S6 - Math 8601 REAL ANALYSIS Fall 2010 Homework#6 Problems and Solutions#1(Borel-Cantelli Lemma Let X M,μ be a measure space and A n be a sequence

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Unformatted text preview: Math 8601: REAL ANALYSIS. Fall 2010 Homework #6. Problems and Solutions. #1 (Borel-Cantelli Lemma). Let ( X, M ,μ ) be a measure space, and { A n } be a sequence of sets in M . Show that if ∞ X n =1 μ ( A n ) < ∞ , then μ limsup n →∞ A n = 0 , where limsup n →∞ A n := ∞ \ k =1 ∞ [ n = k A n . Proof. We have for k = 1 , 2 ,... , μ limsup n →∞ A n ≤ μ ∞ [ n = k A n ≤ ∞ X n = k μ ( A n ) → 0 as k → ∞ . Since the left side is non-negative and does not depend on k , it must be 0. #2. The symmetric difference of two sets A and B is defined as A Δ B = ( A- B ) ∪ ( B- A ) = ( A ∪ B )- ( A ∩ B ) . Let A 1 ,A 2 , ··· be a sequence of measurable sets in a measure space ( X, M ,μ ) satisfying lim n →∞ sup j,k ≥ n μ ( A j Δ A k ) = 0 . Show that there exists a measurable set A , such that lim k →∞ μ ( A k Δ A ) = 0 . Proof. For every ε ∈ (0 , 1), μ χ A j- χ A k ≥ ε = μ χ A j- χ A k = ± 1 = μ ( A j Δ A k ) → 0 as j,k → ∞ ....
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This note was uploaded on 02/15/2012 for the course MATH 8601 taught by Professor Staff during the Fall '08 term at Minnesota.

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S6 - Math 8601 REAL ANALYSIS Fall 2010 Homework#6 Problems and Solutions#1(Borel-Cantelli Lemma Let X M,μ be a measure space and A n be a sequence

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