Editing Fubini's theorem (section)

==Tonelli's theorem for non-negative measurable functions==
'''{{visible anchor|Tonelli's theorem}}''', named after [[Leonida Tonelli]], is a successor of Fubini's theorem. The conclusion of Tonelli's theorem is identical to that of Fubini's theorem, but the assumption that <math>|f|</math> has a finite integral is replaced by the assumption that <math>f</math> is a non-negative measurable function.

Tonelli's theorem states that if <math> (X, A, \mu) </math> and <math> (Y, B, \nu) </math> are [[Sigma-finite measure|σ-finite measure spaces]], while <math> f:X\times Y \to [0,\infty] </math> is a non-negative 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>

A special case of Tonelli's theorem is in the interchange of the summations, as in <math display="inline">\sum_x \sum_y a_{xy} = \sum_y \sum_x a_{xy}</math>, where <math>a_{xy}</math> are non-negative for all ''x'' and ''y''. The crux of the theorem is that the interchange of order of summation holds even if the series diverges. In effect, the only way a change in order of summation can change the sum is when there exist some subsequences that diverge to <math>+\infty</math> and others diverging to <math>-\infty</math>. With all elements non-negative, this does not happen in the stated example.

Without the condition that the measure spaces are σ-finite, all three of these integrals can have different values. Some authors give generalizations of Tonelli's theorem to some measure spaces that are not σ-finite, but these generalizations often add conditions that immediately reduce the problem to the σ-finite case.  For example, one could take the σ-algebra on ''A'' × ''B'' to be that generated by the product of subsets of finite measure, rather than that generated by all products of measurable subsets, though this has the undesirable consequence that the projections from the product to its factors ''A'' and ''B'' are not measurable. Another way is to add the condition that the support of ''f'' is contained in a countable union of products of sets of finite measures. {{harvtxt|Fremlin|2003}} gives some rather technical extensions of Tonelli's theorem to some non σ-finite spaces. None of these generalizations have found any significant applications outside of abstract measure theory, largely because almost all measure spaces of practical interest are σ-finite.