View source for Essential supremum and essential infimum

{{bots|deny=ClueBot NG}}
In [[mathematics]], the concepts of '''essential supremum''' and '''essential infimum''' are related to the notions of [[supremum]] and [[infimum]], but  adapted to [[measure theory]] and [[functional analysis]], where one often deals with statements that are not valid for ''all'' elements in a [[Set (mathematics)|set]], but rather ''[[almost everywhere]]'', i.e., except on a [[null set|set of measure zero]].

==Definition==

Let ''f''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;'''R''' be a [[real number|real]] valued  [[function (mathematics)|function]] defined on a set ''X''. A real number ''a'' is called an ''[[upper bound]]'' for ''f'' if ''f''(''x'')&nbsp;&le;&nbsp;''a'' for all ''x'' in ''X'', i.e., if the set

:<math>f^{-1}(a, \infty) = \{x\in X: f(x)>a\} </math>

is [[empty set|empty]]. Let

:<math> U_f = \{a \in \mathbb{R}: f^{-1}(a, \infty)  = \varnothing\}  \,</math>

be the set of upper bounds of ''f''. Then the supremum of ''f'' is defined by 
 
:<math>  \sup f=\inf {U_f} \, </math>

if the set of upper bounds <math> U_f</math> is nonempty, and <math>\sup f = + \infty</math> otherwise. 

Alternatively, if for some <math> a \in \mathbb{R}</math> we have <math> f(x) \le a </math> for ''all'' <math>x \in X</math> then <math>\sup f \le a</math>. 

Now assume in addition that <math>(X,\Sigma;\mu)</math> is a [[measure (mathematics)|measure]] space and, for simplicity, assume that the function <math>f</math> is measurable. A number <math>a</math> is called an ''essential upper bound'' of ''f'' if the measurable set <math>f^{-1}(a, \infty)</math> is  a set of measure zero,{{efn|For non measurable functions the definition has to be modified  by assuming that <math>f^{-1}(a, \infty)</math> is ''contained'' in a set of measure zero. Alternatively, one can assume that the measure is [[complete measure|complete]]}} i.e., if <math>f(x)\le a</math> for ''almost all'' <math>x</math> in <math>X</math>.  Let

:<math>U^{\operatorname{ess}}_f = \{a \in \mathbb{R}:  \mu(f^{-1}(a, \infty)) = 0\}\,</math>
 
be the set of essential upper bounds. Then the essential supremum is defined similarly as

:<math> \operatorname{ess} \sup f=\inf U^{\mathrm{ess}}_f  \, </math>

if <math> U^{\operatorname{ess}}_f \ne \varnothing</math>, and <math>\operatorname{ess}\sup f = + \infty</math> otherwise.

Alternatively, if for some <math> a \in \mathbb{R}</math> we have <math> f(x) \le a </math> for ''almost all'' <math>x \in X</math> then <math>\operatorname{ess} \sup f \le a</math>. 

Exactly in the same way one defines the '''essential infimum''' as the supremum of the  ''essential lower bounds'', that is,

:<math> \operatorname{ess} \inf f=\sup \{b \in \mathbb{R}: \mu(\{x: f(x) < b\}) = 0\}\, </math>

if the set of essential lower bounds is nonempty, and as <math>-\infty</math> otherwise.

==Examples==

On the real line consider the [[Lebesgue measure]] and its corresponding σ-algebra Σ. Define a function ''f'' by the formula

:<math> f(x)= \begin{cases} 5, & \text{if }  x=1  \\ 
                            -4, & \text{if }  x = -1 \\
                            2, & \text{otherwise. }
 \end{cases} </math>

The supremum of this function (largest value) is 5, and the infimum (smallest value) is −4. However, the function takes these values only on the sets {1} and {−1} respectively, which are of measure zero. Everywhere else, the function takes the value 2. Thus, the essential supremum and the essential infimum of this function are both 2.

As another example, consider the function 
:<math> f(x)= \begin{cases} x^3, & \text{if }  x\in \mathbb Q  \\ 
                            \arctan x, & \text{if } x\in \mathbb R\smallsetminus \mathbb Q \\
 \end{cases} </math>
where '''Q''' denotes the [[rational number]]s. This function is unbounded both from above and from below, so its supremum and infimum are ∞ and −∞ respectively. However, from the point of view of the Lebesgue measure, the set of rational numbers is of measure zero; thus, what really matters is what happens in the complement of this set, where the function is given as&nbsp;arctan&nbsp;''x''. It follows that the essential supremum is {{pi}}/2 while the essential infimum is&nbsp;−{{pi}}/2.

On the other hand, consider the function ''f''(''x'')&nbsp;=&nbsp;''x''<sup>3</sup> defined for all real ''x''. Its essential supremum is <math>+\infty</math>, and its essential infimum is <math>-\infty</math>.

Lastly, consider the function 
:<math> f(x)= \begin{cases} 1/x, & \text{if }  x \ne 0  \\ 
                            0, & \text{if } x = 0. \\
 \end{cases} </math>
Then for any <math>\textstyle a \in \mathbb R</math>, we have <math>\textstyle \mu(\{x \in \mathbb R : 1/x > a\}) \geq \tfrac{1}{|a|}</math> and so <math>\textstyle U_f = \varnothing</math> and <math>\operatorname{ess} \sup f = + \infty</math>.

==Properties==
* If <math> \mu(X)>0 </math> we have <math>\inf f \le \operatorname{ess} \inf f \le \operatorname{ess}\sup f \le \sup f</math>. If <math> X </math> has measure zero <math>\operatorname{ess}\sup f=-\infty</math> and <math>\operatorname{ess}\inf f = +\infty</math>.<ref>Dieudonne J.: Treatise On Analysis, Vol. II. Associated Press, New York 1976. p 172f.</ref>
* <math>\operatorname{ess}\sup (fg) \le (\operatorname{ess}\sup f)(\operatorname{ess}\sup g)</math> whenever both terms on the right are nonnegative.

==See also==

* [[Lp spaces|''L''<sup>''p''</sup> spaces]]

==Notes==
{{notelist}}

== References ==

{{reflist}}

{{PlanetMath attribution|urlname=EssentialSupremum|title=Essential supremum}}

[[Category:Measure theory]]
[[Category:Integral calculus]]