The questions was asked by me on Math StackExchange, but no answer appears, so I ask for help again.

Let $(X, d)$ be a complete (Hausdorff, separable, local compact and other nice properties you want) metric space and $\mathcal{M}$ be the space of local finite **full supported** Borel probability measures on $X$ (with Borel sigma-algebra $\mathcal{B}$). Then the total variation metric can be defined by $d_{TV}:= \sup_\limits{A}|\mu(A)- \nu(A)|$ where $A$ runs over $\mathcal{B}$ and $\mu \in\mathcal{M}, \nu\in \mathcal{M}$.

Questions:

- Is $d_{TV}$ a complete metric on $\mathcal{M}$? Is $(\mathcal{M}, d_{TV})$ a Hausdorff/separable space?
When the subset $\mathcal{N}$ ($\subset \mathcal{M}$) is pre-compact?

Let $(X,d)=([0, 1], | |)$, where $| |$ is the Euclidean metric, then if it has sequence $\{\mu_i\}$ such that $\{\mu_i\}$ weak converge to $\mu$, but not $d_{TV}$-converge to $\mu$? Here $\mu_i, \mu \in \mathcal{M} $.

How about the questions when $(X, d)$ is a Riemannian manifold or require the meassures are Radon measures?