I am reading Royden Real Analysis Forth edition chapter 19.3 named: "The Kantrovitch Representation Theorem for the dual of $L^\infty$" He gives defenition of bounded finitely additive signed measure $v$ as:
set function on measurable space which is finitely additive and has finite total variation of $v$ over whole space $X$.
I am wondering why does this imply that this is in fact a signed measure. So far, I've derived that $v$ of empty set is zero, and some other facts but I can't show countable additivity.
Is this even true?