So I am reading a paper on theoretical many particles physics, and I came across a part where they want to prove that the probability measures they constructed converge to each other in weak sense.
The norm that they use is total variation norm, i.e., $$ \|P\|_{Tv} = \int d|P|. $$
Now before reading this paper, I always thought one way to see the convergence in measure if to use Wasserstein norm (or Monge-Kantorovich).
So what is the essential advantages/disadvantages in using TV-norm over Wasserstein?