Real Analysis: Part II William G. Faris June 3, 2004 ii Contents 1 Function spaces 1 1.1 Spaces of continuous functions . 1 1.2 Pseudometrics and seminorms . 2 1.3 Lp spaces . 2 1.4 Dense subspaces of Lp ........................ 5 1.5 The quotient space Lp ........................ 6 1.6 Duality of Lp spaces . 7 1.7 Orlicz spaces . 8 1.8 Problems . 10 2 Hilbert space 13 2.1 Inner products . 13 2.2 Closed subspaces . 16 2.3 The projection theorem . 17 2.4 The Riesz-Fr´echet theorem . 18 2.5 Bases . 19 2.6 Separable Hilbert spaces . 21 2.7 Problems . 23 3 Differentiation 27 3.1 The Lebesgue decomposition . 27 3.2 The Radon-Nikodym theorem . 28 3.3 Absolutely continuous increasing functions . 29 3.4 Problems . 31 4 Conditional Expectation 33 4.1 Hilbert space ideas in probability . 33 4.2 Elementary notions of conditional expectation . 35 4.3 The L2 theory of conditional expectation . 35 4.4 The L1 theory of conditional expectation . 37 4.5 Problems . 39 iii iv CONTENTS 5 Fourier series 41 5.1 Periodic functions . 41 5.2 Convolution . 42 5.3 Approximate delta functions . 42 5.4 Summability . 43 5.5 L2 convergence . 44 5.6 C(T) convergence . 46 5.7 Pointwise convergence . 48 5.8 Problems . 49 6 Fourier transforms 51 6.1 Fourier analysis . 51 6.2 L1 theory . 52 6.3 L2 theory . 53 6.4 Absolute convergence . 54 6.5 Fourier transform pairs . 55 6.6 Poisson summation formula . 56 6.7 Problems . 57 7 Topology 59 7.1 Topological spaces . 59 7.2 Measurable spaces . 61 7.3 The Borel σ-algebra . 62 7.4 Comparison of topologies . 63 7.5 Bases and subbases . 64 7.6 Compact spaces . 65 7.7 The one point compactification . 66 7.8 Nets . 68 7.9 Problems . 69 8 Product and weak∗ topologies 71 8.1 Introduction . 71 8.2 The Tychonoff product theorem . 72 8.3 Banach spaces and dual Banach spaces . 73 8.4 Weak∗ topologies on dual Banach spaces . 75 8.5 The Alaoglu theorem . 76 8.6 Appendix: The Bourbaki fixed point theorem . 79 8.7 Appendix: Zorn's lemma . 80 9 Radon measures 83 9.1 Topology and measure . 83 9.2 Borel isomorphisms . 84 9.3 Metric spaces . 85 9.4 Riesz representation . 86 9.5 Lower semicontinuous functions . 87 9.6 Weak* convergence . 88 CONTENTS v 9.7 Central limit theorem for coin tossing . 90 9.8 Wiener measure . 91 9.9 Appendix: Measure theory on non-metric spaces . 92 9.10 Problems . 94 10 Distributions 95 10.1 Fr´echet spaces . 95 10.2 Distributions . 96 10.3 Operations on distributions . 97 10.4 Mapping distributions . 98 10.5 The support of a distribution . 99 10.6 Radon measures . 99 10.7 Approximate delta functions . 100 10.8 Tempered distributions . 101 10.9 Poisson equation . 105 10.10Problems . 107 11 Appendix: Standard Borel measurable spaces 109 11.1 Complete separable metric spaces . 109 11.2 Embedding a Cantor set . 110 11.3 Placement in a cube . 111 11.4 A cube inside a Cantor set . 111 11.5 A dynamical system . 112 11.6 The unique Borel structure . 112 11.7 The unique measure structure with given mass . 113 vi CONTENTS Chapter 1 Function spaces 1.1 Spaces of continuous functions This section records notations for spaces of real functions. In some contexts it is convenient to deal instead with complex functions; usually the changes that are necessary to deal with this case are minor. Let X be a topological space. The space C(X) consists of all continuous functions. The space B(X) consists of all bounded functions. It is a Banach space in a natural way. The space BC(X) consists of all bounded continuous functions. It is a somewhat smaller Banach space. Consider now the special case when X is a locally compact Hausdorff space. Thus each point has a compact neighborhood. For example X could be Rn. The space Cc(X) consists of all continuous functions, each one of which has compact support. The space C0(X) is the closure of Cc(X) in BC(X). It is itself a Banach space. It is the space of continuous functions that vanish at infinity. The relation between these spaces is that Cc(X) ⊂ C0(X) ⊂ BC(X). They are all equal when X compact. When X is locally compact, then C0(X) is the best behaved. Recall that a Banach space is a normed vector space that is complete in the metric associated with the norm. In the following we shall need the concept of the dual space of a Banach space E. The dual space E∗ consists of all continuous linear functions from the Banach space to the real numbers. (If the Banach space has complex scalars, then we take continuous linear function from the Banach space to the complex numbers.) The dual space E∗ is itself a Banach space, where the norm is the Lipschitz norm. For certain Banach spaces E of functions the linear functionals in the dual ∗ space E may be realized in a more concrete way. For example, If E = C0(X), then its dual space E∗ = M(X) is a Banach space consisting of signed Radon measures of finite variation. (A signed measure σ of finite variation is the difference σ = σ+−σ of two positive measures σ± that each assign finite measure 1 2 CHAPTER 1. FUNCTION SPACES to the space X. Radon measures willR be discussed later.) If σ is in M(X), then it defines the linear functional f 7! f(x) dσ(x), and all elements of the dual space E∗ arise in this way. 1.2 Pseudometrics and seminorms A pseudometric is a function d : P × P ! [0; +1) that satisfies d(f; f) ≥ 0 and d(f; g) ≤ d(f; h) + d(h; g) and such that d(f; f) = 0. If in addition d(f; g) = 0 implies f = g, then d is a metric. The theory of pseudometric spaces is much the same as the theory of metric spaces. The main difference is that a sequence can converge to more than one limit. However each two limits of the sequence have distance zero from each other, so this does not matter too much. Given a pseudometric space P , there is an associated metric space M. This is defined to be the set of equivalence classes of P under the equivalence relation fEg if and only if d(f; g) = 0. In other words, one simply defines two points r; s in P that are at zero distance from each other to define the same point r0 = s0 in M. The distance dM (a; b) between two points a; b in M is defined by taking 0 0 representative points p; q in P with p = a and q = b. Then dM (a; b) is defined to be d(p; q). A seminorm is a function f 7! kfk ≥ 0 on a vector space E that satisfies kcfk = jcjkfk and the triangle inequality kf + gk ≤ kfk + kgk and such that k0k = 0. If in addition kfk = 0 implies f = 0, then it is a norm. Each seminorm on E defines a pseudometric for E by d(f; g) = kf − gk. Similarly, a norm on E defines a metric for E. Suppose that we have a seminorm on E. Then the set of h in E with khk = 0 is a vector subspace of E. The set of equivalence classes in the construction of the metric space is itself a vector space in a natural way. So for each vector space with a seminorm we can associate a new quotient vector space with a norm. 1.3 Lp spaces In this and the next sections we introduce the spaces Lp(X; F; µ) and the cor- responding quotient spaces Lp(X; F; µ). Fix a set X and a σ-algebra F of measurable functions. Let 0 < p < 1. Define p 1 kfkp = µ(jfj ) p : (1.1) p Define L (X; F; µ) to be the set of all f in F such that kfkp < 1. Theorem 1.1 For 0 < p < 1, the space Lp is a vector space. Proof: It is obvious that Lp is closed under scalar multiplication. The problem is to prove that it is closed under addition. However if f, g are each in 1.3. LP SPACES 3 Lp, then jf + gjp ≤ [2(jfj _ jgj)]p ≤ 2p(jfjp + jgjp): (1.2) Thus f + g is also in Lp. 2 The function xp is increasing for every p > 0. In fact, if φ(p) = xp for x ≥ 0, then φ0(p) = pxp−1 ≥ 0. However it is convex only for p ≥ 1. This is because in that case φ00(x) = p(p − 1)xp−2 ≥ 0. Let a ≥ 0 and b ≥ 0 be weights with a + b = 1. For a convex function we have the inequality φ(au + bv) ≤ aφ(u) + bφ(v). This is the key to the following fundamental inequality. Theorem 1.2 (Minkowski's inequality) If 1 ≤ p < 1, then kf + gkp ≤ kfkp + kgkp: (1.3) p Proof: Let c = kfkp and d = kgkp. Then by the fact that x is increasing and convex p p p f + g c f d g c f d g p ≤ + ≤ + : (1.4) c + d c + d c c + d d c + d c c + d d Integrate. This gives f + g p µ ≤ 1: (1.5) c + d Thus kf + gkp ≤ c + d.