Definition:Additive Function (Measure Theory)

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

Let $$f: \mathcal A \to \overline {\R}$$ be a real-valued 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:
 * $$f \left({\bigcup_{i=1}^n A_i}\right) = \sum_{i=1}^n f \left({A_i}\right)$$

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.