\begin{definition}[Definition:Essentially Bounded Function]
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $f : X \to \R$ be a $\Sigma$-measurable function. 
We say that $f$ is '''essentially bounded''' {{iff}} there exists a real number $c$ such that: 
:$\ds \map \mu {\set {x \in X : \size {\map f x} > c} } = 0$
\end{definition}