(Uniform) continuity of push-forward operator

Question

I am wondering about extension of the the answer given here.

Namely, suppose $U$, $V$ are Polish spaces and $F:U→V$ is uniformly continuous. Does this mean that the push-forward operator $F_*: \mathcal{P}(U) \rightarrow \mathcal{P}(V)$ is also uniformly continuous when each $\mathcal{P}(.)$ is endowed with weak* topology?

Answer 1

If you use the Levy-Prohorov metric (https://en.wikipedia.org/wiki/L%C3%A9vy%E2%80%93Prokhorov_metric) on both $\mathcal P(U)$ and $\mathcal P(V)$, then $F_*$ is indeed uniformly continuous. The proof is a matter of chasing through the definitions, the key being to observe that $$ \{x\in U: d_U[x, F^{-1}(A)]<\delta\}\subset \{x\in U: d_V[F(x),A]<\epsilon\}. $$ provided $A$ is a Borel subset sof $V$. Here, $\epsilon>0$, and $\delta$ is such that $$ d_U[x,y]<\delta \Longrightarrow d_V[F(x),F(y)]<\epsilon, $$ per the uniform continuity of $F$.