Definition:Subadditive Function

Definition

Algebra

Let $\struct {S, +_S}$ and $\struct {T, +_T, \preceq}$ be semigroups such that $\struct {T, +_T, \preceq}$ is ordered.

Let $f: S \to T$ be a mapping from $S$ to $T$ which satisfies the relation:

$\forall a, b \in S: \map f {a +_S b} \preceq \map f a +_T \map f b$

Then $f$ is defined as being subadditive.

The usual context in which this is encountered is where $S$ and $T$ are both the set of real numbers $\R$ (or a subset of them).

Measure Theory

Let $\SS$ be an algebra of sets.

Let $f: \SS \to \overline \R$ be a function, where $\overline \R$ denotes the extended set of real numbers.

Then $f$ is defined to be subadditive (or sub-additive) if and only if:

$\forall S, T \in \SS: \map f {S \cup T} \le \map f S + \map f T$

That is, for any two elements of $\SS$, $f$ applied to their union is not greater than the sum of $f$ of the individual elements.

Also see

• Definition:Countably Subadditive Function.

• Definition:Additive Function
• Definition:Additive

• Results about subadditive functions can be found here.