Total variation and probability measures

Question

I'm currently reading some notes about optimal transport and here there is the definition of total variation:$$\Vert\mu\Vert_{TV}=2\sup\vert\mu(A)\vert$$where $\mu$ is a probability measure of the Wasserstein space $W_2$ over a Polish space $X$. Later there's written that given two probability measures $\mu,\nu\in W_2$ then $$\Vert\mu-\nu\Vert_{TV}=2(\nu-\mu)_+(X)$$where $(\cdot)_+$ is the positive part of the Hahn decomposition, but I think that this is not true for any probability measure, maybe the $W_2$ space is involved but I don't see how.

Answer 1

Your second formula is true according the standard definition of total variation found, for example, in Ruidn's RCA. But the definition given in the first formula is not standard and it is not compatible with the second formula.

Let $\tau=\mu -\lambda$. Note that $\tau (X)=0$ so $\tau^{+}(X)=\tau^{-}(X)$. Hence, $\|\tau\|=|\tau |(X)=\tau^{+}(X)+\tau^{-}(X)=2\tau^{+}(X)$.