I am studying a paper where the author is trying to optimize a cost $C$ on the set of probability measures $\gamma\in P(\Omega \times \Omega)$ such that $\pi^{1}_{\sharp}(\gamma) = \mu$, where
- $\pi^1(x,y) = x$ for all $x,y\in \Omega$, is the projection onto the first component,
- $\sharp$ symbol denotes the push-back measure,
- $\mu\in P(\Omega)$ is a probability measure, and
- $\Omega$ is the closure of an open, bounded and convex subset of $\mathbb{R}^d.$
To put everything together, is the following family of measures is tight: \begin{align} \{\gamma \in P(\Omega \times \Omega): \pi^{1}_{\sharp}(\gamma) = \mu\}? \end{align}
In the article the author says that we can solve the minimization problem since $\Omega$ is compact, but I don't understand how this works, exactly since if we denote, \begin{align} \alpha = \inf\{C(\gamma): \gamma \in P(\Omega \times \Omega), \pi^{1}_{\sharp}(\gamma) = \mu\} \end{align} and take a sequence $\gamma_n$ such that $C(\gamma_n)\to \alpha$ then how do we extract a subsequence $\gamma_{n_k}$ that will allow us to proceed further as we apply the direct method from calculus of variations?