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Sheet4 - 2 F A measure space X A,µ 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 ) , for 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
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Unformatted text preview: 2 F . A measure space ( X , A ,µ) is said to be sigma-±nite if there are sets { X i : i ∈ N } ⊆ A with ∪ i ∈ N X i = X and µ X i < ∞ for each i. (4.5) Let µ and ν be sigma-±nite measures on B ( R ) . Suppose µ is nonatomic, that is, µ { x } = 0 for each x in R . Show that µ ⊗ ν { ( x , y ) ∈ R 2 : x = y } = 0. *(4.6) Let ( X , A ,µ) and ( Y , B ,ν) be two measure spaces, with both µ and ν sigma-±nite. For each f in L 1 ( X × Y , A ⊗ B ,µ ⊗ ν) and each ² > 0 show that there exist sets A i ∈ A and B i ∈ B and real numbers α i , for i = 1 , 2 ,..., k , such that µ ⊗ ν | f ( x , y ) − ∑ i α i { x ∈ A i , y ∈ B i }| < ² . Hint: Consider ±rst the case where both measures are ±nite....
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