solns9b - Mathematics 105 Spring 2004 M. Christ Problem Set...

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Unformatted text preview: Mathematics 105 Spring 2004 M. Christ Problem Set 9 Solutions to Selecta, part 2 3.3.21(i) Let K be a family of measurable functions on a measure space ( E, B , ). Show that K is uniformly -integrable if it is uniformly -absolutely continuous and satisfies sup f K k f k L 1 ( ) < . Conversely, show that if K is uniformly -integrable then it is uniformly -absolutely continuous. Show that if in addition ( E ) < , then sup f K k f k L 1 ( ) < . Solution. Suppose that K is uniformly -absolutely continouus and satisfies sup f K k f k L 1 ( ) = C < . For any positive number R < and any f K , ( { x : | f ( x ) | R } ) R- 1 k f k L 1 CR- 1 by Markovs inequality. Let > 0. By hypothesis there exists > 0 such that R A | f | d < for all A B satisfying ( A ) < . If R is chosen to be sufficiently large that C/R < then the set A = { x : | f ( x ) | R } satisfies ( A ) < and hence R { x : | f ( x ) | R } | f | d < ....
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This note was uploaded on 05/07/2008 for the course MATH 105 taught by Professor Michaelchrist during the Spring '04 term at University of California, Berkeley.

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solns9b - Mathematics 105 Spring 2004 M. Christ Problem Set...

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