Definition:Additive Function (Measure Theory)

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

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

Then $f$ is defined as additive iff:
 * $\forall A, B \in \mathcal A: A \cap B = \varnothing \implies f \left({A \cup B}\right) = f \left({A}\right) + f \left({B}\right)$

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

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

where $A_1, A_2, \ldots, A_n$ is any finite collection of pairwise disjoint elements of $\mathcal A$.

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

 * Countably Additive Function;


 * Subadditive Function.

Warning
In the field of analysis, an additive function has a different defintion.

In the field of number theory, an additive function has another different defintion.