Consider the set of probability measures on a countable space $X$. I am wondering if such sets are compact?
As a hint, I know that for finite $X$, the set is indeed compact as a closed and bounded subset in $R^{|X|}$.
However, I am having trouble for countable space as now in infinite dimension.
Consider $\mathbb N$ with its usual metric. $(\delta_n)$ is a counter-example. [$\delta_n(A)=1$ if $n \in A$ and $0$ otherwise]. $\int f d\delta_n=f(n)$ and $\lim f(n)$ need not exits: take $f(n)=1$ for $n$ even and $f(n)=0$ for $n$ odd. [Recall that any real valued function on $\mathbb N$ is continuous].