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Math Help - Lebesgue Integral

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    Lebesgue Integral

    If f is measurable function such that \int_B f =0 for all measurable sets B \subset A ,show that f =0 almost everywhere on A.
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    Quote Originally Posted by problem View Post
    If f is measurable function such that \int_B f =0 for all measurable sets B \subset A ,show that f =0 almost everywhere on A.
    For n=1,2,3,..., let B_n = \{x\in A:f(x)\geqslant1/n\}. Then B_n is a measurable subset of A, and so \int_{B_n} \!\!f\,d\mu =0 (where \mu denotes the measure). But \int_{B_n}\!\!f\,d\mu \geqslant\mu(B_n)/n. Therefore \mu(B_n)=0. Since this holds for all n, it follows that \textstyle\mu\Bigl(\bigcup_nB_n\Bigr)=0, which implies that f is nonpositive almost everywhere. Now do the same for the sets on which f  \leqslant-1/n, to conclude that f is nonnegative almost everywhere. Hence f=0 almost everywhere.
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