Definition:Additive Function (Measure Theory)

Definition
Let $\mathcal S$ be an algebra of sets.

Let $f: \mathcal S \to \overline{\R}$ be a function, where $\overline{\R}$ denotes the set of extended real numbers.

Then $f$ is defined to be additive iff:
 * $\forall S, T \in \mathcal S: S \cap T = \varnothing \implies f \left({S \cup T}\right) = f \left({S}\right) + f \left({T}\right)$

That is, for any two disjoint elements of $\mathcal S$, $f$ of their union equals the sum of $f$ of the individual elements.

Note from Finite Union of Sets in Additive Function that:
 * $\displaystyle f \left({\bigcup_{i \mathop = 1}^n S_i}\right) = \sum_{i \mathop = 1}^n f \left({S_i}\right)$

where $S_1, S_2, \ldots, S_n$ is any finite collection of pairwise disjoint elements of $\mathcal S$.

Such a function is also referred to as a finitely additive function to distinguish it, when necessary, from a countably additive function.

Context
This definition is usually made in the context of measure theory, but the concept reaches a wider field than that.

Also see

 * Definition:Countably Additive Function
 * Definition:Finitely Additive Function
 * Definition:Subadditive Function (Measure Theory)