# 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