I want to prove the existence of a solution to a special case of the Kantorovich problem:
Let $(X,\mu)$ and $(Y,\nu)$ be compact probability spaces, i.e. $X$ and $Y$ are compact metric spaces and $\mu$ and $\nu$ are probability measures on the Borel $\sigma$-Algebra. Let $f: X \times Y \to \mathbb R$ be continuous.
We define $\Pi(\mu,\nu)$ to be the set of probability measures $\gamma$ on $X\times Y$ such that $\mu$ and $\nu$ are the push forward measures of $\gamma$ with respect to the projection on $X$ and $Y$, respectively: $(\pi_x)_{\#}\gamma=\mu$ and $(\pi_y)_{\#}\gamma=\nu$.
Then the optimality problem (Kantorovich problem) $$\mathrm{(KP)} \quad \inf_{\gamma \in \Pi(\mu,\nu)} K(\gamma) := \int_{X\times Y} f\, \mathrm d\gamma $$ has a solution.
By Weierstrass' criterion I need to show that $\Pi(\mu,\nu)$ is compact and that $K$ defined above is lower semi-continuous. Considering as mode of convergence the weak convergence of probability measures, then the continuity of $K$ follows immediately because $f$ is continuous and $C(X\times Y)=C_b(X\times Y)$ in this case.
But what about the compactness of $\Pi(\mu,\nu)$?
The weak convergence encourages to work with sequence compactness. So let $(\gamma_n)$ be a sequence in $\Pi(\mu,\nu)$. If the sequence converges weakly to some probability measure $\gamma$ on $X\times Y$, I can show that $\gamma\in \Pi(\mu,\nu)$: Let $g\in C(X)=C_b(X)$, then $(x,y)\mapsto g(x)$ is continuous (and bounded) and thus \begin{align} \int_X g(x) \, \mathrm d (\pi_x)_{\#}\gamma(x) &= \int_{X\times Y} g \circ \pi_x (x,y) \, \mathrm d \gamma(x,y) \\ &= \lim_{n\to\infty} \int_{X\times Y} g \circ \pi_x (x,y) \, \mathrm d \gamma_n(x,y) \\ &= \lim_{n\to\infty} \int_X g(x) \, \mathrm d (\pi_x)_{\#}\gamma_n(x) \\ &= \int_X g(x) \, \mathrm d \mu(x) \end{align} where I used weak convergence and change-of-variables. For $\nu$ it is the same, obviously.
All would be well, but how do I come up with the weak limit $\gamma$? This is my question.
In the proof for the general Kantovorich problem (where $X$ and $Y$ are not compact, but merely polish and $f$ lower semi-continuous) the sequence is derived by tightness of $\Pi(\mu,\nu)$ and employing Prokhorov's theorem. Here I'm looking for another way.