Tonelli’s theorem: Let $(X,\Sigma,\mu)$ and $(Y,\mathcal T,\nu)$ be $\sigma$-finite measure spaces, and let $f:X\times Y\to[0,\infty]$ be $(\Sigma\otimes\mathcal T)$-measurable. Then the functions $x\mapsto \int_Y f(x,y)\,d\nu(y) \quad\text{and}\quad y\mapsto \int_X f(x,y)\,d\mu(x)$ are measurable and [ \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), ] where all three integrals are allowed to be $+\infty$.

Here $\mu\times\nu$ is the product measure on the Cartesian product , and the integrals are Lebesgue integrals of a nonnegative measurable function . Tonelli’s theorem is the nonnegative-function counterpart of Fubini's theorem , which applies under an absolute integrability hypothesis.