I have a question about the compactness of a set of martingale measures. Let $\Omega=\mathcal{C}[0,1]$ be the space of continuous functions on $[0,1]$ and $\mathcal{M}_{\Omega}$ be the family of martingale measures $\mathbb{P}$ on $\Omega$, i.e.

$$E^{\mathbb{P}}[X_t|\mathcal{F}_s]=X_s,\ \ \forall 0\leq s<t\leq 1$$

Here $X_t(\omega)=\omega_t$ is the canonical process and $\{\mathcal{F}_t\}$ is the natural filtration. Now we are given a sequence of probability distribution on $\mathbb{R}$: $\mu=(\mu_t)_{0\leq t\leq 1}$ and we define

$$\mathcal{M}_{\Omega}(\mu)=\{\mathbb{P}\in\mathcal{M}_{\Omega}: X_t \stackrel{\mathbb{P}}{\sim}\mu_t\}$$

In other words, $X_t$ has the marginal distribution $\mu_t$ under $\mathbb{P}$. Certainly we impose some conditions on $\mu$ such that $\mathcal{M}_{\Omega}(\mu)$ is not empty. My question is whether we can prove that $\mathcal{M}_{\Omega}(\mu)$ is compact. If yes, can we find some simple condtions on $\mu$ such that $\mathcal{M}_{\Omega}(\mu)$ is compact; if not, can we find a counterexample? Many thanks for your reply.