essential supremum
Essential supremum of a function
Let $(\mathrm{\Omega},\mathcal{F},\mu )$ be a measure space^{} and let $f$ be a Borel measurable function from $\mathrm{\Omega}$ to the extended real numbers $\overline{\mathbb{R}}$. The essential supremum^{} of $f$ is the smallest number $a\in \overline{\mathbb{R}}$ 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 $\mathrm{ess}supf$ 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
$$\mathrm{ess}supf:=inf{A}_{0}.$$ 
The supremum is taken in the set of extended real numbers so, $\mathrm{ess}supf=\mathrm{\infty}$ if ${A}_{0}=\mathrm{\varnothing}$ and $\mathrm{ess}supf=\mathrm{\infty}$ if ${A}_{0}=\mathbb{R}$.
Essential supremum of a collection of functions
Let $(\mathrm{\Omega},\mathcal{F},\mu )$ be a measure space, and $\mathcal{S}$ be a collection^{} of measurable functions $f:\mathrm{\Omega}\to \overline{\mathbb{R}}$. The Borel $\sigma $algebra on $\overline{\mathbb{R}}$ 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 $f:\mathrm{\Omega}\to \overline{\mathbb{R}}$ satisfying the following.

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

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if $g:\mathrm{\Omega}\to \overline{\mathbb{R}}$ is measurable and $g\ge h$ ($\mu $a.e.) for every $h\in \mathcal{S}$, then $g\ge f$ ($\mu $a.e.).
Similarly, the essential infimum, $\mathrm{ess}inf\mathcal{S}$ is defined by replacing the inequalities^{} ‘$\ge $’ by ‘$\le $’ in the above definition.
Note that if $f$ is the essential supremum and $g:\mathrm{\Omega}\to \overline{\mathbb{R}}$ 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, $f\le g$ and $g\le f$, 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  20130322 12:21:29 
Last modified on  20130322 12:21:29 
Owner  gel (22282) 
Last modified by  gel (22282) 
Numerical id  9 
Author  gel (22282) 
Entry type  Definition 
Classification  msc 28C20 
Synonym  esssup 
Synonym  ess sup 
Related topic  Supremum 
Related topic  LpSpace 
Related topic  ExistenceOfTheEssentialSupremum 
Defines  essential infimum 
Defines  essinf 
Defines  ess inf 