Definition:Subadditive 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 subadditive 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:
 * $$f \left({\bigcup_{i=1}^n A_i}\right) \le \sum_{i=1}^n f \left({A_i}\right)$$

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.