Pushforward measure - Knowlpedia

A pushforward measure transports a measure along a measurable map. 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 $T_\#\mu$ or $T_*\mu$ , is the measure on $(Y,\mathcal B)$ defined by

(T_\#\mu)(B)=\mu(T^{-1}(B))\qquad(B\in\mathcal B).

The definition measures subsets of $Y$ by pulling them back to $X$ . Pushforwards are the natural language for the change-of-variables formula.

Examples

Let $\lambda$ be Lebesgue measure on $[0,1]$ and let $T(x)=x^2$ . Then $\nu=T_\#\lambda$ satisfies $\nu([0,t])=\sqrt t$ for $0\le t\le1$ , so $\nu$ has density $1/(2\sqrt y)$ on $(0,1]$ .
If $\pi_X:X\times Y\to X$ is projection and $\mu\otimes\nu$ is a product measure with $\nu(Y)<\infty$ , then $(\pi_X)_\#(\mu\otimes\nu)(A)=(\mu\otimes\nu)(A\times Y)=\mu(A)\nu(Y)$ for $A\in\mathcal A$ . Thus $(\pi_X)_\#(\mu\otimes\nu)=\nu(Y)\mu$ ; in particular, it equals $\mu$ when $\nu$ is a probability measure.