TITLE: Chicken-Egg problem with Fubini’s theorem
QUESTION [10 upvotes]: 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!

REPLY [6 votes]: 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.