Fubini's Theorem - Tonelli's Theorem

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 assumptions are different. Tonelli's theorem states that on the product of two σ-finite measure spaces, a product measure integral can be evaluated by way of an iterated integral for nonnegative measurable functions, regardless of whether they have finite integral.

In fact, the existence of the first integral above (the integral of the absolute value), can be guaranteed by Tonelli's theorem (see below).

A formal statement of Tonelli's theorem is identical to that of Fubini's theorem, except that the requirements are now that (X, A, μ) and (Y, B, ν) are σ-finite measure spaces, while f maps X×Y to [0,∞).

An example where Tonelli's theorem is used is in the interchange of the summations, as in, where 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 which diverge to and others diverging to . With all non-negative elements, this does not happen in the stated example.