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, but rather almost everywhere, that is, 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.
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
The essential infimum is defined in a similar way.
As is often the case in measure-theoretic questions, the definition of essential supremum and infimum does not start by asking what a function
equals a specific value
be a real valued function defined on a set
More concretely, a real number
is called an upper bound for
be the set of upper bounds of
and define the infimum of the empty set by
if the set of upper bounds
is a measure space and, for simplicity, assume that the function
is called an essential upper bound of
be the set of essential upper bounds.
Then the essential supremum is defined similarly as
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
On the real line consider the Lebesgue measure and its corresponding đ-algebra
The supremum of this function (largest value) is 5, and the infimum (smallest value) is â4.
However, the function takes these values only on the sets
Everywhere else, the function takes the value 2.
denotes the rational numbers.
This function is unbounded both from above and from below, so its supremum and infimum are
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
If the essential supremums of two functions
consisting of all of measurable functions that are bounded almost everywhere is a seminormed space whose seminorm
is the essential supremum of a function's absolute value when
[nb 1] This article incorporates material from Essential supremum on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.