The concepts of essential supremum and essential infimum are related to the notions of supremum and infimum, but the former are more relevant in measure theory, where, often times one is not that interested in a property holding all the time, that is for all elements in a set, but rather almost all the time, that is, except on a set of measure zero. In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ... In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is smaller than all other elements of the subset. ... In mathematics, a measure is a function that assigns a number, e. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In measure theory, a null set is a set that is negligible for the purposes of the measure in question. ...

Let (X,Σ,μ) be a measure space and let be a function defined on X and with real values, which is not necessarily measurable. A real number a is called an upper bound for f if f(x)≤ a for all x in X, that is, if the set: Measure can mean: To perform a measurement. ... Partial plot of a function f. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, measurable functions are well-behaved functions between measurable spaces. ...

is empty. In contrast, a is called an essential upper bound if the set In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ...

is of measure zero, in other words, if f(x)≤ a for almost all x in X. Then, in the same way as the supremum of f is defined to be the smallest upper bound, the essential supremum is defined as the smallest essential upper bound.

More formally, we define the essential supremum as follows. Let , and define

the subset of X where f(x) is greater than a. Let A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ...

the set of real numbers for which M_a has measure zero. If , then the essential supremum is defined to be Otherwise, the essential supremum of f is defined as In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

Exactly in the same way one defines the essential infimum as the largest essential lower bound.

Examples

On the real line consider the Lebesgue measure and its corresponding σ-algebra Σ. Define a function f by the formula In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. ...

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 {1} and {−1} respectively, which are of measure zero. Everywhere else, the function takes the value 2. Thus, the essential supremum and the essential infimum of these functions are both 2.

As another example, consider the function

where Q denotes the rational numbers. 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 turns out then that the essential supremum is π/2 while the essential infimum is −π/2. In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...

Properties

See also

This article incorporates material from Essential supremum on PlanetMath, which is licensed under the GFDL. PlanetMath is a free, collaborative, online mathematics encyclopedia. ...