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Unformatted text preview: Mathematics 105 — Spring 2004 — M. Christ Problem Set 10 For Friday April 30: Continue to study § 4.1 of our text. Solve the following problems from Stroock § 4.1: 4.1.8, 10, 11, 12. (For 4.1.10, note that the σalgebras in question are the Borel algebras B R k rather than the associated Lebesgue algebras B R k .) For 4.1.11, the things you are asked to show in the sentence “Solve also the following problems . . . ” were (or will be) shown in class; you need not reprove them but should just prove what is asked in the sentence beginning “Finally”. X.A Here are two associative laws: Let ( E j , A j , μ j ), for j = 1 , 2 , 3, be three σfinite measure spaces. Prove that the two σalgebras ( A 1 × A 2 ) × A 3 and A 1 × ( A 2 × A 3 ) (both of which are σalgebras of subsets of E 1 × E 2 × E 3 ) are equal. Prove that the two measures ( μ 1 × μ 2 ) × μ 3 and μ 1 × ( μ 2 × μ 3 ) on E 1 × E 2 × E 3 are equal. X.B Another example: Let ( E j , A j , μ j ) equal N equipped with the σalgebra of all its subsets, and with counting measure. (Thus A 1 ×A 2 consists of all subsets of N × N .) Define f ( x 1 , x 2 ) to be 1 if x 1 = x 2 ∈ N , to be 1 if x 2 = x 1 + 1, and to be 0 otherwise. Show that RR f dμ 1 dμ 2 6 = RR f dμ 2 dμ 1 . Which hypothesis of Fubini’s theorem is not satisfied? X.C As a complement to problem 4.1.10, show that if ( E j , A j , μ j ) = ( R 1 , B R 1 , λ ) where λ denotes Lebesgue measure on R 1 , then the product measure μ 1 × μ 2 on A 1 × A 2 is not complete. (Hint: Consider { } × A where A ⊂ R 1 is not measurable.) X.D Earlier in the course we sweated quite a bit to prove that  T ( A )  =  det( T )  ·  A  for any measurable set A ⊂ R n and any linear transformation...
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 Spring '04
 MichaelChrist
 Algebra, measure

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