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Unformatted text preview: Math 8601: REAL ANALYSIS. Fall 2010 Homework #5. Problems and Solutions. #1. Let E be a Lebesgue measurable set in R 1 with Lebesgue measure m ( E ) > 0. Show that for any α < 1, there is an open interval I = ( a,b ) such that m ( E ∩ I ) > αm ( I ). Proof. Suppose that this property is not true, i.e. m ( E ∩ I ) ≤ αm ( I ) for any interval I = ( a,b ) , where α = const < 1 . Fix a small ε > 0, such that α (1+ ε ) < 1. By Theorem 1.18, there is an open set U ⊇ E such that m ( U ) < (1 + ε ) m ( E ). By Proposition 0.21, the open set U is represented as a finite or countable union of disjoint open intervals I j . Therefore, m ( U ) = m ( E ∩ U ) = m E ∩ [ j I j = m [ j ( E ∩ I j ) = X j m ( E ∩ I j ) ≤ α X j m ( I j ) = α m ( U ) < α (1 + ε ) m ( E ) < m ( E ) . This contradiction proves the desired property. #2. Let ( X, M ) be a measurable space, and let f n : X → R 1 be a sequence of Mmeasurable functions on X . Show that the set E := { x ∈ X : ∃ lim...
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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.
 Fall '08
 Staff
 Math

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