MATH501_HW5 - Serge Ballif MATH 501 Homework 5 October 31,...

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Unformatted text preview: Serge Ballif MATH 501 Homework 5 October 31, 2007 In all problems ( X, M , ) is a measure space 1. Let f : X [ a,b ] R where [ a,b ] is a finite interval. Assume that the function x 7 f ( x,y ) is integrable for every y [ a,b ] . Assume also that y f ( x,y ) exists for every ( x,y ) and there exists a function g L 1 such that | y f ( x,y ) | g ( x ) for all ( x,y ) . Show that the function x 7 y f ( x,y ) is measurable for every y and y Z f ( x,y ) d ( x ) = Z y f ( x,y ) d ( x ) . For a fixed y [ a,b ] define h n ( x,y ) := f ( x,y +1 /n )- f ( x,y ) 1 /n for n N . Then h n is measurable as a multiple of a difference of integrable functions. Moreover, lim n h n ( x ) exists and is equal to y f ( x,y ). The inequality | h n ( x ) | = | y f ( x,y ) | g ( x ) allows us to invoke the dominated convergence theorem as follows. y Z f ( x,y ) = lim h R f ( x,y + h )- R f ( x,y ) h = lim n Z h n = Z lim n h n = Z lim h f ( x,y + h )- f ( x,y ) h = Z y f ( x,y ) . 2. Prove: (a) k f n- f k if and only if there exists E M such that ( E c ) = 0 and f n f uniformly on E . (b) ( L , k k ) is a Banach space. (a) ( ) Suppose that k f n- f k 0. Then for any > 0, there exists N such that for n N , k f n- f k = inf { a 0 : ( { x X : | f n ( x )- f ( x ) | > a } ) = } < . Then for each m N , there exists N such that whenever n > N the set E m := { x...
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MATH501_HW5 - Serge Ballif MATH 501 Homework 5 October 31,...

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