Probability pushforward embedding theorem ($i:X\hookrightarrow Z⟹i_{\ast }:P(X)\hookrightarrow P(Z)$)

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.