NationMaster - Encyclopedia: Fubini's theorem

In mathematical analysis, Fubini's theorem, named in honor of Guido Fubini, states that if Image File history File links Please see the file description page for further information. ... It has been suggested that this article or section be merged into Fubinis theorem. ... Analysis is the generic name given to any branch of mathematics that depends upon the concepts of limits and convergence. ... Guido Fubini (January 19, 1879 - June 6, 1943) was an Italian mathematician, best known for Fubinis theorem. ...

$int_{Atimes B} |f(x,y)|,d(x,y)<infty,$

the integral being taken with respect to a product measure on the space over $Atimes B$ , then In mathematics, given two measurable spaces and measures on them, one can obtain the product measurable space and the product measure on that space. ...

$int_Aleft(int_B f(x,y),dyright),dx=int_Bleft(int_A f(x,y),dxright),dy=int_{Atimes B} f(x,y),d(x,y),$

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

$int_A f(x), dx int_B g(y), dy = int_{Atimes B} f(x)g(y),d(x,y)$

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 a counterexample related to Fubini's theorem for an illustration of this possibility. It has been suggested that this article or section be merged into Fubinis theorem. ...

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: The integral of any Gaussian function (named after Carl Friedrich Gauss) is quickly reducible to the Gaussian integral This integral cannot be computed by elementary means since the function has no simple antiderivative. ...

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

To see how Fubini's theorem is used to prove this, see Gaussian integral. The integral of any Gaussian function (named after Carl Friedrich Gauss) is quickly reducible to the Gaussian integral This integral cannot be computed by elementary means since the function has no simple antiderivative. ...

Guido Fubini at AllExperts (363 words)
	Born in Venice, he was steered towards mathematics at an early age by his teachers and his father, who was himself a teacher of mathematics.
	In 1939, when Fubini at the age of 60 was nearing retirement, Benito Mussolini's Fascists adopted the anti-Jewish policies advocated for several years by Adolf Hitler's Nazis.
	As a Jew, Fubini feared for the safety of his family, and so accepted an invitation by Princeton University to teach there; he died in New York City four years later.

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Applications

See also

Tonelli's theorem