TITLE: Prokhorov metric vs. total variation norm
QUESTION [12 upvotes]: Let $(S,d)$ be a metric space and let $\mathcal P(S)$ denote the space of Borel probability measures on $S$ endowed with the Prokhorov metric $\pi:\mathcal P(S)\times \mathcal P(S)\to \mathbb R_+$ given by
$$
  \pi(P,Q):=\inf\{\varepsilon\geq 0:P(F)\leq Q(F^\varepsilon)+\varepsilon \text{ for all closed } F\subset S\}
$$
where the $\varepsilon$-inflation of a set is given by  $  F^{\varepsilon} = \{x\in S:d(x,F)<\varepsilon\}.$
Another useful metric on $\mathcal P(S)$ is induced by the total variation norm, i.e. 
$$
  \rho(P,Q):=\sup\limits_{A\in \mathfrak B(S)}|P(A) - Q(A)|
$$
where $\mathfrak B(S)$ is the Borel $\sigma$-algebra on $(S,d)$. I wonder if there are any interesting relations between these two metrics, $\pi$ and $\rho$. In particular, I know that convergence in $\rho$ implies the weak convergence and hence if $S$ is separable than it implies the convergence in $\pi$. 
I wonder, however, if under some additional assumptions it is possible to derive some non-trivial bounds on $\rho$ if I know upper bounds on $\pi$. Or at least, if it is possible to upper-bound $|P(F) - Q(F)|$ for closed $F$ using $\pi$.

REPLY [2 votes]: Here is a simple inequality relating the Prohorov metric $\pi$ and the total variation norm $\|\;\|_{TV}$for probability measures:
Suppose $\mu$ and $\nu$ are probability measures on a metric space $(S,d)$. For any Borel set $A\subset S$,
\begin{aligned}
\mu(A)&=\mu(A)-\nu(A)+\nu(A)\leq \|\mu-\nu\|_{TV}+\nu(A) \\
&\leq \|\mu-\nu\|_{TV} + \nu\big(A^{\|\mu-\nu\|_{TV}}\big)
\end{aligned}
whence we obtained that
$$\pi(\mu,\nu)\leq\|\mu-\nu\|_{TV}
$$

Another relevant metric is the Kantorovih-Rubinstein metric
$$\beta(\mu,\nu)=\sup\left\{\Big|\int f(d\mu-d\nu)\Big|: f\in\operatorname{Lip}_1\right\}
$$
where $\operatorname{Lip}_1$ is the space of Lipschitz functions on $S$ such that $\|f\|_{BL}=\|f\|_\infty+\|f\|_L\leq1$
where $\|f\|_L=\sup_{x\neq y}\frac{|f(x)-f(y)|}{d(x,y)}$. It is a well know fact that $\beta$ and $\pi$ produce the same topology on the space of probability measures, namely the weak topology $\sigma(\mathcal{M}(S),\mathcal{C}_b(S))$ restricted to $\mathcal{M}^+_1(S)$ the space of probabilities on $(S,d)$. The following is also well known:
\begin{aligned}
\frac12\beta(\mu,\nu)&\leq \pi(\mu,\nu)\leq\Big(\frac32\beta(\mu,\nu)\Big)^{1/2}\\
\beta(\mu,\nu)&\leq \|\mu-\nu\|_{TV}
\end{aligned}
Dudley's book Real Analysis and Probability chapter 11  is a good reference.