\begin{definition}[Definition:Signed Measure]
Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu : \Sigma \to \overline \R$ be an extended real-valued function such that: 
:if $\map \mu A = +\infty$ for some $A \in \Sigma$, then $\map \mu B > -\infty$ for all $B \in \Sigma$.
and:
:if $\map \mu A = -\infty$ for some $A \in \Sigma$, then $\map \mu B < +\infty$ for all $B \in \Sigma$.
We say that $\mu$ is a '''signed measure''' on $\struct {X, \Sigma}$ {{iff}}: 
{{begin-axiom}}
{{axiom | n = 1
        | ml= \map \mu \O
        | mo= =
        | mr= 0
}}
{{axiom | n = 2
        | q = \forall \sequence {S_n}_{n \mathop \in \N} \subseteq \Sigma: \forall i, j \in \N: S_i \cap S_j = \O
        | ml= \map \mu {\bigcup_{n \mathop = 1}^\infty S_n}
        | mo= =
        | mr= \sum_{n \mathop = 1}^\infty \map \mu {S_n}
        | rc= that is, $\mu$ is a countably additive function
}}
{{end-axiom}}
\end{definition}