Definition:Essentially Bounded Function

Definition

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 if and only if there exists a real number $c$ such that:

$\map \mu {\set {x \in X : \size {\map f x} > c} } = 0$

Essential Supremum

The essential supremum of $\size {\map f x}$ is the supremum of all possible $c$, and is written $\essup \size {\map f x}$.

Also see

• Definition:Lebesgue Infinity-Space

• Results about essentially bounded functions can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): essentially bounded function
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): essentially bounded function
• 2013: Francis Clarke: Functional Analysis, Calculus of Variations and Optimal Control ... (previous) ... (next): $1.1$: Basic Definitions