Loterre: Mathematics: Fubini's theorem

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Preferred term

Fubini's theorem

Definition(s)

In mathematical analysis, Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if the double integral yields a finite answer when the integrand is replaced by its absolute value.
${\\displaystyle \\,\\iint \\limits _{X\ imes Y}f(x,y)\\,{\ ext{d}}(x,y)=\\int _{X}\\left(\\int _{Y}f(x,y)\\,{\ ext{d}}y\ ight){\ ext{d}}x=\\int _{Y}\\left(\\int _{X}f(x,y)\\,{\ ext{d}}x\ ight){\ ext{d}}y\\qquad {\ ext{ if }}\\qquad \\iint \\limits _{X\ imes Y}|f(x,y)|\\,{\ ext{d}}(x,y)<+\\infty .}$
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 their domains.
A related theorem is often called Fubini's theorem for infinite series, which states that if ${\ extstyle \\{a_{m,n}\\}_{m=1,n=1}^{\\infty }}$ is a doubly-indexed sequence of real numbers, and if ${\ extstyle \\sum _{(m,n)\\in \\mathbb {N} \ imes \\mathbb {N} }a_{m,n}}$ is absolutely convergent, then
${\\displaystyle \\sum _{(m,n)\\in \\mathbb {N} \ imes \\mathbb {N} }a_{m,n}=\\sum _{m=1}^{\\infty }\\sum _{n=1}^{\\infty }a_{m,n}=\\sum _{n=1}^{\\infty }\\sum _{m=1}^{\\infty }a_{m,n}}$
Although Fubini's theorem for infinite series is a special case of the more general Fubini's theorem, it is not appropriate to characterize it as a logical consequence of Fubini's theorem. This is because some properties of measures, in particular sub-additivity, are often proved using Fubini's theorem for infinite series. In this case, Fubini's general theorem is a logical consequence of Fubini's theorem for infinite series
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Fubini%27s_theorem)