Real Analysis Jesse Peterson November 8, 2016 2 Contents 1 Preliminaries 5 1.1 Sets . 5 1.1.1 Countability . 6 1.1.2 Transfinite induction . 7 1.1.3 The axiom of choice . 10 1.1.4 Ordinals and Cardinals . 11 1.1.5 Exercises . 13 1.2 Metric spaces . 14 1.2.1 Exercises . 19 1.3 Normed spaces . 20 1.3.1 Algebras . 20 1.3.2 Exercises . 21 2 Measure and integration 23 2.1 Measurable sets and functions . 24 2.1.1 Exercises . 26 2.2 Measures . 27 2.2.1 Outer measures . 31 2.2.2 Carath´eodory's extension theorem . 32 2.2.3 Exercises . 34 2.3 Borel measures on R ......................... 34 2.3.1 Lebesgue measure on R ................... 36 2.3.2 Regularity of Borel measures . 38 2.3.3 Exercises . 40 2.4 Integration . 41 2.4.1 Integrable functions . 41 2.4.2 Properties of integration . 43 2.4.3 Functions which agree almost everywhere . 45 2.4.4 Convergence properties . 45 2.4.5 Exercises . 48 2.5 Product spaces . 48 2.5.1 Exercises . 53 2.6 Signed and complex measures . 53 2.6.1 Signed measures . 53 3 4 CONTENTS 2.6.2 Complex measures . 56 2.6.3 Exercises . 56 2.7 The Radon-Nikodym Theorem . 57 2.7.1 Exercises . 60 3 Point set topology 63 3.1 Topological spaces . 63 3.1.1 Exercises . 65 3.2 Continuous maps . 65 3.2.1 Exercises . 69 3.3 Compact spaces . 70 3.3.1 Exercises . 74 3.4 The Stone-Weierstrass Theorem . 75 3.4.1 Exercises . 76 3.5 The Stone-Cechˇ compactification . 77 3.5.1 Exercises . 80 3.6 The property of Baire . 80 3.6.1 Exercises . 82 3.7 Cantor spaces . 84 3.7.1 Exercises . 86 3.8 Standard Borel spaces . 87 3.8.1 Exercises . 93 Chapter 1 Preliminaries 1.1 Sets We assume that the reader is familiar with the basic language and concepts of set theory. We use the notation N; Z; Q; R; C to denote respectively the non- negative integers (including zero), the integers, the rational numbers, the real numbers, and the complex numbers. If A is a collection of sets then we denote their union by [A2AA = fa j a 2 A for some A 2 Ag, and their intersection by \A2AA = fa j a 2 A for all A 2 Ag. If the family of sets is indexed A = fAigi2I then we also denote the union and intersection respectively by [i2I Ai and \i2I Ai. The difference of two sets A and B is A n B = fa j a 2 A and a 62 Bg, and their symmetric difference is A∆B = (A n B) [ (B n A). If A is a subset of a set X, and we write Ac for the complement of A in X, i.e., Ac = X n A. The power set of a set X is denoted by 2X and is the collection of all subsets of X, i.e., 2X = fA j A ⊂ Xg. The Cartesian product X × Y of two sets X and Y consists of all ordered pairs (x; y) such that x 2 X and y 2 Y . A function (or mapping) f : X ! Y from X to Y is a a subset of X × Y which has the property that for each x 2 X there exists a unique y 2 Y such that the pair (x; y) is contained in this subset. In this case we write y = f(x) (or sometimes y = fx) for each x 2 X. If A ⊂ X and B ⊂ Y , then the image of A is denoted by f(A) = ff(a) j a 2 Ag, and the inverse image of B is denoted by f −1(B) = fx 2 X j f(x) 2 Bg. If f : X ! Y and g : Y ! Z, the composition of f and g is denoted by g ◦ f, and is defined by the formula (g ◦ f)(x) = g(f(x)). A function f is injective (or 1-1) if f(x) = f(y) only when x = y, and f is surjective (or onto) if f(X) = Y . f is bijective if it is both injective and surjective, and in this case f has a unique inverse map f −1 : Y ! X such that f −1 ◦ f and f ◦ f −1 are the identity maps on X and Y respectively. A sequence in a set X is a function from N to X. If f : N ! X is a sequence and g : N ! N is such that g(n) < g(m) whenever n < m, then we 5 6 CHAPTER 1. PRELIMINARIES say that f ◦ g is a subsequence of f. Through abuse of notation, we will often identify a sequence with its range, for instance, we may say \let fangn2N ⊂ X be a sequence". Q If A is a family of sets, their Cartesian product A2A A consists of all functions f : A![a2AA such that f(A) 2 A for each A 2 A. Similar to unions and intersections, if A is an indexed family A = fAigi2I then the Cartesian Q Q product is written i2I Ai. If X = i2I Ai, and i 2 I, then the coordinate map πi : X ! Ai is given by πi(x) = xi, and we call xi the ith coordinate of x. I If each Ai is a fixed set A, then we denote πi2I Ai by A . If I = f1; 2; : : : ; ng, then we denote AI by An and identify this with the set of ordered n-tuples of elements of A. 1.1.1 Countability If X and Y are sets we write jXj ≤ jY j (resp. jXj = jY j) if there exists an injective (resp. bijective) map f : X ! Y . We also write jXj < jY j if jXj ≤ jY j, and there is no bijection from X to Y . Theorem 1.1.1 (The Cantor-Schr¨oder-BernsteinTheorem). If jXj ≤ jY j and jY j ≤ jXj then jXj = jY j. Proof. Suppose f : X ! Y , and g : Y ! X are both injective. Set B = n [n2N(f ◦ g) (Y n f(X)), and set A = X n g(B). Then we have g(B) = X n A, and f(A) = f(X) n (f ◦ g)(B) = Y n ((Y n f(X)) [ (f ◦ g)(B)) = Y n B: f(x) if x 2 A; Hence if we define θ : X ! Y by θ(x) = g−1(x) if x 2 Y n A = g(B); then θ gives a bijection. A set X is countable if jXj ≤ jNj. We say that X is uncountable if it is not countable. Proposition 1.1.2. 1. If X and Y are countable, then so is X × Y . 2. If I is countable and Xi is countable for each i 2 I then [i2I Xi is also countable. Proof. We let p : N ! N be the map which takes n, to the nth prime number. Suppose f : X ! N, and g : Y ! N are injective, and consider h : X×Y ! N by g(y) g(y1) g(y2) h(x; y) = p(f(x)) . If p(f(x1)) = h(x1; y1) = h(x2; y2) = p(f(x2)) then by uniqueness or prime factorization we have p(f(x1)) = p(f(x2)) and g(y1) = g(y2). As p; f, and g are injective we then have x1 = x2 and y1 = y2. Thus, h is injective. Similarly, if I is countable, and Xi is countable for each i 2 I, then consider f : I ! N injective, and for each i 2 I consider fi : Xi ! N injective. We fi(x) define g : [i2I Xi ! N, by setting g(x) = p(f(i)) where f(i) is the smallest number so that x 2 Xi. Then similar to above it is easy to check that g is injective and hence [i2I Xi is countable. 1.1. SETS 7 Corollary 1.1.3. Z and Q are countable. Proof. We have Z = N [ f0g [ −N showing that Z is countable. Also, writing any rational number in reduced fraction form a=b with a 2 Z and b 2 N n f0g, defines an injective function f(a=b) = (a; b) 2 Z × N. Since Z × N is countable, so is Q. Proposition 1.1.4 (Cantor's diagonalization method). Let X be a set, then jXj < j2X j. Proof. The injective map f : X ! 2X given by f(x) = fxg shows that we have jXj ≤ j2X j. Now, suppose we have an injective function g : X ! 2X . We let A = fx 2 X j x 62 g(x)g. Then, if x 2 X and x 2 g(x) we have x 62 A and hence g(x) 6= A. Similarly, if x 2 X and x 62 g(x) then x 2 A and hence g(x) 6= A. We therefore have produced a set which is not in the range of g showing that g X is not surjective. As g was arbitrary we then have jXj 6= j2 j. Proposition 1.1.5. jRj = j2Nj, and hence R is uncountable. Proof. Note that we have j2Nj = j2Zj. Writing each real number in its binary expansion (If there is ambiguity we choose the representation which ends in zeros) gives an injective map from R to 2Z. On the other hand, each sequence in 2N we may view as a decimal expansion, and this gives an injective map from 2N into R.