= Probability pushforward embedding theorem
{title2=$i:X\hookrightarrow Z\ \Longrightarrow\ i_*:P(X)\hookrightarrow P(Z)$}
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.
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