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Measure Theory Notes for Pure Math 451

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Measure Theory Notes for Pure Math 451 Kenneth R. Davidson University of Waterloo CONTENTS Chapter 1. Measures1 1.1. Introduction1 1.2. σ-Algebras3 1.3. Construction of Measures6 1.4. Premeasures8 1.5. Lebesgue-Stieltjes measures on R 10 Chapter 2. Functions 16 2.1. Measurable Functions 16 2.2. Simple Functions 18 2.3. Two Theorems about Measurable Functions 19 Chapter 3. Integration 21 3.1. Integrating positive functions 21 3.2. Two limit theorems 23 3.3. Integrating complex valued functions 26 3.4. The space L1(µ) 28 3.5. Comparison with the Riemann Integral 29 3.6. Product Measures 34 3.7. Product Integration 36 n 3.8. Lebesgue Measure on R 43 3.9. Infinite Product Measures 46 Chapter 4. Differentiation and Signed Measures 49 4.1. Differentiation 49 4.2. Signed Measures 57 4.3. Decomposing Measures 60 Chapter 5. Lp spaces 64 5.1. Lp as a Banach space 64 5.2. Duality for Normed Vector Spaces 68 5.3. Duality for Lp 69 i ii Contents Chapter 6. Some Topology 74 6.1. Topological spaces 74 6.2. Continuity 76 6.3. Compactness 78 6.4. Separation Properties 81 6.5. Nets* 85 Chapter 7. Functionals on Cc(X) and C0(X) 88 7.1. Radon measures 88 7.2. Positive functionals on Cc(X) 90 7.3. Linear functionals on C0(X) 94 Chapter 8. A Taste of Probability* 97 8.1. The language of probability 97 8.2. Independence 98 8.3. The Law of Large Numbers 100 Bibliography 105 Index 106 CHAPTER 1 Measures 1.1. Introduction The idea of Riemann integration is the following: Convergence of the upper and lower sums to the same limit is required for Riemann integrability. The integral FIGURE 1.1. Riemann Integral n Z b X is approximated by Riemann sums f(x) dx ≈ Li(ti − ti−1): The function a i=1 f(x) has to be ‘almost continuous’ in order for this to succeed. It is too restrictive. Moreover there are no good limit theorems. (Again one needs uniform convergence for most results,) Lebesgue’s approach was to chop up the range instead. Then one gets the FIGURE 1.2. Lebesgue integral Z b n X approximations f(x) dx ≈ yi fx : yi−1 < f(x) ≤ yig : The notation a i=1 1 2 Measures jAj is meant to convey the length of a set A. The complication is that, even for continuous functions, the sets fx : yi−1 < f(x) ≤ yig are neither open nor closed, and can be complicated. It is a Borel set, and that will be seen to be good enough. Lebesgue’s idea was to extend the notion of length or measure from open intervals to a function m on a much larger class of sets so that (1) m((a; b)) = b − a (2) m(A + x) = m(A) for sets A and x 2 R (translation invariance) S˙ 1 P1 (3) if A is the disjoint union A = n=1An, then m(A) = n=1 m(An). (countable additivity) It turns out that it is not possible to define such a function on all subsets of R. To see this, we define an equivalence relation on [0; 1) by x ∼ y if x−y 2 Q. Then use the Axiom of Choice to select a set A which contains exactly one element from each equivalence class. Enumerate Q \ [0; 1) = f0 = r0; r1; r2;::: g and define An = A + rn(mod 1) = (A + rn) [ (A + rn − 1) \ [0; 1) for n ≥ 0: Now by translation invariance and finite additivity, m(A) = m(A + rn) = m (A + rn) \ [0; 1) + m (A + rn) \ [1; 2) = m( A + rn) \ [0; 1) + m (A + rn − 1) \ [0; 1) = m(An): S Observe that Am \ An = ? if m 6= n. By construction, n≥1 An = [0; 1). Hence by countable additivity, 1 1 X X 1 = m([0; 1)) = m(An) = m(A): n=0 n=0 There is no value that can be assigned to m(A) to make sense of this. The way we deal with this is to declare that the set A is not measureable. We will only assign a value to m(A) for a class of ‘nice’ sets. 1.1.1. REMARK. Banach showed that it is possible to define a finitely additive, n translation invariant function on all subsets of R. However in R for n ≥ 3, even this is not possible. The Banach-Tarski paradox shows that if A and B are two bounded subsets of R3 with interior, then there is a finite decomposition of each: S˙ n S˙ n A = i=1Ai and B = i=1Bi and rigid motions (a combination of a rotation and a translation) which carry Ai onto Bi. For example let A be the sphere of radius 1 and let B be the sphere of radius 1010. Clearly this does not preserve volume, in spite of what common sense suggests. This decomposition requires the Axiom of Choice, and can’t be accomplished by hand. 1.2 σ-Algebras 3 1.2. σ-Algebras 1.2.1. DEFINITION. If X is a set, an algebra of subsets is a non-empty col- lection of subsets of X, i.e., A ⊂ P(X), which contains the empty set, ?, and is closed under complements and finite unions. A σ-algebra is an algebra B of sub- sets of X which is closed under countable unions. We say that (X; B) is a measure space. 1.2.2 SIMPLE PROPERTIES OF σ-ALGEBRAS. Let A be an algebra of sub- sets A ⊂ P(X); and let B be a σ-algebra of subsets of X. c c (1) A is non-empty, so suppose that E 2 A. Then E , ? and X = ? all belong to A. (2) If E; F 2 A, then E [ F , E \ F = (Ec [ F c)c, E n F = E \ F c and E4F = (E n F ) [ (F n E) belong to A. T S c c (3) If En 2 B, then n≥1 En = n≥1 En 2 B. S S˙ Sn−1 (4) If En 2 B, then n≥1 En = n≥1Fn where Fn = En n i=1 Ei are disjoint elements of B. T (5) If Bλ are σ-algebras for λ 2 L, then B = λ2L Bλ is a σ-algebra. So if E ⊂ P(X) is any collection of subsets, the intersection of all σ-algebras containing E is the unique smallest σ-algebra containing E. This is called the σ-algebra generated by E. 1.2.3. DEFINITION. If X is a topological space, the Borel sets are the elements of the σ-algebra BorX or Bor(X) generated by the collection of open subsets of X. A Gδ set is a countable intersection of open sets. An Fσ set is a countable union of closed sets. 1.2.4. DEFINITION. A measure on (X; B) is a map µ : B! [0; 1) [ f1g =: S˙ [0; 1] such that µ(?) = 0 and is countably additive: if A = n≥1An where P An \ Am = ? if m 6= n, then µ(A) = i≥1 µ(Ai). S A measure µ if finite if µ(X) < 1 and σ-finite if X = n≥1 En such that µ(En) < 1 for n ≥ 1. It is called a probability measure if µ(X) = 1. A measure µ is semi-finite if for every F 2 B with µ(F ) 6= 0, there is an E 2 B with E ⊂ F and 0 < µ(E) < 1. 1.2.5 SIMPLE PROPERTIES OF MEASURES. Let µ be a measure on (X; B). (1) (monotonicity) If E; F 2 B with E ⊂ F , then µ(F ) = µ(E) + µ(F n E) ≥ µ(E): 4 Measures S S˙ (2) (subadditivity) If En 2 B, write n≥1 En = n≥1Fn where Fn ⊂ En. Then X X µ(E) = µ(Fn) ≤ µ(En): n≥1 n≥1 (3) (continuity from below) If En 2 B with En ⊂ En+1 for n ≥ 1, then setting E0 = ?, [ [˙ X µ En = µ En n En−1 = µ(En n En−1) n≥1 n≥1 n≥1 n X = lim µ(Ei n Ei−1) = lim µ(En): n!1 n!1 i=1 (4) (continuity from above) If En ⊃ En+1 for n ≥ 0 and µ(E0) < 1, then \ µ En = lim µ(En): n!1 n≥1 1.2.6 SIMPLE EXAMPLES OF MEASURES. Let µ be a measure on (X; B). ( jEj if E is finite (1) Counting measure on (X; P(X)) is µc(E) = 1 if E is infinite. This measure is semi-finite. It is σ-finite if and only if X is countable. ( 1 if x 2 E (2) point mass on (X; P(X)) for x 2 X is δx(E) = 0 if x 62 E: This is a probability measure. ( 0 if E = (3) Define µ on (X; P(X)) by µ(E) = ?. This is a valid 1 if E 6= ? measure, but it is essentially useless. Note that it is not even semifinite. 1.2.7. DEFINITION. A measure µ on (X; B) is complete if E 2 B, µ(E) = 0 and F ⊂ E implies that F 2 B. 1.2.8. THEOREM. If (X; B; µ) is a measure, then B = fE [ F : E; N 2 B;F ⊂ N; µ(N) = 0g is a σ-algebra and µ¯(E [ F ) := µ(E) is a complete measure on (X; B) such that µ¯jB = µ.
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