Can I define on the space of probability measure on some complete separable metric space the following metric?
$d_1(\mu, \nu)=\|\mu-\nu\|_{TV}=\frac{1}{2}\sup\limits_{f: X\to [-1.1]}[\int f d\mu -\int f d\nu]$ (Total Variation Norm, distance)?
Now, is that metric is equivalent to the following one?
$d_2(\mu, \nu)=\sup\limits_{f\in A}[\int f d\mu -\int f d\nu]$ where
$A=\{f: X\to \mathbb R: |f(x)-f(y)|\le |x-y|\}$ and $f$ is measurable.
Thanks for confirming.
Somewhat inspired from this question
Two notions of total variation norms
Best Answer
The total variation distance is a metric on the space of all complex Borel measures and its restriction to the class of probability measures is a nice complete metric. [It is not separable in general].