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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 Spring '09
 DavidPollard
 Statistics, Probability, Continuous function, Ri, measure, Lebesgue measure

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