hw8solutions

hw8solutions - Math 318 HW #8 Solutions 1. (a) Prove...

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Math 318 HW #8 Solutions 1. (a) Prove Chebyshev’s inequality, which says that if f is nonnegative and measurable on the bounded, measurable set A , then m { x A : f ( x ) c } ≤ 1 c Z A f dm. Proof. Let c R and define A c := { x A : f ( x ) c } , which is measurable since f is a measurable function. Then f ( x ) c on A c , so, by Theorem 25.4(2), Z A c f dm cm { x A : f ( x ) c ) } . Therefore, m { x A : f ( x ) c } ≤ 1 c Z A c f dm 1 c Z A f dm, where the last inequality follows from Theorem 25.3 and the fact that f is nonnegative. (b) Show that if R A | f | dm = 0, then f = 0 a.e. on A . Proof. For any n N , Chebyshev’s inequality implies that m { x A : | f ( x ) | > 1 /n } ≤ n Z A | f | dm = 0 . Letting B n = { x A : | f ( x ) | > 1 /n } and B = { x A : | f ( x ) | > 0 } , we have that B = [ n =1 B n , and B 1 B 2 ⊂ ··· . Therefore, Corollary 10.9 implies that m ( B ) = lim n →∞ m ( B n ) = lim n
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hw8solutions - Math 318 HW #8 Solutions 1. (a) Prove...

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