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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 Fall '08
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 Math, measure, Lebesgue measure, Lebesgue integration, measure space, E M

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