View source for Essential infimum and essential supremum

{{Short description|Infimum and supremum almost everywhere}}
In [[mathematics]], the concepts of '''essential infimum''' and '''essential supremum''' are related to the notions of [[infimum and supremum]], 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]]'', that is, except on a [[null set|set of measure zero]].

While the exact definition is not immediately straightforward, intuitively the essential supremum of a function is the smallest value that is greater than or equal to the function values everywhere while ignoring what the function does at a set of points of measure zero. For example, if one takes the function <math>f(x)</math> that is equal to zero everywhere except at <math>x = 0</math> where <math>f(0) = 1,</math> then the supremum of the function equals one. However, its essential supremum is zero since (under the [[Lebesgue measure]]) one can ignore what the function does at the single point where <math>f</math> is peculiar. The essential infimum is defined in a similar way.

==Definition ==

As is often the case in measure-theoretic questions, the definition of essential supremum and infimum does not start by asking what a function <math>f</math> does at points <math>x</math> (that is, the ''image'' of <math>f</math>), but rather by asking for the set of points <math>x</math> where <math>f</math> equals a specific value <math>y</math> (that is, the [[preimage]] of <math>y</math> under <math>f</math>).

Let <math>f : X \to \Reals</math> be a [[real number|real]] valued [[function (mathematics)|function]] defined on a set <math>X.</math> The [[supremum]] of a function <math>f</math> is characterized by the following property: <math>f(x) \leq \sup f \leq \infty</math> for ''all'' <math>x \in X</math> and if for some <math>a \in \Reals \cup \{+\infty\}</math> we have <math>f(x) \leq a</math> for ''all'' <math>x \in X</math> then <math>\sup f \leq a.</math> 
More concretely, a real number <math>a</math> is called an ''[[upper bound]]'' for <math>f</math> if <math>f(x) \leq a</math> for all <math>x \in X;</math> that is, if the set
<math display=block>f^{-1}(a, \infty) = \{x \in X : f(x) > a\}</math>
is [[empty set|empty]]. Let
<math display=block>U_f = \{a \in \Reals : f^{-1}(a, \infty) = \varnothing\}\,</math>
be the set of upper bounds of <math>f</math> and define the [[infimum]] of the empty set by <math>\inf \varnothing = +\infty.</math> Then the supremum of <math>f</math> is 
<math display=block>\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.

Now assume in addition that <math>(X, \Sigma, \mu)</math> is a [[measure space]] and, for simplicity, assume that the function <math>f</math> is [[measurable function|measurable]]. Similar to the supremum, the essential supremum of a function is characterised by the following property: <math>f(x) \leq \operatorname{ess} \sup f \leq \infty</math> for <math>\mu</math>-''almost all'' <math>x \in X</math> and if for some <math>a \in \Reals \cup \{+\infty\}</math> we have <math>f(x) \leq a</math> for <math>\mu</math>-''almost all'' <math>x \in X</math> then <math>\operatorname{ess} \sup f \leq a.</math> More concretely, a number <math>a</math> is called an ''{{visible anchor|essential upper bound}}'' of <math>f</math> if the measurable set <math>f^{-1}(a, \infty)</math> is a set of <math>\mu</math>-measure zero,{{efn|For nonmeasurable 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]].}} That is, if <math>f(x) \leq a</math> for <math>\mu</math>-''almost all'' <math>x</math> in <math>X.</math> Let
<math display=block>U^{\operatorname{ess}}_f = \{a \in \Reals : \mu(f^{-1}(a, \infty)) = 0\}</math>
be the set of essential upper bounds. Then the '''{{visible anchor|essential supremum}}''' is defined similarly as
<math display=block>\operatorname{ess} \sup f = \inf U^{\mathrm{ess}}_f</math>
if <math>U^{\operatorname{ess}}_f \neq \varnothing,</math> and <math>\operatorname{ess}\sup f = +\infty</math> otherwise.

Exactly in the same way one defines the '''{{visible anchor|essential infimum}}''' as the supremum of the ''{{visible anchor|essential lower bound}}s'', that is,
<math display=block>\operatorname{ess} \inf f = \sup \{b \in \Reals : \mu(\{x: f(x) < b\}) = 0\}</math>
if the set of essential lower bounds is nonempty, and as <math>-\infty</math> otherwise; again there is an alternative expression as 
<math>\operatorname{ess} \inf f = \sup\{a \in \Reals : f(x) \geq a \text{ for almost all } x \in X\}</math> 
(with this being <math>-\infty</math> if the set is empty).

==Examples==

On the real line consider the [[Lebesgue measure]] and its corresponding [[sigma-algebra|{{sigma}}-algebra]] <math>\Sigma.</math> Define a function <math>f</math> by the formula
<math display=block>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 <math>\{1\}</math> and <math>\{-1\},</math> 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 display=block>f(x) = \begin{cases}
x^3, & \text{if } x \in \Q \\
\arctan x, & \text{if } x \in \Reals \smallsetminus \Q \\
\end{cases}</math>
where <math>\Q</math> denotes the [[rational number]]s. This function is unbounded both from above and from below, so its supremum and infimum are <math>\infty</math> and <math>-\infty,</math> 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 <math>\arctan x.</math> It follows that the essential supremum is <math>\pi / 2</math> while the essential infimum is <math>-\pi / 2.</math> 

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

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

==Properties==

If <math>\mu(X) > 0</math> then 
<math display=block>\inf f ~\leq~ \operatorname{ess} \inf f ~\leq~ \operatorname{ess}\sup f ~\leq~ \sup f.</math> 
and otherwise, if <math>X</math> has measure zero then <ref>[[Jean Dieudonné|Dieudonné J.]]: Treatise On Analysis, Vol. II. Associated Press, New York 1976. p 172f.</ref>
<math display=block>+\infty ~=~ \operatorname{ess}\inf f ~\geq~ \operatorname{ess}\sup f ~=~ -\infty.</math>

If <math>f</math> and <math>g</math> are measurable, then 
:<math display=block>
  \operatorname{ess\,sup} (f + g)
  \le  \operatorname{ess\,sup} f + \operatorname{ess\,sup} g
</math>
and 
:<math display=block>
  \operatorname{ess\,inf} (f + g)
  \ge  \operatorname{ess\,inf} f + \operatorname{ess\,inf} g.
</math>

If <math>f</math> and <math>g</math> are measurable and if <math>f \le g</math> almost everywhere, then 
:<math display=block>
  \operatorname{ess\,sup} f
  \le  \operatorname{ess\,sup} g
</math>
and 
:<math display=block>
  \operatorname{ess\,inf} f
  \le  \operatorname{ess\,inf} g.
</math>

If the essential supremums of two functions <math>f</math> and <math>g</math> are both nonnegative, then 
<math>\operatorname{ess}\sup (f g) ~\leq~ (\operatorname{ess}\sup f) \, (\operatorname{ess}\sup g).</math>

The essential supremum of a function is not just the infimum of the essential lower bounds, but also their minimum. A similar result holds for the essential infimum.

Given a [[measure space]] <math>(S, \Sigma, \mu),</math> the [[Lp space|space <math>\mathcal{L}^\infty(S, \mu)</math>]] consisting of all of measurable functions that are bounded almost everywhere is a [[seminormed space]] whose [[seminorm]]
<math display="block">\|f\|_\infty = \inf \{C \in \Reals_{\geq 0} : |f(x)| \leq C \text{ for almost every } x\} 
= \begin{cases}
\operatorname{ess}\sup |f| & \text{ if } 0 < \mu(S),\\
0 & \text{ if } 0 = \mu(S),
\end{cases}</math>
is the essential supremum of a function's absolute value when <math>\mu(S) \neq 0.</math>{{efn|If <math>\mu(S) = 0</math> then <math>\operatorname{ess}\sup|f| = -\infty.</math>}}

==See also==
* [[Essential range]]
* {{annotated link|Lp space|<math>L^p</math> space}}

==Notes==

{{notelist|30em}}

==References==

{{reflist}}

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

{{Lp spaces}}
{{Measure theory}}

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