Fubini's theorem

In mathematical analysis, Fubini's theorem characterizes the conditions under which it is possible to compute a double integral by using an iterated integral. It was introduced by Guido Fubini in 1907. The theorem states that if a function is Lebesgue integrable on a rectangle ${\displaystyle X\times Y}$, then one can evaluate the double integral as an iterated integral:$${\displaystyle \,\iint \limits _{X\times Y}f(x,y)\,{\text{d}}(x,y)=\int _{X}\left(\int _{Y}f(x,y)\,{\text{d}}y\right){\text{d}}x=\int _{Y}\left(\int _{X}f(x,y)\,{\text{d}}x\right){\text{d}}y.}$$ This formula is generally not true for the Riemann integral, but it is true if the function is continuous on the rectangle. In multivariable calculus, this weaker result is sometimes also called Fubini's theorem, although it was already known by Leonhard Euler.

Tonelli's theorem, introduced by Leonida Tonelli in 1909, is similar but is applied to a non-negative measurable function rather than to an integrable function over its domain. The Fubini and Tonelli theorems are usually combined and form the Fubini-Tonelli theorem, which gives the conditions under which it is possible to switch the order of integration in an iterated integral.

A related theorem is often called Fubini's theorem for infinite series,^[1] although it is due to Alfred Pringsheim.^[2] It states that if ${\textstyle \{a_{m,n}\}_{m=1,n=1}^{\infty }}$ is a double-indexed sequence of real numbers, and if ${\textstyle \displaystyle \sum _{(m,n)\in \mathbb {N} \times \mathbb {N} }a_{m,n}}$ 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 necessarily appropriate to characterize the former as being proven by the latter because the properties of measures needed to prove Fubini's theorem proper, in particular subadditivity of measure, may be proven using Fubini's theorem for infinite series.^[3]