Tonelli's theorem - Imported Knowlpedia Content

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.