Indeed, applying Theorem 7 in Strassen's paper, one has $\int \psi(y)\nu(dy)\le \int \psi\left(\int y K_x(dy)\right)\mu(dx)$, for all $K_x$ satisfying $\int \left|\int y K_x(dy)-x\right|\mu(dx)\le\varepsilon$. However, it seems not trivial to simplify the above expression. Do you have any idea?

What is the right condition for $\mu$ and $\nu$ if the inequality $(\ast)$ is replaced by $\mathbb E\big[\big|\mathbb E[N|M]-M\big|\big]\le \varepsilon$?

Thank you very much for the literature. But what I'm interested in is slightly different from the quantization. Indeed, I'd like to find $\mu_n$ s.t. the computation of $\mu_n[\{\vec{k}/n\}]$ is tractable and the Wasserstein distance $W_1(\mu,\mu_n)$ is easy to estimate. Of course, the quantisation approach provides a good upper bound for $W_1(\mu,\mu_n)$, but the computation of $\mu_n[\{\vec{k}/n\}]$ is not obvious.

@AnthonyQuas Could you please explain a bit more how the above arguments could be applied to the case $p\neq 1/2$? By the way, why do you assume the slope $|\epsilon|<1$? Thank you very much!

@AnthonyQuas Actually, this problem is linked to the Voronoi diagram that is used in solving an (semi-discrete) optimal transport problem, where the marginal distributions are given by $\mu=p\delta_{(x_1,y_1)}+(1-p)\delta_{(x_2,y_2)}$ and $\nu=\mathcal{Les}$

@AnthonyQuas Thanks a lot for the quick reply. Could you please specify a bit more?

Oh, it's absolutely true! So sorry for my stupid question, and thanks again for your patience and kindness!

Basically, if one has a $\pi$ s.t. $\pi$ has marginals $\mu$ and $\nu$ and satisfies the above condition, then $\int \varphi d\nu\le \sup \int \varphi(y)\gamma(dx,dy)$ among $\gamma(dx,dy)=\mu(dx)K_x(dy)$ satisfying $|\int y K_x(dy)-x|\le \epsilon$. My question is how to show the inverse direction?

Thank you so much for the prompt reply. I think I didn't formulate my question correctly. Let me reformulate: Why there exists $\pi$ s.t. $\pi$ has marginals $\mu$ and $\nu$ and satisfies the above condition, if and only if, one has $\int \varphi d\nu\le \sup \int \varphi(y)\gamma(dx,dy)$ among $\gamma(dx,dy)=\mu(dx)K_x(dy)$ satisfying $|\int y K_x(dy)-x|\le \epsilon$?

Actually this is the only part where I don't know how to adapt the arguments of Strassen to my case. Could you please explain a bit more? Thank you very much!

Thank you very much for the reply. I've read the paper by Strassen actually and I was confused with the Hanh-Banach theorem. I've one question related to your argument: in the second paragraph, why the existence of $\gamma$ is equivalent to $\int \varphi d\nu\le \int h_{\varphi} d\mu$?