I have started to study optimal transport, and it stuck me with the wealth of applications it opens up. In particular, I have been reading Villani's book Lecture Notes on Optimal Transport. There, in the appendix to chapter 1, the following result is proven ($\tau^*$ denotes the collection of closed sets in a topological spaces).
(Monge-Kantorovich duality for $\{0,1\}$-valued cost) Let $X,Y$ be Polish space, $\mu\in\Delta(X),\nu\in\Delta(Y)$,$\emptyset\neq C\in\tau(X\times Y)$. Then: $$\inf_{\pi\in\mathcal{M}(\mu,\nu)}\pi(C)=\sup_{A\in\tau^*(X)}\{\mu(A)-\nu(A_C)\}$$ where $A_C=\{y\in\ Y\ |\ \exists x\in A\ (x,y)\notin C\}$
As a corollary one can prove the following theorem from Strassen
Let $X$ be a Polish space, $\mu,\nu\in\Delta(X)$, $\epsilon\geq 0$. Then: $$\inf_{\pi\in\mathcal{M}(\mu,\nu)}\pi(\{d(x,y)>\epsilon\})=\sup_{A\in\tau^*(X)}\{\mu(A)-\nu(A_\epsilon)\}$$ where $X$ is metrized by $d$ and $A_\epsilon=\{y\in\ X\ |\ d(y,A)\leq\epsilon\}$
Another interesting feature is that:
Let $$\mathcal{T}_c(\mu,\nu)=\inf_{\pi\in\mathcal{M}(\mu,\nu)}\pi(C)=\sup_{A\in\tau^*(X)}\{\mu(A)-\nu(A_C)\}$$ be the total transportation cost of the problem, then $$\mathcal{T}_c(\mu,\nu)=\frac{1}{2}\Vert\mu-\nu\Vert_{TV}$$ where, given a signed measure $\psi\in\Delta_s(X)$, $\Vert\psi\Vert_{TV}$ is the total variation norm. Hence, $\mathcal{T}_c$ metrizes the weak topology on $\Delta(X)$
This kind of simple results has found application in econometrics: https://arxiv.org/abs/2102.12257
I would be curious about other context where this simple but apparently informative costs shows up and if other interesting results are related to it.