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 $g:Y\to[0,\infty]$ ,

\int_Y g\,d\nu \;=\; \int_X (g\circ T)\,d\mu.

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.

Interpretation

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.