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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.
 Spring '04
 MichaelChrist
 Math

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