fubini's theorem

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. For an example, see an elegant rearrangement of a conditionally convergent iterated integral.

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}.

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