Suppose that $(M,d)$ is a separable and complete metric space. Define $C_b(M)$ to be the space of all bounded continuous functions $f\colon M\to\mathbb R$ and denote by $\mathcal P(M)$ the space of all Borel probability measures on $M$.
Endow $\mathcal P(M)$ with the weak topology, that is, the coarsest topology that makes $\mu\mapsto \int_M f d\mu$ continuous for all $f\in C_b(M)$.
Suppose that $K\subset \mathcal P(E)$ is relatively weakly compact.
Suppose that $L\subset M$ satisfies that $\delta_x\in K$ for all $x\in L$, where $\delta_x$ denotes the Dirac's delta.
My question is: Is it true that $L$ is relatively compact in $(M,d)$?
YES. Convergence of $\delta_{x_n}$ to some $\mu$ implies that $\mu$ is degenerate. (This can be proved by decomposing $M$ into countable many sets of arbitraily small diameter). From this it follows that if $(x_n)$ is a sequence in $L$ then $\delta_{x_n}$ has a subsequence converging to $\delta_x$ for some $x$ which proves that $L$ is relatively compact.