Fubini’s theorem: Let $(X,\Sigma,\mu)$ and $(Y,\mathcal T,\nu)$ be $\sigma$-finite measure spaces, and let $f:X\times Y\to\mathbb R$ be $(\Sigma\otimes\mathcal T)$-measurable. If $\int_{X\times Y} |f|\,d(\mu\times\nu)<\infty,$ then for $\mu$-almost every $x\in X$ the section $y\mapsto f(x,y)$ is $\nu$-integrable, and for $\nu$-almost every $y\in Y$ the section $x\mapsto f(x,y)$ is $\mu$-integrable. Moreover, the iterated integrals exist as finite numbers and satisfy [ \int_{X\times Y} f,d(\mu\times\nu)

\int_X\Big(\int_Y f(x,y),d\nu(y)\Big),d\mu(x)

\int_Y\Big(\int_X f(x,y),d\mu(x)\Big),d\nu(y). ]

This theorem applies to integrable functions on a product measure space and justifies computing a Lebesgue integral by iterated integration. Compare Tonelli's theorem , which gives the same conclusion for nonnegative functions without assuming $\int|f|<\infty$.