Let $(X,d)$ be a separable metric space, let $\mathcal B(X)$ be its Borel $\sigma$-algebra, and let $D\in[0,\infty)$ satisfy
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\begin{align*}
d(x,y)\le D
\end{align*}
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for every $x,y\in X$. Let $\rho$ be a Borel probability measure on $(X,\mathcal B(X))$. Then, for every Borel probability measure $\nu$ on $(X,\mathcal B(X))$,
where $W_1$ is the $1$-Wasserstein distance induced by $d$ on Borel probability measures on $X$, and $H(\nu\mid\rho)$ is the relative entropy of $\nu$ with respect to $\rho$, with the convention $H(\nu\mid\rho)=+\infty$ when $\nu$ is not absolutely continuous with respect to $\rho$.