\begin{definition}[Definition:Measure (Measure Theory)]
Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu: \Sigma \to \overline \R$ be a mapping, where $\overline \R$ denotes the set of extended real numbers.
Then $\mu$ is called a '''measure''' on $\Sigma$ {{iff}} $\mu$ has the following properties:
{{begin-axiom}}
{{axiom | n = 1
        | q = \forall E \in \Sigma
        | ml= \map \mu E
        | mo= \ge
        | 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
}}
{{axiom | n = 3
        | q = \exists E \in \Sigma
        | ml= \map \mu E
        | mo= \in
        | mr= \R
        | rc= that is, there exists at least one $E \in \Sigma$ such that $\map \mu E$ is finite
}}
{{end-axiom}}
\end{definition}