# The properties of total variation metric

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?

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).

Concerning Question 2: $$\mathcal N$$ will be pre-compact if and only if $$\mathcal N$$ is totally bounded 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.