Fubini's theorem: Difference between revisions

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{{technical|date=August 2020}}

In [[mathematical analysis]] '''Fubini's theorem''', introduced by [[Guido Fubini]] in 1907, is a result that gives conditions under which it is possible to compute a [[double integral]] by using an [[iterated integral]]. One may switch the order of integration if the double integral yields a finite answer when the integrand is replaced by its absolute value.

: <math> \int_Xint_{X\left(\int_Ytimes Y} f(x,y)\,\text{d}(x,y\right)\,\text{d}x = \int_Yint_X\left(\int_Xint_Y f(x,y)\,\text{d}xy\right)\,\text{d}yx=\int_{Xint_Y\times Y}left(\int_X f(x,y)\,\text{d}(x\right)\,\text{d}y) </math>

As a consequence, it allows the [[Order of integration (calculus)|order of integration]] to be changed in certain iterated integrals.

Fubini's theorem implies that two iterated integrals are equal to the corresponding double integral across its integrands. '''Tonelli's theorem''', introduced by [[Leonida Tonelli]] in 1909, is similar, but applies to a non-negative measurable function rather than one integrable over its ___domain.