Change of variables for pushforward measures

Change of variables for pushforward measures: Let $(X,\Sigma,\mu)$ be a measure space , let $(Y,\mathcal T)$ be a measurable space , and let $T:X\to Y$ be a measurable function . Let $\nu = T_*\mu$ be the pushforward measure of $\mu$ by $T$. Then for every nonnegative measurable function $g:Y\to[0,\infty]$,

If $g$ is Lebesgue integrable with respect to $\nu$, then $g\circ T$ is Lebesgue integrable with respect to $\mu$ and the same identity holds with finite values.

This formula packages the defining property of the pushforward measure into an equality of Lebesgue integrals and is a standard “change of variables” principle for transporting a measure along a map. It is frequently used together with constructions such as the product measure .