We have a particular set-up of the Monge-Kantorovich problem:
Let $\mu$ and $\nu$ be probability measures on a compact spaces $\mathcal{X}$ and $\mathcal{Y}$ respectively ($\mathcal{X}$ and $\mathcal{Y}$ have the Borel $\sigma$-algebra induced by their topologies). Let $c: \mathcal{X} \times \mathcal{Y} \rightarrow \mathbb{R} \cup \{\infty\}$ be continuous. Let $\Gamma(\mu,\nu)$ be the set of probability measures on $\mathcal{X} \times \mathcal{Y}$ whose marginals are $\mu$ and $\nu$. We know that in this setting, there exists a minimizer to the Monge-Kantorovich problem:
$$\mathcal{L}_c(\mu,\nu) = \min_{\gamma \in \Gamma(\mu,\nu)} \int_{\mathcal{X} \times \mathcal{Y}} c(x,y) \; d\gamma(x,y)$$
My questions are:
- If there is at least one pair of points $(x,y) \in \mathcal{X} \times \mathcal{Y}$ such that $c(x,y) < \infty$, then is it true that $\mathcal{L}_c(\mu,\nu) < \infty$?
- If $c: \mathcal{X} \times \mathcal{Y} \rightarrow \mathbb{R}$ then is it true that $\mathcal{L}_c(\mu,\nu) < \infty$?
I've realised there are some properties of $c$ that immediately follow from the problem statement itself.
First of all, if $\mathcal{X} \times \mathcal{Y}$ is compact and $c \in \mathcal{C}(\mathcal{X} \times \mathcal{Y})$ is continuous, then $c$ cannot attain the value $\infty$. As a result, the first question is redundant as such cases always happen. From this, there exists an $M > 0$ such that $|c(x,y)| < M$ for all $(x,y) \in \mathcal{X} \times \mathcal{Y}$. Hence:
$$\int_{\mathcal{X}\times\mathcal{Y}}c(x,y) \; d\gamma(x,y) \leq \left|\int_{\mathcal{X}\times\mathcal{Y}}c(x,y) \; d\gamma(x,y)\right| \leq \int_{\mathcal{X}\times\mathcal{Y}}|c(x,y)| \; d\gamma(x,y) < \int_{\mathcal{X}\times\mathcal{Y}} M \; d\gamma(x,y)$$
$$M \int_{\mathcal{X}\times\mathcal{Y}} 1 \; d\gamma(x,y) = M \cdot 1 = M$$
so the integral is always finite.