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.

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