essential supremum

Essential supremum of a function

Let $(\Omega,\mathcal{F},\mu)$ be a measure space and let $f$ be a Borel measurable function from $\Omega$ to the extended real numbers $\mathbb{\bar{R}}$ . The essential supremum of $f$ is the smallest number $a\in\mathbb{\bar{R}}$ for which $f$ only exceeds $a$ on a set of measure zero. This allows us to generalize the maximum of a function in a useful way.

More formally, we define $\mathrm{ess}\sup f$ as follows. Let $a\in\mathbb{R}$ , and define

M_{a}=\{x:f(x)>a\},

the subset of $X$ where $f(x)$ is greater than $a$ . Then let

A_{0}=\{a\in\mathbb{R}:\mu(M_{a})=0\},

the set of real numbers for which $M_{a}$ has measure zero. The essential supremum of $f$ is

\mathrm{ess}\sup f:=\inf A_{0}.

The supremum is taken in the set of extended real numbers so, $\mathrm{ess}\sup f=\infty$ if $A_{0}=\emptyset$ and $\mathrm{ess}\sup f=-\infty$ if $A_{0}=\mathbb{R}$ .

Essential supremum of a collection of functions

Let $(\Omega,\mathcal{F},\mu)$ be a measure space, and $\mathcal{S}$ be a collection of measurable functions $f\colon\Omega\rightarrow\mathbb{\bar{R}}$ . The Borel $\sigma$ -algebra on $\mathbb{\bar{R}}$ is used.

If $\mathcal{S}$ is countable then we can define the pointwise supremum of the functions in $\mathcal{S}$ , which will itself be measurable. However, if $\mathcal{S}$ is uncountable then this is often not useful, and does not even have to be measurable. Instead, the essential supremum can be used.

The essential supremum of $\mathcal{S}$ , written as ${\mathrm{ess}\sup}\,\mathcal{S}$ , if it exists, is a measurable function $f\colon\Omega\rightarrow\mathbb{\bar{R}}$ satisfying the following.

•

$f\geq g$ , $\mu$ -almost everywhere (http://planetmath.org/AlmostSurely), for any $g\in\mathcal{S}$ .
•

if $g\colon\Omega\rightarrow\mathbb{\bar{R}}$ is measurable and $g\geq h$ ( $\mu$ -a.e.) for every $h\in\mathcal{S}$ , then $g\geq f$ ( $\mu$ -a.e.).

Similarly, the essential infimum, ${\mathrm{ess}\inf}\mathcal{S}$ is defined by replacing the inequalities ‘ $\geq$ ’ by ‘ $\leq$ ’ in the above definition.

Note that if $f$ is the essential supremum and $g\colon\Omega\rightarrow\mathbb{\bar{R}}$ is equal to $f$ $\mu$ -almost everywhere, then $g$ is also an essential supremum. Conversely, if $f, g$ are both essential supremums then, from the above definition, $f\leq g$ and $g\leq f$ , so $f=g$ ( $\mu$ -a.e.). So, the essential supremum (and the essential infimum), if it exists, is only defined almost everywhere.

It can be shown that, for a $\sigma$ -finite measure $\mu$ , the essential supremum and essential infimum always exist (http://planetmath.org/ExistenceOfTheEssentialSupremum). Furthermore, they are always equal to the supremum or infimum of some countable subset of $\mathcal{S}$ .

Title	essential supremum
Canonical name	EssentialSupremum
Date of creation	2013-03-22 12:21:29
Last modified on	2013-03-22 12:21:29
Owner	gel (22282)
Last modified by	gel (22282)
Numerical id	9
Author	gel (22282)
Entry type	Definition
Classification	msc 28C20
Synonym	ess-sup
Synonym	ess sup
Related topic	Supremum
Related topic	LpSpace
Related topic	ExistenceOfTheEssentialSupremum
Defines	essential infimum
Defines	ess-inf
Defines	ess inf