Definition of (Finite) Signed Measure

Question

My class notes define a signed measure on a measurable space $(X, \mathcal{R})$ as a $\sigma$-additive function $\nu : \mathcal{R} \to \mathbb{R}$. (I take this to mean we're only considering finite measures.) I'm confused on what I'm supposed to interpret $\sigma$-additive to mean here. My first guess would be that if $A = \bigcup_nA_n$ where $A_1,A_2,\ldots$ are disjoint, then $$\nu(A) = \sum_{n=1}^\infty \nu(A_n);$$ i.e., just the usual definition of $\sigma$-additivity. But this seems problematic when the series on the right doesn't converge absolutely, because then its value depends on the order of the $A_i$, which my gut tells me shouldn't be the case. Does $\sigma$-additivity here also involve the claim that the series on the right always converges absolutely? The wikipedia page on signed measures does require this, but no other source I found online, or my class notes, explicitly states it.

Answer 1

$\nu$ takes only finite values. So disjointness of $(A_n)$ implies that $\sum \nu(A_n)$ is a convergent series. So is any rearrangement of terms since disjointness still holds. If a series of real numbers converges whenever the terms are permuted then the series is absolutely convergent. Hence $\sum |\nu (A_n)| <\infty$.