Fubini's theorem

Fubini's theorem For the Fubini theorem for category, see Kuratowski–Ulam theorem. This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (August 2020) (Learn how and when to remove this template message) 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 _{Xtimes Y}f(x,y),{text{d}}(x,y)=int _{X}left(int _{Y}f(x,y),{text{d}}yright),{text{d}}x=int _{Y}left(int _{X}f(x,y),{text{d}}xright),{text{d}}yqquad {text{ if }}qquad iint limits _{Xtimes Y}|f(x,y)|,{text{d}}(x,y)<+infty .} As a consequence, it allows the 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 their domains. A related theorem is often called Fubini's theorem for infinite series,[1] which states that if {textstyle {a_{m,n}}_{m=1,n=1}^{infty }} is a doubly-indexed sequence of real numbers, and if {textstyle sum _{(m,n)in mathbb {N} times mathbb {N} }a_{m,n}} is absolutely convergent, then {displaystyle sum _{(m,n)in mathbb {N} times 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.[2] In this case, Fubini's general theorem is a logical consequence of Fubini's theorem for infinite series. Contents 1 History 2 Product measures 3 For integrable functions 4 Tonelli's theorem for non-negative measurable functions 5 Fubini–Tonelli theorem 6 For complete measures 7 Proofs 7.1 Riemann integrals 8 Counterexamples 8.1 Failure of Tonelli's theorem for non σ-finite spaces 8.2 Failure of Fubini's theorem for non-maximal product measures 8.3 Failure of Tonelli's theorem for non-measurable functions 8.4 Failure of Fubini's theorem for non-measurable functions 8.5 Failure of Fubini's theorem for non-integrable functions 9 See also 10 References 11 Further reading 12 External links History The special case of Fubini's theorem for continuous functions on a product of closed bounded subsets of real vector spaces was known to Leonhard Euler in the 18th century. Henri Lebesgue (1904) extended this to bounded measurable functions on a product of intervals.[3] Levi (1906) conjectured that the theorem could be extended to functions that were integrable rather than bounded, and this was proved by Fubini (1907).[4] Leonida Tonelli (1909) gave a variation of Fubini's theorem that applies to non-negative functions rather than integrable functions.[5] Product measures If X and Y are measure spaces with measures, there are several natural ways to define a product measure on their product. The product X × Y of measure spaces (in the sense of category theory) has as its measurable sets the σ-algebra generated by the products A × B of measurable subsets of X and Y. A measure μ on X × Y is called a product measure if μ(A × B) = μ1(A)μ2(B) for measurable subsets A ⊂ X and B ⊂ Y and measures µ1 on X and µ2 on Y. In general there may be many different product measures on X × Y. Fubini's theorem and Tonelli's theorem both need technical conditions to avoid this complication; the most common way is to assume all measure spaces are σ-finite, in which case there is a unique product measure on X×Y. There is always a unique maximal product measure on X × Y, where the measure of a measurable set is the inf of the measures of sets containing it that are countable unions of products of measurable sets. The maximal product measure can be constructed by applying Carathéodory's extension theorem to the additive function μ such that μ(A × B) = μ1(A)μ2(B) on the ring of sets generated by products of measurable sets. (Carathéodory's extension theorem gives a measure on a measure space that in general contains more measurable sets than the measure space X × Y, so strictly speaking the measure should be restricted to the σ-algebra generated by the products A × B of measurable subsets of X and Y.) The product of two complete measure spaces is not usually complete. For example, the product of the Lebesgue measure on the unit interval I with itself is not the Lebesgue measure on the square I × I. There is a variation of Fubini's theorem for complete measures, which uses the completion of the product of measures rather than the uncompleted product. For integrable functions Suppose X and Y are σ-finite measure spaces, and suppose that X × Y is given the product measure (which is unique as X and Y are σ-finite). Fubini's theorem states that if f is X × Y integrable, meaning that f is a measurable function and {displaystyle int _{Xtimes Y}|f(x,y)|,{text{d}}(x,y)

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