Definition:Additive Function (Measure Theory)

This page is about additive function in the context of measure theory. For other uses, see Additive Function.

Definition

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 an additive function if and only if:

$\forall S, T \in \SS: S \cap T = \O \implies \map f {S \cup T} = \map f S + \map f T$

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

$\ds \map f {\bigcup_{i \mathop = 1}^n S_i} = \sum_{i \mathop = 1}^n \map f {S_i}$

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

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