Consider the space $L_p(\mathbb{R})$ of probability measures on $\mathbb{R}$ with $p$th moments. This set is a metric space under the $p$-Wasserstein metric $$W_p(\mu,\nu)^p=\inf_{Z=(X,Y):X\sim\mu,Y\sim\nu}{\mathbb{E}[d(X,Y)^p]}$$ Now, pick your favorite bounded continuous function $f\in C_b(\mathbb{R})$. Then $f$ defines a function $f^*$ on $L_p$ via integration: $$f^*(\mu)=\mathbb{E}_{X\sim\mu}[f(X)]$$ (Fun exercise: prove that $f(X)$ has a first moment.) In the course of solving this problem, I found myself asking: is $f^*$ always continuous?
If $f$ is Lipschitz, then the answer is "yes", by the argument given at the end of my solution to that problem. But is the claim true for non-Lipschitz functions? I am not sure.
On the one hand, my proof uses the Lipschitz constant in an essential way.
On the other, take $p=1$ for simplicity. Choose $\mu_n(dx)=\frac{dx}{|nx|^3+1}$, so that $\mu$ only has moments of order $<2$ and let $f:C(\mathbb{R})$; $f(x)=|x|^{\alpha}$ for $\alpha<1$. If there were a counterexample in the non-Lipschitz case, I would think that those choices of $f$ and $\mu_n$ should be it, for they are as badly-behaved as can be while still having enough moments. Yet $\mu_n\overset{W_1}{\longrightarrow}\delta_0$ and $\mathbb{E}_{X\sim\mu_n}[|X|^{\alpha}]\to0$.
Perhaps the problem is that the above non-counterexample is Lipschitz away from $0$, and I should look at functions that aren't Lipschitz on a more generic region — say, $f:C((-1,1))$; $f(x)=\sin{\frac{1}{1-x^2}}$, with $\mu_n\overset{W_1}{\longrightarrow}\mathcal{U}(\{-1,1\})$? But I am not sure how to impose cutoff to regularize $\mathbb{E}_{X\sim\mathcal{U}(\{-1,1\})}[f(X)]$.
Best Answer
As Tobsn explained to me in the comments, it is a classical result that convergence in $W_p$ is equivalent to (combined) weak convergence (as $C_b^*$) and convergence of the $p$th moment.
The key idea missing from my argument is the following:
($\mathrm{Lip}_b$ denotes the space of bounded Lipschitz functions.)
That claim follows from e.g. supremal/infimal convolution, and implies weak convergence for $f$, via Fatou's lemma.
(H/T Villani, Topics in Optimal Transportation (2003), p. 217.)