1 INTRODUCTION TO MEASURE THEORY AND LEBESGUE INTEGRATION Eduard EMELYANOV Ankara | TURKEY 2007 2 FOREWORD This book grew out of a one-semester course for graduate students that the author have taught at the Middle East Technical University of Ankara in 2004- 06. It is devoted mainly to the measure theory and integration. They form the base for many areas of mathematics, for instance, the probability theory, and at least the large part of the base of the functional analysis, and operator theory. Under measure we understand a σ-additive function with values in R+ [ f1g defined on a σ-algebra. We give the extension of this basic notion to σ-additive vector-valued measures, in particular, to signed and complex measures and refer for more delicate topics of the theory of vector-valued measures to the excellent book of Diestel and Uhl [3]. The σ-additivity plays a crucial role in our book. We do not discuss here the theory of finitely-additive functions defined on algebras of sets which are sometimes referred as finitely-additive measures. There are two key points in the measure theory. The Caratheodory extension theorem and construction of the Lebesgue integral. Other results are more or less technical. Nevertheless, we can also emphasize the importance of the Jor- dan decomposition of signed measure, theorems about convergence for Lebesgue integral, Cantor sets, the Radon { Nikodym theorem, the theory of Lp-spaces, the Liapounoff convexity theorem, and the Riesz representation theorem. We provide with proofs only basic results, and leave the proofs of the others to the reader, who can also find them in many standard graduate books on the measure theory like [1], [4], and [5]. Exercises play an important role in this book. We expect that a potential reader will try to solve at least half of them. Exercises marked with ∗ are more difficult, and sometimes are too difficult for the first reading. In the case if someone will use this book as a basic text-book for the analysis graduate course, the author recommends to study subsections marked with ◦, omitting the material of subsections marked with !. I am indebted to many. I thank Professor Safak Alpay for reading the manuscript and offering many valuable suggestions. I thank to our students of 2004-2006 METU graduate real analysis classes, especially Tolga Karayayla, who made many corrections. Finally, I thank to my wife Svetlana Gorokhova for helping in preparing of the manuscript. 3 Contents Introduction Chapter 1 1.1. σ-Algebras, Measurable Functions, Measures, the Egoroff Theorem, Ex- haustion Argument 1.2. Vector-valued, Signed and Complex Measures, Variation of a Vector-valued Measure, Operations with Measures, the Jordan Decomposition Theorem, Ba- nach Space of Signed Measures of Bounded Variation 1.3. Construction of the Lebesgue Integral, the Monotone Convergence Theo- rem, the Dominated Convergence Theorem, Chapter 2 2.1. The Caratheodory Theorem, Lebesgue Measure on R, Lebesgue { Stieltjes Measures, the Product of Measure Spaces, the Fubini Theorem 2.2. Lebesgue Measure on Rn, Lebesgue Integral in Rn, the Lusin Theorem, Cantor Sets Chapter 3 3.1. The Radon { Nikodym Theorem, Continuity of a Measure with Respect to another Measure, the Hahn Decomposition Theorem 3.2. H¨older'sand Minkowski's Inequalities, Completeness, Lp-Spaces, Duals 3.3. The Liapounoff Convexity Theorem Chapter 4 4.1. Vector Spaces of Functions on Rn, Convolutions 4.2. Radon Measures, the Riesz Representation Theorem Bibliography Index 4 Chapter 1 1.1 σ-Algebras, Measurable Functions, Measures Here we define the notion of σ-algebra which plays the key role in the measure theory. We study basic properties of σ-algebras and measurable functions. In the end of this section we define and study the notion of measure. 1.1.1.◦ σ-Algebras. The following notion is the principal in the measure theory. Definition 1.1.1 A collection A of subsets of a set X is called a σ-algebra if (a) X 2 A; (b) if A 2 A then X n A 2 A; S (c) given a sequence (Ak)k ⊆ A, we have Ak 2 A. k It follows from this definition that the empty set ; belongs to A since X 2 A and ; = X n X. Further, given a sequence (Ak)k ⊆ A, we have \ [ Ak = X n (X n Ak) 2 A: k k Finally, the difference A n B := A \ (X n B) and the symmetric difference A4B := (A n B) [ (B n A) both belong to A. Any σ-algebra of subsets of a set X has at least two elements: ; and X itself. One of the main and mostly obvious examples of a σ-algebra is P(X) is the set of all subsets of X. The following simple proposition shows us how to construct new σ-algebras from a given family of σ-algebras. 5 6 CHAPTER 1. Proposition 1.1.1 Let fΩαgα2A be a nonempty family of σ-algebras in P(X), then Ω = \αΩα is also a σ-algebra. 2 This proposition leads to the following definition. Definition 1.1.2 Let G ⊆ P(X), then the set of all σ-algebras containing G is nonempty since it contains P(X). Hence we may talk about the minimal σ- algebra containing G. This σ-algebra is called the σ-algebra generated by G. An important special case of this notion is the following. Definition 1.1.3 Let X be a topological space and let G be the family of all open subsets of X. The σ-algebra generated by G is called the Borel algebra of X and denoted by B(X). 1.1.2.◦ Measurable functions. Let f be a real-valued function defined on a set X. We suppose that some σ-algebra Ω ⊆ P(X) is fixed. Definition 1.1.4 We say that f is measurable, if f −1([a; b]) 2 Ω for any reals a < b. The following three propositions are obvious. Proposition 1.1.2 Let f : X ! R be a function. Then the following conditions are equivalent: (a) f is measurable; (b) f −1([0; b)) 2 Ω for any real b; (c) f −1((b; 1)) 2 Ω for any real b; (d) f −1(B) 2 Ω for any B 2 B(R). 2 Proposition 1.1.3 Let f and g be measurable functions, then (a) α · f + β · g is measurable for any α; β 2 R; (b) functions maxff; gg and f · g are measurable. In particular, functions f + := maxff; 0g, f − := (−f)+, and jfj := f + + f − are measurable. 2 1.1. σ-ALGEBRAS, MEASURABLE FUNCTIONS, MEASURES 7 1 Proposition 1.1.4 Let (fn)n=1 be a sequence of measurable functions, then lim sup fn and lim inf fn n!1 n!1 are measurable. 2 1.1.3.◦ Measures. The main definition here is the following. Definition 1.1.5 Let A be a σ-algebra. A function µ : A! R [ f1g is called a measure, if: (a) µ(;) = 0; (b) µ(A) ≥ 0 for all A 2 A; and S P (c) µ( k Ak) = k µ(Ak) for any sequence (Ak)k of pairwise disjoint sets from A, that is Ai \ Aj = ; for i 6= j. The axiom (c) is called σ-additivity of the measure µ. As usual, we will also assume that any measure under consideration satisfies the following axiom: (d) for any subset A 2 A with µ(A) = 1, there exists B 2 A such that B ⊆ A and 0 < µ(B) < 1. This axiom allows to avoid a pathological case that for some A, with µ(A) = 0 it holds either µ(B) = 0 or µ(B) = 1 for any B ⊆ A, B 2 A. We will use often the following simple proposition, the proof of which is left to the reader. Proposition 1.1.5 Let µ be a measure on a σ-algebra A, An 2 A, and An ! A. 1 Then A 2 A and µ(A) = lim µ(An). In particular, if (Bn)n=1 is a decreasing n!1 T1 sequence of elements of A such that n=1 Bn = ;, then µ(Bn) ! 0. 2 1.1.4.◦ Measure spaces. We consider a fixed but arbitrary σ-algebra with a measure. Definition 1.1.6 If A is a σ-algebra of subsets of X and µ is a measure on A, then the triple (X; A; µ) is called a measure space. The sets belonging to A are called measurable sets because the measure is defined for them. 8 CHAPTER 1. Now we present several examples of measure spaces. Example 1.1.1 Let X = fx1; : : : ; xN g be a finite set, A be the σ-algebra of all subsets of X and a measure is defined by setting to each xi 2 X a nonnegative number, say pi. This follows that the measure of a subset fxα1 ; : : : ; xαk g ⊆ X is just pα1 + ··· + pαk . If pi = 1 for all i, then the measure is called a counting measure because it counts the number of elements in a set. Example 1.1.2 If X is a topological space, then the most natural σ-algebra of subsets of X is the Borel algebra B(X). Any measure defined on a Borel algebra is called Borel measure. In Section 2:1, we will prove that on the Borel σ- algebra B(R) there exists a unique measure such that µ([a; b]) = b − a for any interval [a; b] ⊆ R. Whenever we consider spaces X = [a; b], X = R, or X = Rn, we usually assume that the measure under consideration is the Borel measure.