0809IntegrationSlides - WLLN Borel-Cantelli lemma A.s...

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Unformatted text preview: WLLN Borel-Cantelli lemma A.s. convergence Relations Integration Integration of functions Want: an integral f mapsto→ integraltext f which is a) linear: for a , b ∈ R , integraltext ( af + bg ) = a integraltext f + b integraltext g ; b) monotone: if f ≤ g , then integraltext f ≤ integraltext g ; c) respects limits: if f n → f “nicely”, then integraltext f n → integraltext f . Riemann integral satisfies a) and b) only! E.g. the Dirichlet function. WLLN Borel-Cantelli lemma A.s. convergence Relations Integration Lebesgue construction We need: a measure space ( E , A , μ ), where E is a set, A is a σ-field of subsets of E , and μ is a measure . Recall: A collection A of subsets of E is a σ-field if: 1. ∅ ∈ A ; 2. if A 1 , A 2 , ··· ∈ A , then uniontext ∞ k =1 A k ∈ A ; 3. if A ∈ A , then A c ∈ A . A set function μ : A → R + ≡ [0 , ∞ ] is called σ-additive or a measure , if 1. μ ( ∅ ) = 0; 2. for every sequence ( A k ) k ≥ 1 , of disjoint sets in A , μ parenleftBig ∞ uniondisplay k =1 A k parenrightBig = ∞ summationdisplay k =1 μ ( A k ) . WLLN Borel-Cantelli lemma A.s. convergence Relations Integration Simple non-negative functions Let SF + = SF + ( E , A , μ ) be the collection of finite sums k summationdisplay j =1 a j 1I A j ( x ) , x ∈ E , k ∈ N , with a j ∈ [0 , ∞ ] pairwise different (ie., a i = a j iff i = j ), and braceleftbig A 1 , .. . , A k bracerightbig ⊆ A being a finite partition of...
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This note was uploaded on 05/12/2010 for the course APPLIED ST 2010 taught by Professor Various during the Spring '10 term at Universidad Nacional Agraria La Molina.

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0809IntegrationSlides - WLLN Borel-Cantelli lemma A.s...

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