Editing Fubini's theorem (section)

==Fubini–Tonelli theorem==
Combining Fubini's theorem with Tonelli's theorem gives the Fubini–Tonelli theorem. Often just called Fubini's theorem, it states that if <math>X</math> and <math>Y</math> are [[σ-finite measure]] spaces, and if <math>f</math> is a measurable function, 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>
Furthermore, if any one of these integrals is finite, 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 absolute value of <math>f</math> in the conditions above can be replaced by either the positive or the negative part of <math>f</math>; these forms include Tonelli's theorem as a special case as the negative part of a non-negative function is zero and so has finite integral. Informally, all these conditions say that the double integral of <math>f</math> is well defined, though possibly infinite.

The advantage of the Fubini–Tonelli over Fubini's theorem is that the repeated integrals of <math>|f|</math> may be easier to study than the double integral. As in Fubini's theorem, the single integrals may fail to be defined on a measure 0 set.