Pushforward measure §

AkA image measure.

Let $(X,\mathcal{F})$ and $(Y,\mathcal{E})$ be two measurable spaces, and a measurable map $h:X\to Y$.

Suppose we have a measure $\mu :\mathcal{F}\to [0,\infty ]$ on the first set, we can define a new measure $\lambda :\mathcal{E}\to [0,\infty ]$ for $Y$ using the function $h$, in the following manner:

$$\lambda (E):=\mu (h^{-1}(E)),E\in \mathcal{E}$$

this is well defined, since $h$ is measurable, so that $h^{-1}(E)\in \mathcal{F}$, and $\lambda$ is a measure since the preimage $h^{-1}$ behaves “well” w.r.t. set operations.

Other notations used are:

$$h_{\ast }\mu \mu \circ h^{-1}h\#\mu$$