Definition:Subadditive 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 subadditive (or sub-additive) iff:
 * $\forall A, B \in \mathcal A: f \left({A \cup B}\right) \le f \left({A}\right) + f \left({B}\right)$

That is, for any two elements of $\mathcal A$, $f$ of their union is less than or equal to the sum of $f$ of the individual elements.

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

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

Such a function is also referred to as a finitely subadditive function to distinguish it, when necessary, from a countably subadditive function.

Context
This definition is usually made in the context of measure theory, but the concept reaches a wider field than that.

Note
There is no requirement that the sets involved have to be disjoint, as they have to be when considering an additive function.

Also See

 * Additive Function;


 * Countably Subadditive Function.