Chicken-Egg problem with Fubini’s theorem

Solution 1:

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.