# 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.