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Math Help - lebesgue integral exercise, it's easy but I still require assistance

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    lebesgue integral exercise, it's easy but I still require assistance

    Hi guys. I'm just starting to work with Lebesgue integrals, and so even the easy stuff is awkward for me.

    Let f be a nonnegative measurable function. Show that \int f=0 implies f=0 almost everywhere (i.e. except on a set of measure zero).
    EDIT: Okay, I cracked it. Here's my solution, for reference purposes:

    Let D denote the domain of f, and put A_n=\{x:f(x)>1/n\} for each positive integer n and A=\{x:f(x)>0\}. Then D-A_n=\{x:f(x)\in[0,1/n]\} is measurable, and so by prop. 4.12 of Royden, Real Analysis 3rd Ed (p87), 0\leq mA_n/n\leq \int_{D-A_n}f+\int_{A_n}f=\int f=0. Thus we have mA_n=0 for all n, which means 0\leq mA=m(\bigcup A_n)\leq\sum mA_n=0 and therefore mA=0. The conclusion follows. \blacksquare
    Last edited by hatsoff; October 28th 2010 at 09:05 AM.
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