In mathematical analysis, Fubini's theorem, named in honor of Guido Fubini, states that if

the integral being taken with respect to a product measure on the space over $A\backslash times\; B$, then

$\backslash int\_A\backslash left(\backslash int\_B\; f(x,y)\backslash ,dy\backslash right)\backslash ,dx=\backslash int\_B\backslash left(\backslash int\_A\; f(x,y)\backslash ,dx\backslash right)\backslash ,dy=\backslash int\_\{A\backslash times\; B\}\; f(x,y)\backslash ,d(x,y),$

the first two integrals being iterated integrals, and the third being an integral with respect to a product measure. Also,

$\backslash int\_A\; f(x)\backslash ,\; dx\; \backslash int\_B\; g(y)\backslash ,\; dy\; =\; \backslash int\_\{A\backslash times\; B\}\; f(x)g(y)\backslash ,d(x,y)$

the third integral being with respect to a product measure.

If the above integral of the absolute value is not finite, then the two iterated integrals may actually have different values. See a counterexample related to Fubini's theorem for an illustration of this possibility.

Tonelli's theorem

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Tonelli's theorem (named after Leonida Tonelli) is a predecessor 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 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), is guaranteed by Tonelli's theorem (see below).

Applications

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One of the most beautiful applications of Fubini's theorem is the evaluation of the Gaussian integral which is the basis for much of probability theory:

$\backslash int\_\{-\backslash infty\}^\backslash infty\; e^\{-x^2\}\backslash ,dx\; =\; \backslash sqrt\{\backslash pi\}.$

To see how Fubini's theorem is used to prove this, see Gaussian integral.

Another nice use of Tonelli's theorem is to apply it to $|f(x,y)|$ for a complex valued function $f$.

It use useful to note that if

$\backslash varphi(x)=\backslash int\; |f(x,y)|\backslash ,dy$ and

then

This is often a useful way to check the conditions of Fubini's theorem.

See also

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Clairaut's theorem

Mathematical theorems

Satz von Fubini | Théorème de Fubini | Teorema di Fubini | Fubinin lause | Теорема Тонелли — Фубини | 富比尼定理

References

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W. Rudin "Real and Complex Analysis", McGraw Hill, 1987.