pushforward measure

A pushforward measure is a measure induced by a measurable function between measurable spaces.

Formally,
Let $(X,\mathrm{\Sigma })$ and $(Y,T)$ be measurable spaces with measurable function $f:X\to Y$.
Given measure $\mu :\mathrm{\Sigma }\to \mathbb{R}$,
the pushforward measure $f_{\ast }\mu :T\to \mathbb{R}$ of $\mu$ induced by $f$ is defined for all $E\in T$ by

such that $f^{-1}(E)\in \mathrm{\Sigma }$ (by construction).