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Fubini's theorem states that if you have a double integral over a function $f(x,y)$, then you can compute the integral as an iterated integral, if $f(x,y)$ is in $\mathcal{L}^1$. But to find out if $f$ is in $\mathcal{L}^1$ you need to compute the double integral.

What am I missing? The examples I found all apply Fubini's theorem without checking that $f(x,y)$ is in $\mathcal{L}^1$. Many thanks for any clarification!

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    Thank you, Qiaochu Yuan. I would like to accept your comment as the answer to my question, if you don't mind re-posting it as answer.2011-01-13

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The comment made by Qiaochu Yuan answered my question but because it's a comment I can't accept it and close this question.

So hereby I declare this question as solved. Thanks to all for your help.

Here is the comment:

You don't have to compute the double integral; you just have to bound it. For example if both of the underlying measure spaces are finite and f is bounded then this is automatically true. Similarly if you are computing an integral over R^2 it suffices to bound f on a sequence of compact subsets of R^2, say concentric circles or unit squares.

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For Fubini to apply you need $f$ to be in $L^1$ as a function of $x$ for almost every $y$ and as a function of $y$ for almost every $x$.

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    That's right. Thanks for the correction.2011-01-13