Pushforward measure ($f_{\ast }\mu$)

For a measurable map $f:X\to Y$, the pushforward of $\mu$ is the measureIt is the distribution induced on $Y$ by mapping points distributed according to $\mu$ through $f$.

A homeomorphic embedding of a metric space induces a homeomorphic embedding of its probability measures for their weak topologies. The inverse continuity follows from the open-set Portmanteau theorem, since every relatively open set is the trace of an ambient open set. This does not give surjectivity onto $P(Z)$ when points of $Z$ are outside the embedded image.