MATH 245A (17F) (L) M. Bonk / (TA) A. Wu Real Analysis Contents 1 Measure Theory 3 1.1 σ-algebras . .3 1.2 Measures . .4 1.3 Construction of non-trivial measures . .5 1.4 Lebesgue measure . 10 1.5 Measurable functions . 14 2 Integration 17 2.1 Integration of simple non-negative functions . 17 2.2 Integration of non-negative functions . 17 2.3 Integration of real and complex valued functions . 19 2.4 Lp-spaces . 20 2.5 Relation to Riemann integration . 22 2.6 Modes of convergence . 23 2.7 Product measures . 25 2.8 Polar coordinates . 28 2.9 The transformation formula . 31 3 Signed and Complex Measures 35 3.1 Signed measures . 35 3.2 The Radon-Nikodym theorem . 37 3.3 Complex measures . 40 4 More on Lp Spaces 43 4.1 Bounded linear maps and dual spaces . 43 4.2 The dual of Lp ....................................... 45 4.3 The Hardy-Littlewood maximal functions . 47 5 Differentiability 51 5.1 Lebesgue points . 51 5.2 Complex Borel measures on R ............................... 54 5.3 The fundamental theorem of calculus . 58 6 Functional Analysis 61 6.1 Locally compact Hausdorff spaces . 61 6.2 Weak topologies . 62 6.3 Some theorems in functional analysis . 65 6.4 Hilbert spaces . 67 1 CONTENTS MATH 245A (17F) 7 Fourier Analysis 73 7.1 Trigonometric series . 73 7.2 Fourier series . 74 7.3 The Dirichlet kernel . 75 7.4 Continuous functions and pointwise convergence properties . 77 7.5 Convolutions . 78 7.6 Convolutions and differentiation . 78 7.7 Translation operators . 79 7.8 The Fourier transform . 79 8 The Theory of Distributions 83 8.1 Sobolev functions and weak derivatives . 83 8.2 Distributions . 84 8.3 Localization and support of distributions . 85 8.4 Fourier transforms of distributions . 85 2 MATH 245A (17F) (L) M. Bonk / (TA) A. Wu Real Analysis 1 MEASURE THEORY 1.1 σ-ALGEBRAS Definition 1.1.1 (σ-algebra). Let X be a set and A be a family of subsets of X. We say that A is an algebra on X if (i) ; 2 A; (ii) if A 2 A, then XnA 2 A; (iii) if A; B 2 A, then A [ B 2 A. We say that A is a σ-algebra on X if closure under finite unions extends to countable unions. Proposition 1.1.2. Let A be an algebra on X. If A; B 2 A, then 1. A \ B 2 A; 2. AnB 2 A. Moreover, if A is a σ-algebra and An 2 A, then 1 \ An 2 A: n=1 Example 1.1.3. 1. For any set X, A = P(X) is a σ-algebra. 2. For any set X, A = fA ⊂ X j A is countable or XnA is countableg is a σ-algebra. T 3. If fAi j i 2 Ig is a family of σ-algebras on X, then so is A = i Ai. Proposition 1.1.4. Let F ⊂ P(X) be a family of subsets of X. Then there is a unique σ-algebra A such that F ⊂ A ⊂ A0, where A0 is any σ-algebra on X with F ⊂ A0. Proof. Let F be the family of σ-algebras A0 containing F. The appropriate A is \ A = A0: A02F Definition 1.1.5 (σ-algebra generated by a family of sets). The σ-algebra A is the σ-algebra generated by F, denoted by σ(F). Definition 1.1.6 (Borel σ-algebra). Let (X; O) be a topological space. The Borel σ-algebra on X is BX = σ(O), the σ-algebra generated by the open sets in X. Example 1.1.7. 1. Every open set is Borel, and by complementation, every closed set is Borel. 2. In R, the set Q is Borel, as it is a countable union of points, which are closed. n 3.A rectangle in R is a set of the form [a1; b1] × · · · × [an; bn] with ai ≤ bi for all i. If R is the n family of all rectangles, then σ(R) = BR . 3 1.2 Measures MATH 245A (17F) 1.2 MEASURES Definition 1.2.1 (Measure). A pair (X; A) of a set X and a σ-algebra A on X is a measurable space.A (positive) measure µ on a measurable space (X; A) is a function µ : A! [0; 1] such that (i) µ(;) = 0; (ii) if An 2 A (n 2 N) are pairwise disjoint, then 1 ! 1 [ X µ An = µ(An): n=1 n=1 The triple (X; A; µ) is a measure space. Example 1.2.2. Let X be a set and A = P(X). 1. For a given a 2 X, the Dirac measure at a is ( 0 a 62 M; δa(M) = 1 a 2 M: 2. The counting measure is ( jMj M is finite; µ(M) = 1 otherwise: 3. If X is a topological space, then a Borel measure is a measure defined on (X; BX ). S Notation. 1. Write An % A if A1 ⊂ A2 ⊂ · · · and A = n An. T 2. Write An & A if A1 ⊃ A2 ⊃ · · · and A = n An. Theorem 1.2.3. Let (X; A; µ) be a measure space. Then 1. (monotonicity) if A; B 2 A and A ⊂ B, then µ(A) ≤ µ(B); 2. (countable subadditivity) if An 2 A, then 1 ! 1 [ X µ An ≤ µ(An); n=1 n=1 3. (continuity from below) if An 2 A and An % A, µ(A) = lim µ(An); n!1 4. (continuity from above) if An 2 A, An & A, and µ(A1) < 1, then µ(A) = lim µ(An): n!1 4 MATH 245A (17F) (L) M. Bonk / (TA) A. Wu Real Analysis Proof. 1. Since A ⊂ B, we have a decomposition B = A [ (BnA) into disjoint subsets. Then µ(B) = µ(A) + µ(BnA) ≥ µ(A): 2. Define -n−1 [ Bn = An An ⊂ An: k=1 S S By construction, the Bn's are pairwise disjoint and n Bn = n An. Then 1 ! 1 ! 1 1 [ [ X X µ An = µ Bn = µ(Bn) ≤ µ(An): n=1 n=1 n=1 n=1 3. This time, define Bn = AnnAn−1 (suppose A0 = ;). We get 1 ! 1 ! 1 N [ [ X X µ An = µ Bn = µ(Bn) = lim µ(Bn) = lim µ(AN ): N!1 N!1 n=1 n=1 n=1 n=1 4. Let Bn = A1nAn. Then Bn % A1nA in A and µ(A1) = µ(An)+µ(Bn), so applying continuity from below gives µ(A1) = µ(A1nA) + µ(A) = lim µ(Bn) + µ(A): n!1 Since µ(A1) is finite, the result follows from writing µ(A1) = lim (µ(A1) − µ(An)) + µ(A) = µ(A1) + µ(A) − lim µ(An): n!1 n!1 Remark 1.2.4. Continuity from above may fail if we allow µ(A1) = 1. For example, on R with T the counting measure, let An = [n; 1). Then n An = ;, but µ(An) = 1 for each n. Definition 1.2.5 (Null set). Let (X; A; µ) be a measure space. A null set is a measurable set N ⊂ X with µ(N) = 0. Definition 1.2.6 (\Almost everywhere"). Let P (x) be a statement about points x in a measure space (X; A; µ). We say that P (x) is true µ-almost everywhere (or for µ-almost every x 2 X), abbreviated µ-a.e., if there exists a null set N for which P (x) is true for all x 2 XnN. 1.3 CONSTRUCTION OF NON-TRIVIAL MEASURES Let (X; A; µ) be a measure space. Definition 1.3.1 (Complete measure space). We say that (X; A; µ) is complete if every subset of a null set is a null set. (In particular, every subset of a null set is measurable.) Theorem 1.3.2. Let A = fA [ B j A 2 A and there exists a null set N such that B ⊂ Ng: Then A is a σ-algebra on X with A ⊂ A. Moreover, µ can be uniquely extended to a measure µ on A which is complete. 5 1.3 Construction of non-trivial measures MATH 245A (17F) Proof. It is clear that A ⊂ A. To see that A is a σ-algebra, we check the axioms. (i) Since A ⊂ A, we have ; 2 A and X 2 A. (ii) Let A [ B 2 A with B ⊂ N for some null set N. By replacing A with AnN and B with B [ (A \ N) if necessary, we can suppose that A \ N = ;. Then Xn(A [ B) = (XnA) \ (XnB) = [(XnA) \ (XnN)] [ [NnB] 2 A: (iii) Let An [ Bn 2 A with Bn ⊂ Nn for null sets Nn, and let 1 1 1 [ [ [ A = An;B = Bn;N = Nn: n=1 n=1 n=1 P Then A; N 2 A and µ(N) ≤ n µ(Nn) = 0, so N is a null set with B ⊂ N, and 1 [ (An [ Bn) = A [ B 2 A: n=1 For existence of an extension µ, we attempt to define µ(A [ B) = µ(A). To show existence, define µ by µ(A [ B) = µ(A). To see that it is well-defined, let A [ B = A0 [ B0 with A; A0 2 A and B; B0 subsets of null sets. The union of two null sets is a null set, so we can take B; B0 ⊂ N for some null set N. Then µ(A [ B) = µ(A) = µ(A [ N) = µ(A0 [ N) = µ(A0) = µ(A0 [ B0): The rest of the proof is omitted.