Essential supremum - Knowlpedia

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

Equivalent characterizations

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

Remarks

Unlike the pointwise supremum, the essential supremum is unchanged if $f$ is modified on a null set. It therefore depends only on the almost-everywhere equivalence class of $f$ , and it defines the $L^\infty$ norm through $\lVert f\rVert_\infty=\operatorname*{ess\,sup}|f|$ .

Examples

On $([0,1],\mathcal B,\lambda)$ , the function $f(x)=x$ has essential supremum $1$ .
On the same space, if $f(0)=1$ and $f(x)=0$ for $x>0$ , then $\sup f=1$ but $\operatorname*{ess\,sup}f=0$ .