In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the concepts of essential infimum and essential supremum are related to the notions of
infimum and
supremum
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 l ...
, but adapted to
measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...
and
functional analysis, where one often deals with statements that are not valid for ''all'' elements in a
set, but rather ''
almost everywhere'', 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
In mathematics, the image of a function is the set of all output values it may produce.
More generally, evaluating a given funct