Suppose we have $(X, M, v)$ as a measure space. $v$ is a signed measure, then it also satisfies sigma additivity, i.e.
If {$E_j$} is a sequence of disjoint sets in $M$ , then $v(\cup_1^\infty E_j)=\sum_1^\infty v(E_j)$
What is the meaning of the series on the right-hand side? Can I always write it as the limit of finite summation?
I know a signed measure can only attain at most one of $+\infty$ and $-\infty$. So, if at least one of the $v(E_i)=+\infty$ or $-\infty$, then I can do it.
If all the $v(E_i)$s are finite, then $\sum_1^\infty |v(E_j)|$ must converge ($+\infty$ or finite).
If it is finite, then the original series is absolute convergent, so convergent, so I can also write it as a limit of finite summation.
If it is infinite, then can I still write the original series as the limit of finite summation? In this case, the limit may not exist, writing in this way makes no sense to me.