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.

then the supremum of the function equals one. However, its essential supremum is zero since (under the Lebesgue measure is peculiar. The essential infimum is defined in a similar way.

Definition

(that is, the image (that is, the preimage).

be a real valued function The supremum for all for all is called an upper bound is empty and define the infimum.

is a measure space is measurable-almost all-almost all is called an -measure zero,-almost all be the set of essential upper bounds. Then the otherwise.

Exactly in the same way one defines the as the supremum of the s if the set is empty).

Examples

On the real line consider the Lebesgue measure and its corresponding -algebra

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.

denotes the rational numbers

Properties

has measure zero then

are measurable, then

and

almost everywhere, then

and

The essential supremum of a function is not just the infimum of the essential upper bounds, but also their minimum. A similar result holds for the essential infimum.

Given a measure space the \mathcal{L}^\infty(S, \mu) consisting of all of measurable functions that are bounded almost everywhere is a seminormed space whose seminorm

See also

• Essential range

Notes

References