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Econ 508B: Lecture 2 Lebesgue Measure and Lebesgue Measurable Functions Econ 508B: Lecture 2 Lebesgue Measure and Lebesgue Measurable Functions Hongyi Liu Washington University in St. Louis July 18, 2017 Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 1 / 35 Outline 1 Review of Real Analysis 2 Lebesgue Outer Measure 3 Lebesgue Measure 4 Lebesgue Measurable Functions Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 2 / 35 Outline 1 Review of Real Analysis 2 Lebesgue Outer Measure 3 Lebesgue Measure 4 Lebesgue Measurable Functions Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 3 / 35 Remark 1.1 The elements of C are called open sets and the collection C is called a topology on S. Using de Morgan's laws, the above axioms defining open sets become axioms defining closed sets. Topological Space Definition 1.1 A topological space is a pair (S; C), where S is a non-empty set and C is a collection of subsets of S such that ;; S 2 C, (finite intersection:)C1;C2 2 C ) C1 \ C2 2 C, and + (finite or infinite union:)fCk : k 2 N g ⊂⇒ [k2N + 2 C Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 4 / 35 Topological Space Definition 1.1 A topological space is a pair (S; C), where S is a non-empty set and C is a collection of subsets of S such that ;; S 2 C, (finite intersection:)C1;C2 2 C ) C1 \ C2 2 C, and + (finite or infinite union:)fCk : k 2 N g ⊂⇒ [k2N + 2 C Remark 1.1 The elements of C are called open sets and the collection C is called a topology on S. Using de Morgan's laws, the above axioms defining open sets become axioms defining closed sets. Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 4 / 35 Union/Intersection Definition 1.2 A sequence fCkg of sets is said to increase to [kCk if Ck ⊂ Ck+1 for all k and to decrease to \kCk if Ck ⊃ Ck+1 for all k; we use the notations Ck %[kCk and Ck &\kCk to denote these two possibilities. 1 If fCkgk=1 is a sequence of sets, we define 1 8 1 9 1 8 1 9 \ > [ > [ > \ > lim supC = > C > ; lim infC = > C > k > k> k > k> j=1 :k=j ; j=1 :k=j ; noting that S1 T1 Uj = k=j Ck and Vj = k=j Ck satisfy Uj & lim supCk and Vj % lim infCk, lim infCk ⊂ lim supCk. Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 5 / 35 Norm on RN (1) jxj ≥ 0 and jxj = 0 , x = 0, N (2) jαxj = jαjjxj; x 2 R ; α 2 R, (3) jx + yj ≤ jxj + jyj; 8x; y 2 R, (4)( Cauchy-Schwarz inequality:)jx· yj ≤ jxjjyj. Proof for (4): 1 1 8x; y 2 ; xy ≤ x2 + y2 R 2 2 X 1 X 1 X 1 1 x · y = x y ≤ x2 + y2 = jxj2 + jyj2 k k 2 k 2 k 2 2 1 x0 = λx, y0 = y; λ 6= 0(to be chosen); x0 · y0 = x · y λ s 1 1 jyj ≤ jλj2jxj2 + jyj2 = jxjjyj (choose λ = ) 2 2jλj2 jxj Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 6 / 35 Metric Space Definition 1.3 A metric space is a pair (S; d) where S is a nonempty set and d is a + function from S × S to R (d is called a metric on S) satisfying (i) d(x; y) = d(y; x) for all x; y 2 S, (ii) d(x; y) = 0 iff x = y, (iii)( triangle inequality:)d(x; z) ≤ d(x; y) + d(y; z) for all x; y; z 2 S. A metric space (S; d) is a topological space where a set C is open if for all x 2 C, 9 an > 0 such that B(x; ) ≡ fy : d(y; x) < g ⊂ C. Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 7 / 35 continued Any one of the following metrics defined on any Euclidean space n R ; 1 ≤ n < 1 is a metric space: (1) For 1 < p < 1, 1 8 n 9 p >X p> dp(x; y) = > jxi − yij > :> ;> i=1 (2) d1(x; y) = max jxi − yij 1≤i≤n (3) For 0 < p < 1, 8 n 9 >X p> dp(x; y) = > jxi − yij > :> ;> i=1 1 Question: why there does not exist power p to (3)? Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 8 / 35 Outline 1 Review of Real Analysis 2 Lebesgue Outer Measure 3 Lebesgue Measure 4 Lebesgue Measurable Functions Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 9 / 35 Lebesgue Outer (Exterior) Measure Define closed n-dimensional intervals I = fx : aj ≤ xj ≤ bj; j = 1; :::; ng Qn and their volumes v(I) = j=1(bj − aj). To define the outer measure n of an arbitrary subset C of R , cover C by a countable collection S of intervals Ik, and let X σ(S) = v(Ik) Ik2S The Lebesgue outer measure of C, denoted as µ∗(C), is defined by µ∗(C) = infσ(S) where the infimum is taken over all covers S of C. Thus, 0 ≤ µ∗(C) ≤ +1 Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 10 / 35 Properties of Outer Measure For an interval I, µ∗(I) = v(I). ∗ ∗ Monotonicity: If C1 ⊂ C2, then µ (C1) ≤ µ (C2). Countable sub-additivity: If C = [Ck is a countable union of ∗ P ∗ sets, then µ (C) ≤ µ (Ck). Empty set: The empty set has outer measure zero, e.g., Q. Remark 2.1 In particular, any subset of a set with outer measure zero has outer measure zero and that the countable union of sets with outer measure zero has outer measure zero as shown by the example of Q. Moreover, there are sets with outer measure zero that are not countable, e.g., Cantor Set. Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 11 / 35 continued Theorem 2.1 (Outer Approximation) n Let C ⊂ R . Then given > 0, 9 an open set G s.t. C ⊂ G and µ∗(G) ≤ µ∗(C) + . Hence, µ∗(C) = infµ∗(G); where the infimum is taken over all open sets G containing C. Theorem 2.2 n ∗ ∗ If C ⊂ R , 9 a set H of type Gδ s.t. C ⊂ H and µ (C) = µ (H) Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 12 / 35 Outline 1 Review of Real Analysis 2 Lebesgue Outer Measure 3 Lebesgue Measure 4 Lebesgue Measurable Functions Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 13 / 35 Definition 3.2 (Measure) If C is measurable, its outer measure is referred to as its Lebesgue measure or simply its measure, and denoted by µ(C) as previously illustrated: µ(C) = µ∗(C); for measurable C: Example 3.1 Every open set is measurable. Every set of outer measure zero is measurable. `Measurable' & `Measure' Definition 3.1 (Lebesgue measurable) n A subset C of R is defined to be Lebesgue measurable, or measurable, if given > 0, 9 an open set G s.t. C ⊂ G; and µ∗(G − C) < Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 14 / 35 Example 3.1 Every open set is measurable. Every set of outer measure zero is measurable. `Measurable' & `Measure' Definition 3.1 (Lebesgue measurable) n A subset C of R is defined to be Lebesgue measurable, or measurable, if given > 0, 9 an open set G s.t. C ⊂ G; and µ∗(G − C) < Definition 3.2 (Measure) If C is measurable, its outer measure is referred to as its Lebesgue measure or simply its measure, and denoted by µ(C) as previously illustrated: µ(C) = µ∗(C); for measurable C: Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 14 / 35 `Measurable' & `Measure' Definition 3.1 (Lebesgue measurable) n A subset C of R is defined to be Lebesgue measurable, or measurable, if given > 0, 9 an open set G s.t. C ⊂ G; and µ∗(G − C) < Definition 3.2 (Measure) If C is measurable, its outer measure is referred to as its Lebesgue measure or simply its measure, and denoted by µ(C) as previously illustrated: µ(C) = µ∗(C); for measurable C: Example 3.1 Every open set is measurable. Every set of outer measure zero is measurable. Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 14 / 35 Remark 3.1 Noting that the condition for measurability should not be confused with theorem 2.1, which only states that 9 an open set G containing C such that µ∗(G) ≤ µ∗(C) + . Since G = C [ (G − C) when C ⊂ G, which only implies that µ∗(G) ≤ µ∗(C) + µ∗(G − C). However, we cannot obtain from µ∗(G) ≤ µ∗(C) + that µ∗(G − C) < . Hongyi Liu (Washington University in St. Louis)Math Camp 2017 Stats July 18, 2017 15 / 35 Properties of Measurable Set (Countable subaddtivity:) The union C = [Ck of a countable P measurable sets is measurable and µ(C) ≤ µ(Ck). Every closed set F is measurable. The complement of a measurable set is measurable.
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