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In mathematics, the concepts of essential infimum and essential supremum are related to the notions of
infimum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
and supremum, but adapted to measure theory and
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
, where one often deals with statements that are not valid for ''all'' elements in a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
, but rather ''
almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
'', i.e., except on a 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 f(x) that is equal to zero everywhere except at x=0 where f(0)=1, then the supremum of the function equals one. However, its essential supremum is zero because we are allowed to ignore what the function does at the single point where f 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 ''f'' does at points ''x'' (i.e., the ''image'' of ''f''), but rather by asking for the set of points ''x'' where ''f'' equals a specific value ''y'' (i.e., the preimage of ''y'' under ''f''). Let f: X \to \mathbb be a
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valued
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defined on a set ''X''. The supremum of a function is characterised by the following property: f(x) \le \sup f \le \infty for ''all'' x \in X and if for some a \in \mathbb \cup +\infty we have f(x) \le a for ''all'' x \in X then \sup f \le a. More concretely, a real number ''a'' is called an ''
upper bound In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an eleme ...
'' for ''f'' if ''f''(''x'') ≤ ''a'' for all ''x'' in ''X'', i.e., if the set :f^(a, \infty) = \ is
empty Empty may refer to: ‍ Music Albums * ''Empty'' (God Lives Underwater album) or the title song, 1995 * ''Empty'' (Nils Frahm album), 2020 * ''Empty'' (Tait album) or the title song, 2001 Songs * "Empty" (The Click Five song), 2007 * ...
. Let : U_f = \ \, be the set of upper bounds of ''f''. Then the supremum of ''f'' is defined by : \sup f=\inf U_f \, if the set of upper bounds U_f is nonempty, and \sup f = + \infty otherwise. Now assume in addition that (X,\Sigma,\mu) is a
measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...
space and, for simplicity, assume that the function f is measurable. Similar to the supremum, the essential supremum of a function is characterised by the following property: f(x) \le \operatorname \sup f \le \infty for \mu-''almost all'' x \in X and if for some a \in \mathbb \cup +\infty we have f(x) \le a for \mu-''almost all'' x \in X then \operatorname \sup f \le a. More concretely, a number a is called an ''essential upper bound'' of ''f'' if the measurable set f^(a, \infty) is a set of \mu-measure zero, i.e., if f(x)\le a for \mu-''almost all'' x in X. Let :U^_f = \\, be the set of essential upper bounds. Then the essential supremum is defined similarly as : \operatorname \sup f=\inf U^_f \, if U^_f \ne \varnothing, and \operatorname\sup f = + \infty otherwise. Exactly in the same way one defines the essential infimum as the supremum of the ''essential lower bounds'', that is, : \operatorname \inf f=\sup \ if the set of essential lower bounds is nonempty, and as -\infty otherwise; again there is an alternative expression as \operatorname \inf f = \sup\ (with this being -\infty if the set is empty).


Examples

On the real line consider the Lebesgue measure and its corresponding σ-algebra Σ. Define a function ''f'' by the formula : f(x)= \begin 5, & \text x=1 \\ -4, & \text x = -1 \\ 2, & \text \end 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 and 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 : f(x)= \begin x^3, & \text x\in \mathbb Q \\ \arctan x, & \text x\in \mathbb R\smallsetminus \mathbb Q \\ \end where Q denotes the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
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 arctan ''x''. It follows that the essential supremum is /2 while the essential infimum is −/2. On the other hand, consider the function ''f''(''x'') = ''x''3 defined for all real ''x''. Its essential supremum is +\infty, and its essential infimum is -\infty. Lastly, consider the function : f(x)= \begin 1/x, & \text x \ne 0 \\ 0, & \text x = 0. \\ \end Then for any \textstyle a \in \mathbb R, we have \textstyle \mu(\) \geq \tfrac and so \textstyle U_f^ = \varnothing and \operatorname \sup f = + \infty.


Properties

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


See also

* ''L''''p'' space


Notes


References

{{Measure theory Measure theory Integral calculus