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Unformatted text preview: 2 F . A measure space ( X , A ,) is said to be sigmanite if there are sets { X i : i N } A with i N X i = X and X i < for each i. (4.5) Let and be sigmanite measures on B ( R ) . Suppose is nonatomic, that is, { x } = 0 for each x in R . Show that { ( x , y ) R 2 : x = y } = 0. *(4.6) Let ( X , A ,) and ( Y , B ,) be two measure spaces, with both and sigmanite. For each f in L 1 ( X Y , A B , ) and each > 0 show that there exist sets A i A and B i B and real numbers i , for i = 1 , 2 ,..., k , such that  f ( x , y ) i i { x A i , y B i } < . Hint: Consider rst the case where both measures are nite....
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This note was uploaded on 11/21/2009 for the course STAT 330 taught by Professor Davidpollard during the Spring '09 term at Yale.
 Spring '09
 DavidPollard
 Statistics, Probability

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