# Definition:Additive Function (Measure Theory)

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*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:

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

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:

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

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

## 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: $17$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**additive**:**2.**(of a set function on a class of sets)