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Just having a little problem solving this, however it probably is pretty easy and I am just being dumb.

Suppose you have a lebesgue integrable function $f$. The goal is, for any $ \epsilon > 0 $, to find a set $C$ with $ \mu (C) < \infty$ such that $\int_{C^c} |f| d \mu < \epsilon$.

Any ideas on how to construct this set? I think it has to do with the Dominated Convergence Theorem, but I don't see it.

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    I think you must have your quantifiers out of order; you probably want to show that for all $\epsilon>0$ there is a set $C$ such that $\int_{C^{c}} |f|\, d\mu < \epsilon$. Otherwise, the function $f(x) = e^{-x^2}$ is a counterexample.2012-11-19
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    Yep, thats what I meant. Sorry.2012-11-19

3 Answers 3