p5 - be a measure space with μ X< ∞ and let f be as in...

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Math 8601: REAL ANALYSIS. Fall 2010 Homework #5 (due on Wednesday, November 10). 50 points are divided between 5 problems, 10 points each. #1 (Problem 30 on p.40. 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 ). #2 (Problem 3 on p.48. Let ( X, M ) be a measurable space, and let f n : X R 1 be a sequence of M -measurable functions on X . Show that the set E := { x X : lim n →∞ f n ( x ) } is M -measurable. #3 (Problem 16 on p.52. Let ( X, M ) be a measure space, and let f L + - the space of all measurable functions from X to [0 , ], with R f dμ < . Show that for every ε > 0, there exists a set E ∈ M such that μ ( E ) < and Z E f dμ > Z f dμ - ε. #4. Let ( X, M
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Unformatted text preview: ) be a measure space with μ ( X ) < ∞ , and let f be as in the previous problem. Give a direct proof, based on the definition of R f dμ on p.50, of the following fact: ∀ ε > , ∃ δ > 0 such that from E ∈ M and μ ( E ) < δ it follows Z E f dμ < ε. Remark. This fact is known as the absolute continuity of the Lebesgue integral, and it is contained in Corollary 3.6 on p.89. #5. Let ( X, M ,μ ) be a measure space with μ ( X ) < ∞ . Show that d ( f + g ) ≤ d ( f ) + d ( g ) for all measurable functions f, g on X, where d ( f ) := inf ε> ± ε + μ { x ∈ X : | f ( x ) | ≥ ε } ² ....
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