I'm reading Real Analysis - Modern Techniques and Their Applications by Folland. On page 85, signed measures are defined as follows.
Let $(X, \mathcal{M})$ be a measurable space. A signed measure on $(X, \mathcal{M})$ is a function $\nu : \mathcal{M} \rightarrow [-\infty,\infty]$ such that:
- $\nu(\varnothing) = 0$;
- $\nu$ assumes at most one of the values $\pm \infty$;
- if $\{E_j\}$ is a sequence of disjoint sets in $\mathcal{M}$, then $\nu(\bigcup_1^\infty E_j) = \sum_1^\infty \nu(E_j)$, where the latter sum converges absolutely if $\nu(\bigcup_1^\infty E_j)$ is finite.
My issue is with the third condition. Specifically, what is it saying if $\nu(\bigcup_1^\infty E_j)$ is not finite? The summation might not even make sense, depending on the values of the $\nu(E_j)$ (e.g. if it is the alternating series $\sum (-1)^j$). Is this condition saying that the sum will always make sense? Or is it only saying the equality holds when the sum makes sense?
How can this condition be formulated correctly? And is there a textbook with a precise, rigorous formulation? (I couldn't find one.)