# Are these two total variation distance definitions equivalent in general for measures over countable sets?

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.