Essential supremum

Least upper bound of a measurable function after ignoring a null set.

Essential supremum

An essential supremum of a measurable function $f:X\to\overline{\mathbb{R}}$ on a measure space $(X,\Sigma,\mu)$ is the number

\operatorname*{ess\,sup}_{x\in X} f(x) := \inf\Bigl\{M\in\mathbb{R} : f(x)\le M \text{ for }\mu\text{-almost every }x\in X\Bigr\}.

Equivalently, $\operatorname*{ess\,sup} f$ is the infimum of all real numbers $M$ that are an upper bound for $f$ outside a null set .

Unlike the pointwise supremum , the essential supremum is unchanged if $f$ is modified on a null set; in particular it depends only on the a.e. equivalence class of $f$ . This notion is used to define the $p=\infty$ case of the $L^p$ norm .

Examples:

On $([0,1],\mathcal{B},\lambda)$ , for $f(x)=x$ one has $\operatorname*{ess\,sup} f=1$ .
On the same space, if $f(0)=1$ and $f(x)=0$ for $x\in(0,1]$ , then $\sup f=1$ but $\operatorname*{ess\,sup} f=0$ since the exceptional point $\{0\}$ is a null set .