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
that is equal to zero everywhere except at
where
, 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
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
be a
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
valued
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
defined on a set ''X''. The supremum of a function is characterised by the following property:
for ''all''
and if for some
we have
for ''all''
then
.
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
:
is
empty
Empty may refer to:
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* ''Empty'' (God Lives Underwater album) or the title song, 1995
* ''Empty'' (Nils Frahm album), 2020
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* "Empty" (The Click Five song), 2007
* ...
. Let
:
be the set of upper bounds of ''f''. Then the supremum of ''f'' is defined by
:
if the set of upper bounds
is nonempty, and
otherwise.
Now assume in addition that
is a
measure space and, for simplicity, assume that the function
is measurable. Similar to the supremum, the essential supremum of a function is characterised by the following property:
for
-''almost all''
and if for some
we have
for
-''almost all''
then
. More concretely, a number
is called an ''essential upper bound'' of ''f'' if the measurable set
is a set of
-measure zero, i.e., if
for
-''almost all''
in
. Let
:
be the set of essential upper bounds. Then the essential supremum is defined similarly as
:
if
, and
otherwise.
Exactly in the same way one defines the essential infimum as the supremum of the ''essential lower bounds'', that is,
:
if the set of essential lower bounds is nonempty, and as
otherwise; again there is an alternative expression as
(with this being
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
:
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
:
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
, and its essential infimum is
.
Lastly, consider the function
:
Then for any
, we have
and so
and
.
Properties
* If
we have
. If
has measure zero
and
.
[ Dieudonné J.: Treatise On Analysis, Vol. II. Associated Press, New York 1976. p 172f.]
*
whenever both terms on the right are nonnegative.
See also
*
''L''''p'' space
Notes
References
{{Measure theory
Measure theory
Integral calculus