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

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