Definition:Subadditive Function (Measure Theory)
Contents
Definition
Let $\mathcal S$ be an algebra of sets.
Let $f: \mathcal S \to \overline \R$ be a function, where $\overline \R$ denotes the extended set of real numbers.
Then $f$ is defined to be subadditive (or sub-additive) iff:
- $\forall S, T \in \mathcal S: f \left({S \cup T}\right) \le f \left({S}\right) + f \left({T}\right)$
That is, for any two elements of $\mathcal S$, $f$ applied to their union is not greater than the sum of $f$ of the individual elements.
Also known as
Such a function is also referred to as a finitely subadditive function to distinguish it, when necessary, from a countably subadditive function.
This arises from Finite Union of Sets in Subadditive Function, where it is shown that:
- $\displaystyle f \left({\bigcup_{i \mathop = 1}^n S_i}\right) \le \sum_{i \mathop = 1}^n f \left({S_i}\right)$
where $S_1, S_2, \ldots, S_n$ is any finite collection of elements of $\mathcal S$.
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.
