Definition:Monotone (Measure Theory)

Let $\mathcal S$ be an algebra of sets.
Let $f: \mathcal S \to \overline \R$ be an extended real-valued function, where $\overline \R$ denotes the set of extended real numbers.
Then $f$ is defined as monotone or monotonic iff:
$\forall A, B \in \mathcal S: A \subseteq B \iff f \left({A}\right) \le f \left({B}\right)$