Finite Signed Measures

In the setting of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation, let $(X,\mathcal{F})$ be a measurable space. For a sequence $(A_{m})$ in $\mathcal{F}$, its union $\bigcup_{m\in\mathbb{N}}A_{m}\in\mathcal{F}$ and the pairwise disjointness of $(A_{m})$ are as in Measure, Measure Space, and Probability Measure.

(i) Finite signed measure. A finite signed measure on $(X,\mathcal{F})$ is a map $\nu:\mathcal{F}\to\mathbb{R}$ such that $\nu(\emptyset)=0$ and, for every pairwise disjoint sequence $(A_{m})$ in $\mathcal{F}$, the series $\sum_{m=1}^{\infty}\nu(A_{m})$ converges and

$$\nu\Bigl(\bigcup_{m\in\mathbb{N}}A_{m}\Bigr)=\sum_{m=1}^{\infty}\nu(A_{m})$$

(countable additivity).