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

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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.
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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 ) .
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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 E .
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