Definition:Subadditive Function (Conventional)

Definition

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

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

$\forall a, b \in S: f \left({a +_S b}\right) \preceq f \left({a}\right) +_T f \left({b}\right)$

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).

Also see

Compare with the field of measure theory, in which the definition of subadditive function is completely different.