Let $(X,d)$ be a Polish metric space, let $\mathcal B(X)$ denote its Borel $\sigma$-algebra, and let $\mathcal P(X)$ denote the set of Borel probability measures on $(X,\mathcal B(X))$. For any measurable space $(E,\mathcal E)$, let $\mathcal P(E,\mathcal E)$ denote the set of probability measures on $(E,\mathcal E)$. For $p\in\{1,2\}$ and $\mu,\nu\in\mathcal P(X)$, define the extended $p$-Wasserstein distance by

\begin{align*} W_p(\mu,\nu):=\inf_{\pi\in\Pi(\mu,\nu)}\left(\int_{X\times X}d(x,y)^p\,d\pi(x,y)\right)^{1/p}, \end{align*}

where $\Pi(\mu,\nu)$ is the set of Borel probability measures $\pi\in\mathcal P(X\times X,\mathcal B(X\times X))$ whose first marginal is $\mu$ and whose second marginal is $\nu$. Let $H(\nu\mid\rho)$ denote 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 $\rho$.

Let $C>0$, and let $\rho\in\mathcal P(X)$ satisfy the transportation inequality $T_2(C)$ in the following normalization: for every $\nu\in\mathcal P(X)$,

\begin{align*} W_2(\nu,\rho)^2 \le 2C\,H(\nu\mid\rho). \end{align*}

Let $L>0$, and let $f:X\to\mathbb R$ be an $L$-Lipschitz Borel function such that $f\in L^1(X,\mathcal B(X),\rho)$. Then, for every $r\ge 0$,

\begin{align*} \rho\left(\left\{x\in X: f(x)-\int_X f\,d\rho(x) \ge r\right\}\right) \le \exp\left(-\frac{r^2}{2CL^2}\right). \end{align*}

Moreover, for every $r\ge 0$,

\begin{align*} \rho\left(\left\{x\in X: \int_X f\,d\rho(x)-f(x) \ge r\right\}\right) \le \exp\left(-\frac{r^2}{2CL^2}\right). \end{align*}