Pushforward measure

A pushforward measure transports a measure along a function . Let $(X,\mathcal{A},\mu)$ be a measure space , let $(Y,\mathcal{B})$ be a measurable space , and let $T:X\to Y$ be a measurable function . The pushforward of $\mu$ by $T$, denoted $T_\#\mu$ (also written $T_*\mu$), is the measure on $(Y,\mathcal{B})$ defined by

Pushforward measures encode how $\mu$ “looks” after applying $T$ and are the natural language for the change-of-variables formula via pushforward . The definition measures subsets of $Y$ by pulling them back to $X$ and then measuring in $X$.

Examples:

• Let $\lambda$ be Lebesgue measure on $[0,1]$ and let $T(x)=x^2$. Then $\nu=T_\#\lambda$ satisfies $\nu([0,t])=\lambda([0,\sqrt{t}])=\sqrt{t}$ for $0\le t\le 1$, so $\nu$ has density $\frac{1}{2\sqrt{y}}$ with respect to Lebesgue measure on $(0,1]$.

• If $\pi_X:X\times Y\to X$ is the projection map and $\mu\otimes\nu$ is a product measure , then for $A\in\mathcal{A}$ one has

  $$(\pi_X)_\#(\mu\otimes\nu)(A)=(\mu\otimes\nu)(A\times Y)=\mu(A)\nu(Y)$$

  whenever $\nu(Y)<\infty$; in particular, if $\nu$ is a probability measure then $(\pi_X)_\#(\mu\otimes\nu)=\mu$.