This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 8601: REAL ANALYSIS. Fall 2010 Homework #6. Problems and Solutions. #1 (BorelCantelli 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 nonnegative 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 → ∞ ....
View
Full
Document
This note was uploaded on 02/15/2012 for the course MATH 8601 taught by Professor Staff during the Fall '08 term at Minnesota.
 Fall '08
 Staff
 Sets

Click to edit the document details