Is this total variation distance is same as
$\|\mu-\nu\|_{TV}=\sup_{A \text{ is Borel}}\{|\mu(A)- \nu(A)|\}$
I am confused as I see many definitions of TV metric. Thanks for helping. If one can refer to a book where I can get the definition of the first type would be good.
Edit: I found this but there is a $\frac{1}{2}$, still confused and need a reference. Thanks!
Best Answer
In the case of $[0,1]$ for example (or any compact space for that matter), we can show that $\rho(\mu, \nu)=\sup \{\sum\limits_{k=1}^{\infty} |\mu(A_n)-\nu(A_n)|\}$ where the supremum is taken over all partitions $(A_n)$ of $[0,1]$ by measurable sets. This is not the same as $\|\mu-\nu\|_{TV}$. However $\|\mu-\nu\|_{TV} \leq \rho(\mu, \nu) \leq 2 \|\mu-\nu\|_{TV}$, so these two metrics are equivalent.