Fubini theorem

A theorem that establishes a connection between a multiple integral and a repeated one. Suppose that $(X,\mathfrak S_X,\mu_x)$ and $(Y,\mathfrak S_Y,\mu_y)$ are measure spaces with $\sigma$-finite complete measures $\mu_x$ and $\mu_y$ defined on the $\sigma$-algebras $\mathfrak S_X$ and $\mathfrak S_Y$, respectively. If the function $f(x,y)$ is integrable on the product $X\times Y$ of $X$ and $Y$ with respect to the product measure $\mu=\mu_x\times\mu_y$ of $\mu_x$ and $\mu_y$, then for almost-all $y\in Y$ the function $f(x,y)$ of the variable $x$ is integrable on $X$ with respect to $\mu_x$, the function $g(y)=\int_Xf(x,y)d\mu_x$ is integrable on $Y$ with respect to $\mu_y$, and one has the equality

$$\int\limits_{X\times Y}f(x,y)d\mu=\int\limits_Yd\mu_y\int\limits_Xf(x,y)d\mu_x.\label{1}\tag{1}$$

Fubini's theorem is valid, in particular, for the case when $\mu_x$, $\mu_y$ and $\mu$ are the Lebesgue measures in the Euclidean spaces $\mathbf R^m$, $\mathbf R^n$ and $\mathbf R^{m+n}$ respectively ($m$ and $n$ are natural numbers), $X=\mathbf R^m$, $Y=\mathbf R^n$, $X\times Y=\mathbf R^m\times\mathbf R^n=\mathbf R^{m+n}$, and $f=f(x,y)$ is a Lebesgue-measurable function on $\mathbf R^{m+n}$, $x\in\mathbf R^m$, $y\in\mathbf R^n$. Under these assumptions, formula \eqref{1} has the form

$$\iint\limits_{\mathbf R^{m+n}}f(x,y)d(x,y)=\int\limits_{\mathbf R^n}dy\int\limits_{\mathbf R^m}f(x,y)dx.\label{2}\tag{2}$$

In the case of a function $f$ defined on an arbitrary Lebesgue-measurable set $E\subset\mathbf R^{m+n}$, in order to express the multiple integral in terms of a repeated one, one must extend $f$ by zero to the whole of $\mathbf R^{m+n}$ and apply \eqref{2}. See also Repeated integral.

The theorem was established by G. Fubini [1].

References

[1]	G. Fubini, "Sugli integrali multipli" , Opere scelte , 2 , Cremonese (1958) pp. 243–249 Zbl 38.0343.02