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Real Analysis II, Winter 2018

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Real Analysis II, Winter 2018 Real Analysis II, Winter 2018 From the Finnish original “Moderni reaalianalyysi”1 by Ilkka Holopainen adapted by Tuomas Hytönen February 22, 2018 1Version dated September 14, 2011 Contents 1 General theory of measure and integration 2 1.1 Measures . 2 1.11 Metric outer measures . 4 1.20 Regularity of measures, Radon measures . 7 1.31 Uniqueness of measures . 9 1.36 Extension of measures . 11 1.45 Product measure . 14 1.52 Fubini’s theorem . 16 2 Hausdorff measures 21 2.1 Basic properties of Hausdorff measures . 21 2.12 Hausdorff dimension . 24 2.17 Hausdorff measures on Rn ...................... 25 3 Compactness and convergence of Radon measures 30 3.1 Riesz representation theorem . 30 3.13 Weak convergence of measures . 35 3.17 Compactness of measures . 36 4 On the Hausdorff dimension of fractals 39 4.1 Mass distribution and Frostman’s lemma . 39 4.16 Self-similar fractals . 43 5 Differentiation of measures 52 5.1 Besicovitch and Vitali covering theorems . 52 1 Chapter 1 General theory of measure and integration 1.1 Measures Let X be a set and P(X) = fA : A ⊂ Xg its power set. Definition 1.2. A collection M ⊂ P (X) is a σ-algebra of X if 1. ? 2 M; 2. A 2 M ) Ac = X n A 2 M; S1 3. Ai 2 M, i 2 N ) i=1 Ai 2 M. Example 1.3. 1. P(X) is the largest σ-algebra of X; 2. f?;Xg is the smallest σ-algebra of X; 3. Leb(Rn) = the Lebesgue measurable subsets of Rn; 4. If M is a σ-algebra of X and A ⊂ X, then MjA = fB \ A : B 2 Mg is a σ-algebra of A. 5. If M is a σ-algebra of X and A 2 M, then MA = fB ⊂ X : B \ A 2 Mg is a σ-algebra of X. Definition 1.4. If F ⊂ P(X) is any family of subsets of X, then \ σ(F) = fM : M is a σ-algebra of X; F ⊂ Mg is the σ-algebra generated by F. It is the smallest σ-algebra that contains F. 2 Example 1.5. A subset I ⊂ Rn is an open n-interval if it has the form I = f(x1; : : : ; xn): aj < xj < bjg; where −∞ ≤ aj < bj ≤ +1. Then n n σ(fI : I is an n-intervalg) = σ(fA : A ⊂ R is openg) =: Bor(R ) is the σ-algebra of Borel sets of Rn. (Think why there is equality on the left.) n We observe that, in R , all open sets, closed sets, Gδ sets (countable inter- sections of open sets), Fσ sets (countable unions of closed sets), Fσδ sets, Gδσ sets, etc., are Borel sets. Thus e.g. the set Q of rational numbers is a Borel set. Remark 1.6. In every topological space X, one can define the Borel sets Bor(X) = σ(fA : A ⊂ X openg: Definition 1.7. Let M be a σ-algebra of X. A mapping µ : M! [0; +1] is a measure, if 1. µ(?) = 0, S1 P1 2. µ( i=1 Ai) = i=1 µ(Ai), if Ai 2 M are disjoint. The triplet (X; M; µ) is called a measure space and the elements of M measur- able sets. Condition (2) above is called countable additivity. The definition implies the monotonicity of the meusure: If A; B 2 M and A ⊂ B, then µ(A) ≤ µ(B). Remark 1.8. 1. If µ(X) < 1, the measure µ is finite. 2. If µ(X) = 1, then µ is a probability measure. S1 3. If X = i=1 Ai, where µ(Ai) < 1 for all i, the measure µ is σ-finite. We also say that X is σ-finite with respect to µ. 4. If A 2 M and µ(A) = 0, then A has (or is of) measure zero. 5. If X is a topological space and Bor(X) ⊂ M (i.e., every Borel set is measurable), then µ is a Borel measure. n n Example 1.9. 1. X = R , M = Leb R , µ = mn = Lebesgue measure. n n n 2. X = R , M = Bor R , µ = mnj Bor R = the restriction of the Lebesgue measure on Borel sets. 3. Let X 6= ? be any set. Fix x 2 X and define for all A ⊂ X ( 1; if x 2 A; µ(A) = 0; if x2 = A: Then µ : P(X) ! [0; +1] is a measure (so called Dirac measure at x). It is often denoted by µ = δx. 3 4. If f : Rn ! [0; 1] is Lebesgue measurable, then µ : Leb(Rn) ! [0; 1] defined by µ(E) = f(x)dmn(x) ˆE is a measure. (See course “Measure and integral”.) 5. If (X; M; µ) is a measure space and A 2 M, then µxA : MA ! [0; 1] defined by (µxA)(B) = µ(B \ A) is a measure. It is called the restriction of µ on A. Theorem 1.10. Let (X; M; µ) be a measure space and A1;A2;::: 2 M. 1 [ 1. If A1 ⊂ A2 ⊂ :::, then µ( Ai) = lim µ(Ai). i!1 i=1 1 \ 2. If A1 ⊃ A2 ⊃ ::: and µ(Ak) < 1 for some k, then µ( Ai) = lim µ(Ai). i!1 i=1 Proof. Measure and integral. 1.11 Metric outer measures Definition 1.12. A mapping µ~ : P(X) ! [0; +1] is an outer measure (or exterior measure) on X if 1. µ~(?) = 0; P1 S1 2. µ~(A) ≤ i=1 µ~(Ai), if A ⊂ i=1 Ai ⊂ X. Remark 1.13. 1. The outer measure is thus defined on all subsets of X. 2. Condition (2) implies the monotonicity of the outer measure, i.e., µ~(A) ≤ µ~(B) if A ⊂ B ⊂ X. 3. Many books (e.g. Evans–Gariepy, Mattila, . ) refer to outer measures simply as measures. 4. Let µ~ be an outer measure on X and A ⊂ X. Then its restriction to A, (~µxA)(B) =µ ~(B \ A); is an outer measure on X. Every outer measure defines a σ-algebra of “measurable” sets through the so called Carathéodory condition: 4 Definition 1.14. Let µ~ be an outer measure on X. A set E ⊂ X is µ~- measurable, or just measurable, if µ~(A) =µ ~(A \ E) +µ ~(A n E) for all A ⊂ X. Theorem 1.15. Let µ~ be an outer measure on X and M = Mµ~ = fE ⊂ X : E is µ~-measurableg: Then 1. M is a σ-algebram and 2. µ =µ ~jM is a measure (i.e., µ is countably additive). Proof. Measure and integral. Definition 1.16. An outer measure µ~ on a topological space X is called a Borel outer measure if every Borel set of X is µ~-measurable (i.e., the measure determined by µ~ is a Borel measure). We next investigate, when a given outer measure on a topological space X has this property. Definition 1.17 (Carathéodory). An outer measure µ~ on a metric space (X; d) is a metric outer measure if µ~(A [ B) =µ ~(A) +µ ~(B) for all A; B ⊂ X such that dist(A; B) = inffd(a; b): a 2 A; b 2 Bg > 0. Theorem 1.18. An outer measure µ~ on a metric space (X; d) is a Borel outer measure, if and only if µ~ is a metric outer measure. We begin with a lemma: Lemma 1.19. Let µ~ be a metric outer measure and A ⊂ G ⊂ X, where G is open. If c Ak = fx 2 A : dist(x; G ) ≥ 1=kg; k 2 N; then µ~(A) = limk!1 µ~(Ak). S1 Proof. Since G is open, we have A = k=1 Ak. Let Bk = Ak+1 n Ak: Then 1 1 [ [ A = A2n [ B2k [ B2k+1 ; k=n k=n so that 1 1 X X µ~(A) ≤ µ~(A2n) + µ~(B2k) + µ~(B2k+1) =µ ~(A2n) + (I)n + (II)n: k=n k=n 5 Consider the limit n ! 1. (1) If (I)n; (II)n ! 0 as n ! 1, then µ~(A) ≤ lim µ~(A2n) ≤ lim µ~(A2n+1) ≤ µ~(A); n!1 n!1 and the claim of the lemma is true. X (2) If (I)n 6! 0 as n ! 1, then µ~(B2n) = 1. n n−1 [ On the other hand, A ⊃ A2n ⊃ B2k; where k=1 1 1 dist(B ;B ) ≥ − > 0: 2k 2k+2 2k + 1 2k + 2 Since µ~ is a metric outer measure, it follows that n−1 n−1 X [ µ~(B2k) =µ ~ B2k ≤ µ~(A2n) ≤ µ~(A2n+1) ≤ µ~(A): k=1 k=1 As n ! 1, we deduce that µ~(A) = limk!1 µ~(Ak) = 1, and the claim of the lemma is true again. If (II)n 6! 0 as n ! 1, the argument is entirely similar. Proof of Theorem 1.18. Let us first assume that µ~ is a metric outer measure and prove that µ~ is a Borel outer measure. Since Bor(X) = σ(fF : F ⊂ X closedg) and Mµ~ is a σ-algebra, it suffices to prove that every closed set F ⊂ X is µ~-measurable. Let E ⊂ X be an arbitrary set, for which we check the Carathéodory condi- tion µ~(E) =µ ~(E \ F ) +µ ~(E n F ). We apply Lemma 1.19 with A = E n F ⊂ X n F = G: When Ak ⊂ A is as in the lemma, we have dist(Ak;F \ E) ≥ dist(Ak;F ) ≥ 1=k > 0; lim µ~(Ak) =µ ~(A) =µ ~(E n F ): k!1 By the assumption that µ~ is a metric outer measure and observing that Ak ⊂ A ⊂ E, we have µ~(Ak) +µ ~(F \ E) =µ ~(Ak [ (F \ E)) ≤ µ~(E): When k ! 1, the left-hand side tends to µ~(E n F ) +µ ~(F \ E), and we get the Carathéodory condition.
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