hw10 - Mathematics 105 Spring 2004 M. Christ Problem Set 10...

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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 -algebras 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 , j ), for j = 1 , 2 , 3, be three -finite measure spaces. Prove that the two -algebras ( A 1 A 2 ) A 3 and A 1 ( A 2 A 3 ) (both of which are -algebras of subsets of E 1 E 2 E 3 ) are equal. Prove that the two measures ( 1 2 ) 3 and 1 ( 2 3 ) on E 1 E 2 E 3 are equal. X.B Another example: Let ( E j , A j , j ) equal N equipped with the -algebra 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 d 1 d 2 6 = RR f d 2 d 1 . Which hypothesis of Fubinis theorem is not satisfied? X.C As a complement to problem 4.1.10, show that if ( E j , A j , j ) = ( R 1 , B R 1 , ) where denotes Lebesgue measure on R 1 , then the product measure 1 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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hw10 - Mathematics 105 Spring 2004 M. Christ Problem Set 10...

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