Sheet4 - 2 F . A measure space ( X , A ,) is said to be...

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Statistics 330/600 Due: Thursday 10 February 2005: Sheet 4 No lecture on Tuesday 8 February *(4.1) (Weierstrass approximation) UGMTP Problem 2.25. Note: You would need this result to understand the handout on λ -spaces. Remember, a continuous function on [0 , 1] is automatically uniformly continuous. *(4.2) (Facts about R 2 ) (i) Show that B ( R 2 ) = B ( R ) B ( R ) . Hint: Show that every open subset of R 2 can be written as a countable union of measurable rectangles. (ii) Show that { ( x , y ) R 2 : x = y }∈ B ( R ) B ( R ) . (iii) Show that every continuous function f : R 2 R is B ( R ) B ( R ) -measurable. (4.3) ( product-measurability of py p 1 { f ( x )> y > 0 } . ) UGMTP Problem 4.2. You may assume the results from Problem 1. Hint: Draw diagrams to keep track of all the measurability assertions. *(4.4) Let m i denote Lebesgue measure on B ( R i ) ,fo r i = 1 , 2. Suppose f M + ( R 2 , B ( R 2 )) . (i) Show that the set F : ={ ( x , t ) R × R + : f ( x ) t 0 } belongs to B ( R 2 ) . (ii) Show that m 1 f =
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Unformatted text preview: 2 F . A measure space ( X , A ,) is said to be sigma-nite 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 sigma-nite 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 sigma-nite. 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.

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