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:


1. Is $d\_{TV}$ a complete metric on $\mathcal{M}$? Is $(\mathcal{M}, d\_{TV})$ a Hausdorff/separable space?
2. When the subset $\mathcal{N}$ ($\subset \mathcal{M}$) is pre-compact?
3. 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?



The properties of total variation metric
**Answer to Question 3:** Yes, there is such a sequence. E.g., for all $n=0,1,\dots$ and all Borel $B\subseteq[0,1]$, let  

$$\mu\_n(B):=\frac12\int\_B(2+\sin2\pi nx)\,dx.$$
Then $\mu\_n$ converges to $\mu\_0$ weakly but not in total variation. (Also, $\mu\_n$ is full supported for each $n$.)


**Concerning Question 1:** In general, $d\_{TV}$ is not a complete metric on $\mathcal M$. Indeed, for all $n=0,1,\dots$ and all Borel $B\subseteq X:=[0,1]$, let  

$$\mu\_n(B):=(2-1/n)|B\cap[0,1/2]|+(1/n)|B\cap[1/2,1]|,$$ 
where $|\cdot|$ is the Lebesgue measure,
and 
$$\mu(B):=2|B\cap[0,1/2]|.$$
Then $\mu\_n\in\mathcal M$ for all $n$, $d\_{TV}(\mu\_n,\mu\_k)\to0$ as $n,k\to\infty$, but $\mu\_n\to\mu$ in total variation as $n\to\infty$ and $\mu\notin\mathcal M$. 


**Also concerning Question 1:** The metric space $(\mathcal{M}, d\_{TV})$ is Hausdorff, as any metric space. 


The metric space $(\mathcal{M}, d\_{TV})$ is not separable in general. Indeed, for all $x\in X:=[0,1]$ and all Borel $B\subseteq[0,1]$, let  

$$\mu\_x(B):=\tfrac12\,|B|+\tfrac12\,1\_B(x).$$ 
Then the subset $\{\mu\_x\colon x\in[0,1]\}$ of $\mathcal{M}$ is uncountable and $d\_{TV}(\mu\_x,\mu\_y)=1/2$ for any distinct $x$ and $y$ in $[0,1]$. So, $(\mathcal{M}, d\_{TV})$ is not separable (cf. e.g. the first paragraph in [this answer](https://math.stackexchange.com/a/97651/96609)). 


**Concerning Question 2:** $\mathcal N$ will be pre-compact if and only if $\mathcal N$ is [totally bounded](https://en.wikipedia.org/wiki/Totally_bounded_space#Definition_for_a_metric_space) and closed in $d\_{TV}$ as a subset of the set of all probability measures on $X$. 


NB: Try to avoid asking multiple questions in one post. 


