Editing Fubini's theorem (section)

===Failure of Fubini's theorem for non-maximal product measures===

Fubini's theorem holds for spaces even if they are not assumed to be σ-finite provided one uses the maximal product measure. In the example above, for the maximal product measure, the diagonal has infinite measure so the double integral of |''f''| is infinite, and Fubini's theorem holds vacuously.
However, if we give ''X''×''Y'' the product measure such that the measure of a set is the sum of the Lebesgue measures of its horizontal sections, then the double integral of |''f''| is zero, but the two iterated integrals still have different values. This gives an example of a product measure where Fubini's theorem fails.

This gives an example of two different product measures on the same product of two measure spaces. For products of two σ-finite measure spaces, there is only one product measure.