E.g., for the Wasserstein distance $W_1(P,Q)$ between two probability measures $P$ and $Q$ one has the following duality formula: \begin{equation} W_1(P,Q)=\sup\Big\{\Big|\int f\,dP-\int f\,dQ\Big|\colon \text{Lip}(f)\le1\Big\}, \end{equation} where Lip$(f)$ is the Lipschitz constant for a function $f$. It then trivially follows that $|\int f\,dP_\epsilon-\int f\,dP|\le\epsilon$ for any $1$-Lipschitz function $f$ if $W_1(P_\epsilon,P)\le\epsilon$. (Of course, $\int f\,dP$ -- or simply $Pf$ or $P(f)$ -- is a simpler and better notation for what people in some research fields write as $\mathbb E_{x \sim P}[f(x)]$. If the probability measure is fixed, then it may be better to write $\mathbb Ef$ instead of $\int f\,dP$.  )

E.g., for the Wasserstein distance $W_1(P,Q)$ between two probability measures $P$ and $Q$ one has the following duality formula: \begin{equation} W_1(P,Q)=\sup\Big\{\Big|\int f\,dP-\int f\,dQ\Big|\colon \text{Lip}(f)\le1\Big\}, \end{equation} where Lip$(f)$ is the Lipschitz constant for a function $f$. It then trivially follows that $|\int f\,dP_\epsilon-\int f\,dP|\le\epsilon$ for any $1$-Lipschitz function $f$ if $W_1(P_\epsilon,P)\le\epsilon$. (Of course, $\int f\,dP$ -- or simply $Pf$ or $P(f)$ -- is a simpler and better notation for what people in some research fields write as $\mathbb E_{x \sim P}[f(x)]$. If a probability measure $\mathbb P$ is fixed, then it may be better to write $\mathbb Ef$ instead of $\int f\,d\mathbb P$.)

E.g., for the Wasserstein distance $W_1(P,Q)$ between two probability measures $P$ and $Q$ one has the following duality formula: \begin{equation} W_1(P,Q)=\sup\Big\{\Big|\int f\,dP-\int f\,dQ\Big|\colon \text{Lip}(f)\le1\Big\}, \end{equation} where Lip$(f)$ is the Lipschitz constant for a function $f$. It then trivially follows that $|\int f\,dP_\epsilon-\int f\,dP|\le\epsilon$ for any $1$-Lipschitz function $f$ if $W_1(P_\epsilon,P)\le\epsilon$.

E.g., for the Wasserstein distance $W_1(P,Q)$ between two probability measures $P$ and $Q$ one has the following duality formula: \begin{equation} W_1(P,Q)=\sup\Big\{\Big|\int f\,dP-\int f\,dQ\Big|\colon \text{Lip}(f)\le1\Big\}, \end{equation} where Lip$(f)$ is the Lipschitz constant for a function $f$. It then trivially follows that $|\int f\,dP_\epsilon-\int f\,dP|\le\epsilon$ for any $1$-Lipschitz function $f$ if $W_1(P_\epsilon,P)\le\epsilon$. (Of course, $\int f\,dP$ -- or simply $Pf$ or $P(f)$ -- is a simpler and better notation for what people in some research fields write as $\mathbb E_{x \sim P}[f(x)]$. If the probability measure is fixed, then it may be better to write $\mathbb Ef$ instead of $\int f\,dP$. )