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Unformatted text preview: Serge Ballif MATH 501 Homework 5 October 31, 2007 In all problems ( X, M ,μ ) 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 ) dμ ( x ) = Z ∂ y f ( x,y ) dμ ( 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 μ ( 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 : μ ( { 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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 Fall '08
 WYSOCKI,KRZYSZTOF
 Math, lim, FN, Dominated convergence theorem, Monotone convergence theorem, Fatou's lemma

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