Yeah, I think that's right.

The idea is to first replace the constraint on gamma (gamma \in pi) with an optimization over a function f, then swap the order of inf_gamma and sup_f, and lastly replace the optimization over gamma with an constraint on f (f is 1-Lipschitz continuous).

So we want to change from inf_gamma sup_f to sup_f inf_gamma. This is only possible if both sup_f is "convex" (or whatever we call it) in gamma, and inf_gamma is "concave" in f. We show that this condition is sufficient in an abstract way in the g(a,b) part. In our case the whole function (all the expectations, but without the inf and sup) is g(gamma, f), and it is relatively easy to see that the conditions are fulfilled.