\begin{definition}[Definition:Additive Function (Measure Theory)]
Let $\SS$ be an algebra of sets.
Let $f: \SS \to \overline \R$ be a function, where $\overline \R$ denotes the set of extended real numbers.
Then $f$ is defined to be '''additive''' {{iff}}:
:$\forall S, T \in \SS: S \cap T = \O \implies \map f {S \cup T} = \map f S + \map f T$
That is, for any two disjoint elements of $\SS$, $f$ of their union equals the sum of $f$ of the individual elements.
Note from Finite Union of Sets in Additive Function that:
:$\ds \map f {\bigcup_{i \mathop = 1}^n S_i} = \sum_{i \mathop = 1}^n \map f {S_i}$
where $S_1, S_2, \ldots, S_n$ is any finite collection of pairwise disjoint elements of $\SS$.
\end{definition}