Fubini's theorem

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

As a consequence it allows the order of integration to be changed in iterated integrals. Fubini's theorem implies that the two repeated integrals of a function of two variables are equal if the function is integrable. Tonelli's theorem introduced by Leonida Tonelli (1909) is similar but applies to functions that are non-negative rather than integrable.

The special case of Fubini's theorem for continuous functions on a product of closed bounded subsets of real vector spaces was known to Euler in the 18th century. Lebesgue (1904) extended this to bounded measurable functions on a product of intervals. Levi (1906) conjectured that the theorem could be extended to functions that were integrable rather than bounded, and this was proved by Fubini (1907). Tonelli (1909) gave a variation of Fubini's theorem that applies to non-negative functions rather than integrable functions.

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.)

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