Definition:Supremum Seminorm

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Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

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

The supremum seminorm of $f$, commonly denoted as $\left\Vert{f}\right\Vert_\infty$, is defined as:

$\displaystyle \left\Vert{f}\right\Vert_\infty = \inf_{\substack {N \mathop \in \Sigma \\ \mu \left({N}\right) \mathop = 0}} \sup \ \left\{ {\left\vert{f \left({x}\right)}\right\vert: x \notin N}\right\}$

Observe that for all $M > \Vert f \Vert_\infty$, then:

$\mu \left({ \left\{ {x \in X: \left\vert{f \left({x}\right)}\right\vert \ge M}\right\} }\right) = 0$

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

Also see