Fubini's theorem

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

\int_{A\times B} |f(x,y)|\,d(x,y)<\infty,

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

\int_A\left(\int_B f(x,y)\,dy\right)\,dx=\int_B\left(\int_A f(x,y)\,dx\right)\,dy=\int_{A\times B} f(x,y)\,d(x,y),

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

\int_A f(x)\, dx \int_B g(y)\, dy = \int_{A\times B} f(x)g(y)\,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 examples for an illustration of this possibility.

Tonelli's theorem

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.

Applications

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:

\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}.

To see how Fubini's theorem is used to prove this, see Gaussian integral.de:Satz von Fubini fi:Fubinin lause zh:富比尼定理

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