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0809LebesgueIntegration - O.H Probability and Markov Chains...

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O.H. Probability and Markov Chains – MATH 2561/2571 E09 Construction of the Lebesgue integral (sketch) The material of this section is optional and thus for fun, not for exam! A.1 Measures and measure spaces Let E be a (fixed) set. We start by recalling the following definitions. Definition A.1. Let A be a collection of subsets of E . We shall call A a field if it has the following properties: 1. ∈ A ; 2. if A 1 , A 2 ∈ A , then A 1 A 2 ∈ A ; 3. if A ∈ A , then A c ∈ A . Remark A.1.1. Obviously, every field is closed w.r.t. taking finite unions or inter- sections. Definition A.2. Let A be a collection of subsets of E . We shall call A a σ -field if it has the following properties: 1. ∈ A ; 2. if A 1 , A 2 , · · · ∈ A , then S k =1 A k ∈ A ; 3. if A ∈ A , then A c ∈ A . Remark A.2.1. Obviously, property 2 above can be replaced by the equivalent con- dition T k =1 A k ∈ A . Definition A.3. Let C be a collection of subsets of E . The intersection of all σ -fields containing C shall be called the σ -field generated by C and denoted σ ( C ) . In other words, every collection C of subsets of E can be completed to a σ -field in a unique way. An important example . Let C 0 be the collection of all open intervals in R 1 , C 0 = n ( a,b ) : −∞ <a<b< o . Definition A.4. The Borel σ -field in R 1 is B 1 = σ ( C 0 ) . Denote C 1 = n [ a,b ) : −∞ <a<b< o , C 2 = n [ a, ) : −∞ <a< o , C 3 = n ( a, ) : −∞ <a< o . Exercise A.5. Show that σ ( C 1 ) = σ ( C 2 ) = σ ( C 3 ) = B 1 . Definition A.6. Let C be a collection of subsets of E . Any set function μ : C → R + [0 , ] is called σ -additive or a measure , if 1. μ ( ) = 0 ; 1
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O.H. Probability and Markov Chains – MATH 2561/2571 E09 2. every sequence A k , k N , of disjoint sets A k ∈ C ( k N ) satisfying k =1 A k ∈ C has the following property μ [ k =1 A k = X k =1 μ ` A k ´ . Recall that a probability measure P on a measurable space (Ω , F ) is a σ -additive set function P : F → [0 , 1] such that P (Ω) = 1. Remark A.6.1. Observe that in general the condition k =1 F k ∈ C cannot be dropped. Theorem A.7 (Carath´ eodory) . Let A be a field of subsets of E and let μ be a σ -additive map μ : A → [0 , ] . If D = σ ( A ) , there exists a measure ¯ μ on D such that A ∈ A = ¯ μ ( A ) μ ( A ) . If μ ( E ) < , then ¯ μ is unique. In particular, if A is a field of subsets of Ω and μ : A → [0 , 1] is a σ -additive set function such that μ (Ω) = 1, then there exists a unique extension P of μ to the generated σ -field F = σ ( A ). Using the Carath´ eodory theorem one can construct various probability spaces (Ω , F , P ). Definition A.8. A triple ( E, A ) where E is a set, A is a σ -field of subsets of E and μ -a measure is called a measure space ; simliarly, ( E, A ) is called a measurable space . A.2 Measurable functions Definition A.9. Let ( E 1 , A 1 ) and ( E 2 , A 2 ) be measurable spaces. A function f : E 1 E 2 is called measurable if for all B ∈ A 2 , f 1 ( B ) def = n x E 1 : f ( x ) B o ∈ A 1 .
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