8 11]. A typical application of Zorn s Lemma is the following: Lemma 0.5. Any linear vector space X contains a subset {x i : i I}, so-called Hamel basis for X, of linear independent elements such that the linear subspace spanned by {x i : i I} coincides with X. Proof. Consider the partially ordered set S whose elements are all the subsets of linearly independent elements in X, with the partial order given by the inclusion. If {A α } is a chain or totally ordered subset of S then A = α A α is an upper bound, since any finite number of element in A are linearly independent. Hence,

22 0.4. Some Basic Lemmas 11 Zorn s Lemma implies the existence of a maximal element, denoted by {x i : i I}. Now, for any x in X the set {x i : i I} {x} has to be linearly dependent and so, x is linear combination of some finite number of elements in {x i : i I}, as desired. For instance, the interested reader may check the books by Berberian [12], DiBenedetto [32], Dshalalow [36], Dudley [37], Hewitt and Stromberg [62], Royden [98], among many others. On the other hand, the reader may take a look (again) at the books by Apostol [4], Dieudonné [33] or Rudin [99]. Essentially, as a background for what follows, the reader should be familiar with most of the basic material covered in Taylor [114, Chapters 1-3, pp ].

23 12 Chapter 0. Background

24 Chapter 1 Measurable Spaces Discrete measures were briefly presented in the introduction, where we were able to measure any subset of the given (possible uncountable) space, essentially, by counting its elements. However, in general (with non-discrete measures), due to the requirement of σ-additivity and some axioms of the set theory, we cannot use (or measure) every element in 2. Therefore, a first point to consider is classes (or family or collection or systems) of subsets (of a given space ) suitable for our analysis. We begin with a general discussion on infinite sets and some basic facts about functions. 1.1 Classes of Sets Let be a nonempty set and 2 be the parts of, i.e., set of all subsets of. Clearly, if has n elements then 2 has 2 n elements, but our interest is when has an infinite number of elements, for instance if is countable infinite (i.e., it is in a one-to-one relation with the positive integers) then 2 has the cardinality of the continuum. A class (collection or family or system) of sets is a subset of 2, that by convenience, we assume it contains the empty set. Note that and, 2. Typical operations between two elements A and B in 2 are the intersection A B, the union A B, the difference A B and the complement A c = A. The union and the intersection can be extended to any number of sets, e.g., if A i 2 for i in some sets of indexes I then we have i I A i and i I A i. Sometimes, to simplify notation we write A + B or i I A i (for disjoint unions) to express the fact that A + B = A B with A B = or i I A i = i I A i with A i A j = if i j. Below we introduce a number of elementary (or intermediary) classes, which are used later in the following Chapters, when the extension questions are addressed. Definition 1.1. Given classes P, L, R and A of subsets of, each containing, we say that P is a π-class if A, B P implies A B P, 13

25 14 Chapter 1. Measurable Spaces L is a l-class (or additive class) if (a) A, B L with A B = implies A B L and (b) A, B L with A B implies B A L, R is a ring if A, B R implies (a) A B R and (b) A B R, A is algebra if (a) A A implies A c A and (b) A, B A implies A B A. Finally, a π-class S is called (1) a semi-ring if A, B S with A B implies B A = n i=1 C i with C i S, (2) a semi-algebra if A S implies A c = n i=1 C i with C i S, and (3) a lattice if A, B S implies A B S. Usually, a field is defined as an algebra, but the name l-class or additive class is not completely standard in the literature. By induction, a π-class is stable under the formation of finite intersections, and a lattice is also stable under the formation of finite unions. Similarly, l-classes are stable under the formation of finite disjoint unions, and rings and algebras are stable under the formation of finite unions. Further, l-classes are stable under the formation of monotone differences, rings are stable under the formation of all differences, and algebras are stable under the formation of complements. Since A B = [A B] [(A B) (B A)], A B = (A c B c ) c and A B = (A c B) c, rings and algebras are also stable under finite intersections, and stable under the formation of complements implies stable under differences. Thus, any algebra is a ring, and every ring is simultaneously a π-class, an l-class, and a lattice. The equality A B = ((A (A B)) B implies that any l-class which is also a π-class is necessarily a ring. Also, a ring containing the whole space is indeed an algebra, and a lattice is not necessarily an l-class, e.g., the class {, F, } (with F not empty and different from ) is a lattice but not an l-class. From the definitions, it is clear that any interception of π-classes, l-classes, lattices, rings or algebras is again a π-class, an l-class, a lattice, a ring or an algebra. Therefore, given any subset G of 2 we may define the π-class, l-class, lattice, ring or algebra generated by G, e.g., the algebra A(G) generated by G is indeed the intersection of all algebras containing G. For instance, if A and B are subsets of then the minimal classes P, L, R and A containing A and B (as the name suggests) are P = {, A, B, A B}, L = {, A, B, A B if B A, B A if A B, A B if A B = }, R = {, A, B, A B, A B, B A, A B, (A B) (B A)}, and A = {, A, B, A B, A B, A c, B c, A c B c, A c B c, A c B, A c B, A B c, A B c, }. Certainly, interesting cases are when an infinite number of subsets of are involved. Exercise 1.1. Prove that the algebra A (ring) generated by a S semi-algebra (semi-ring) is the class of finite disjoint unions of sets in S, i.e., A A if and only if A = n i=1 A i with A i S. Hint: prove first that the class of finite disjoint unions of sets in S is stable under the formation of finite intersections. Similarly, remark that lattice L generated by a π-class P is the class of finite unions of sets in P, i.e., A L if and only if A = n i=1 A i with A i L. As seen later, a lattice of interest is the class L 2 R of all finite unions of closed intervals, while a semi-ring of interest for us is the class S of intervals of the form (a, b], with a, b real numbers, where the previous Exercise 1.1 can be applied.