p6 - Math 8601 REAL ANALYSIS Fall 2010 Homework#6(due on...

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Math 8601: REAL ANALYSIS. Fall 2010 Homework #6 (due on Wednesday, November 24). 50 points are divided between 5 problems, 10 points each. #1 (Borel-Cantelli Lemma). Let ( X, M ) 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 ± lim sup n →∞ A n ² = 0 , where lim sup n →∞ A n := ³ k =1 [ n = k A n . #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 ) 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 . #3. Let f ( x ) be a right continuous function on R 1 , i.e. lim x x 0 ,x>x 0 f ( x ) = f ( x 0 ) for all x 0 R 1 . (a). Show that f is measurable. (b). Show that
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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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