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.