Continuity of pushforward operation

Question

Let $X$ and $Y$ be compact metric spaces and let $f,g:X\rightarrow Y$ be $\epsilon$-uniformly close; i.e.: $$ \sup_{x \in X} d_Y(f(x),g(x))<\epsilon. $$ Then, are their push-forwards close in Wasserstein distance; i.e.: $$ W_1\left(f_{\#}\mathbb{P},g_{\#}\mathbb{P}\right)<\delta(\epsilon) , $$ for every $\mathbb{P}\in \mathcal{P}_1(X)$ for some $\delta(\epsilon)$ depending on $\epsilon,f,g$ but for which $\lim\limits_{\epsilon \to 0} \delta(\epsilon)=0$?

Answer 1

Yes. If the uniform distance of $f$ and $g$ is less than $\epsilon$, simply take your coupling to be the push-forward of the function $x\mapsto\big(f(x),g(x)\big)$. The resulting coupling verifies that the Wasserstein-$1$-distance is less than $\epsilon$.