The set of all probability measures on $(\Xi,\mathcal{B})$ is weakly compact with respect to topology of total variation?

Question

Based on Prokhorov's theorem, a closed set $\mathcal{A}$ of probability measures is weakly compact if and only if it is tight. Hence, if $\Xi$ is a compact set in $R^k$, we will have the set of all probability measures on $(\Xi,\mathcal{B})$ is weakly compact with respect to topology of weak topology?

I want to know whether it is also compact with respect to the total variation topology? Assume $\Xi$ is compact in $R^k$

Answer 1

Certainly not! Consider the sequence $\delta_{1/n}$ of probability measures on $[0,1]$. The total variation norm of $\delta_{1/n}-\delta_{1/m}$ is 2 for $n \neq m$.