Definition:Essentially Bounded Function
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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
- 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