I'm really stuck on the first part of this exercise. I've tried to type it up in a fashion that is as self-contained as possible, but some background in optimal transport is likely required to read this.
Let $P_{ac,2}(\mathbb{R}^n)$ denote the space of absolutely continuous probability measures with finite second moments. Suppose $\sigma \in P_{ac,2}(\mathbb{R}^n)$ is given, as well as a family $\rho_k \in P_{ac,2}(\mathbb{R}^n)$ that weakly converges to some $\rho \in P_{ac,2}(\mathbb{R}^n)$.
Let $\nabla \varphi_k$ be the optimal mapping of the Monge problem with quadratic cost between $\sigma$ and $\rho_k$, i.e. $$\int_{\mathbb{R}^n} |x-\nabla \varphi_k(x)|^2 d\sigma(x) = \inf_{T \# \sigma = \rho_k} \int_{\mathbb{R}^n} |x - T(x)|^2 d\sigma(x)$$ Here, the infimum is taken over functions $T : x \mapsto T(x)$ satisfying the pushforward constraint. Similarly, we can define $\nabla \varphi$ between $\sigma$ and $\rho$.
Let $d\pi_k(x,y) = d\sigma(x) \delta(y = \nabla \varphi_k(x))$, and similarly define $\pi$ between $\sigma$ and $\rho$.
How can I show that $\pi_k$ weakly converges to $\pi$? (The hint given is to note that the optimal transference plan is unique.)