Definition:Supremum Seminorm

Definition

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $f: X \to \R$ be a real-valued function.

The supremum seminorm of $f$, commonly denoted as $\norm f_\infty$, is defined as:

$\ds \norm f_\infty = \inf_{\substack {N \mathop \in \Sigma \\ \map \mu N \mathop = 0} } \sup \set {\size {\map f x}: x \notin N}$

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Observe that for all $M > \norm f_\infty$:

$\map \mu {\set {x \in X: \size {\map f x} \ge M} } = 0$

and that an essentially bounded function is almost everywhere equal to a bounded function.

Also see

• Definition:L-Infinity

Sources

• 1966: Robert G. Bartle: The Elements of Integration and Lebesgue Measure: Definition $6.15$