Pushforward measure - Imported Knowlpedia Content

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

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

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 $X$ .

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

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