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It is known that an integrable function is a.e. finite. Is an a.e. finite function integrable? What if the measure is finite?

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    $\begingroup$ The function $f(x) = 1$ is finite for all $x\in\Bbb R$ but it's not integrable on $\Bbb R$. $\endgroup$ Oct 13, 2015 at 0:30

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No. A characteristic function of a non measurable set is everywhere finite, but not integrable.

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No, just consider the constant function $1$. It is not integrable on the real line.

You don't even need an unbounded domain. Let $f(x) = \frac 1x$ and integrate over $[0,1]$ to find a counterexample to your statement.

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No. $\frac{1}{x}$ is an example.

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