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I found this definition for the total variation distance for measures over countable sets:

The total variation distance between two measures $\mu$, $\nu$ on a countable set $S$ is defined as: $||\mu - \nu|| \equiv 0.5\sum_{z\in S}|\mu(z) - \nu(z)| = \sup_{A \subset S} |\mu(A) - \nu(A)|$.

Here $z$ is every singleton of $S$ and $A$ an arbitrary set from $S$. The proof is straightforward, but uses this statement:

$|\mu(A) - \nu(A)| + |\mu(A^c) - \nu(A^c)| = 2|\mu(A) - \nu(A)|$

which I can prove if both $\mu$ and $\nu$ are probability measures. If, for example, $\mu(S) \neq \nu(S)$ or if both of these measures is not finite, I cannot see the proof holding.

Given that the statement was found on a probability book, I believe this proof is supposed to hold for probability measures. But, as I quoted, it is stated for measures in general. This might just be poor phrasing in the book, so I am asking if this proof holds in general for an arbitrary measure (not necesarily a probability one).

Thank you very much in advance.

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