# Definition:Additive Function (Measure Theory)

*This page is about additive functions in measure theory. For other uses, see Definition:Additive Function.*

## Contents

## 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** if and only if:

- $\forall S, T \in \mathcal S: S \cap T = \varnothing \implies f \left({S \cup T}\right) = f \left({S}\right) + f \left({T}\right)$

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 $\mathcal S$.

## Also known as

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

## Also see

## Sources

- 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $\S 1.11$: Problems: $17$