Fubini's theorem

In mathematical analysis, Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. It states that if a function is integrable on a rectangle $X\times Y$ , then one can evaluate the double integral as an iterated integral:

\,\iint \limits _{X\times Y}f(x,y)\,{\text{d}}(x,y)=\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.

Tonelli's theorem, introduced by Leonida Tonelli in 1909, is similar, but applies to a non-negative measurable function rather than one integrable over their domains. Fubini's and Tonelli's theorems combine to the Fubini-Tonelli theorem (see below), which allows one to switch the order of integration in an iterated integral under certain conditions.

A related theorem is often called Fubini's theorem for infinite series, which states that if ${\textstyle \{a_{m,n}\}_{m=1,n=1}^{\infty }}$ is a doubly-indexed sequence of real numbers, and if ${\textstyle \sum _{(m,n)\in \mathbb {N} \times \mathbb {N} }a_{m,n}}$ is absolutely convergent, then

$\sum _{(m,n)\in \mathbb {N} \times \mathbb {N} }a_{m,n}=\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }a_{m,n}=\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }a_{m,n}$

Although Fubini's theorem for infinite series is a special case of the more general Fubini's theorem, it is not appropriate to characterize it as a logical consequence of Fubini's theorem. This is because some properties of measures, in particular sub-additivity, are often proved using Fubini's theorem for infinite series. In this case, Fubini's general theorem is a logical consequence of Fubini's theorem for infinite series.

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