Editing Fubini's theorem (section)

==For integrable functions==
Suppose ''X'' and ''Y'' are [[sigma-finite measure|σ-finite]] measure spaces and suppose that ''X''&nbsp;×&nbsp;''Y'' is given the product measure (which is unique as ''X'' and ''Y'' are σ-finite). Fubini's theorem states that if ''f'' is ''X''&nbsp;×&nbsp;''Y'' integrable, meaning that ''f'' is a [[measurable function]] and
<math display="block">\int_{X\times Y} |f(x,y)|\,\text{d}(x,y) < \infty,</math>
then
<math display="block">\int_X\left(\int_Y f(x,y)\,\text{d}y\right)\,\text{d}x = \int_Y\left(\int_X f(x,y)\,\text{d}x\right)\,\text{d}y = \int_{X\times Y} f(x,y)\,\text{d}(x,y).</math>

The first two integrals are iterated integrals with respect to two measures, respectively, and the third is an integral with respect to the product measure. The partial integrals <math display="inline">\int_Y f(x,y)\,\text{d}y</math> and <math display="inline">\int_X f(x,y)\,\text{d}x</math> need not be defined everywhere, but this does not matter as the points where they are not defined form a set of measure 0.

If the above integral of the absolute value is not finite, then the two iterated integrals may have different values.  See [[#Counterexamples|below]] for an illustration of this possibility.

The condition that ''X'' and ''Y'' are σ-finite is usually harmless because almost all measure spaces for which one wishes to use Fubini's theorem are σ-finite. 
Fubini's theorem has some rather technical extensions to the case when ''X'' and ''Y'' are not assumed to be σ-finite {{harv|Fremlin|2003}}. The main extra complication in this case is that there may be more than one product measure on ''X''×''Y''. Fubini's theorem continues to hold for the maximal product measure but can fail for other product measures. For example, there is a product measure and a non-negative measurable function ''f'' for which the double integral of |''f''| is zero but the two iterated integrals have different values; see the section on counterexamples below for an example of this. Tonelli's theorem and the Fubini–Tonelli theorem (stated below) can fail on non σ-finite spaces, even for the maximal product measure.