essential supremum

Essential supremum of a function

Let (Ω,,μ) be a measure spaceMathworldPlanetmath and let f be a Borel measurable function from Ω to the extended real numbers ¯. The essential supremumMathworldPlanetmath of f is the smallest number a¯ for which f only exceeds a on a set of measure zeroMathworldPlanetmath. This allows us to generalize the maximum of a function in a useful way.

More formally, we define esssupf as follows. Let a, and define


the subset of X where f(x) is greater than a. Then let


the set of real numbers for which Ma has measure zero. The essential supremum of f is


The supremum is taken in the set of extended real numbers so, esssupf= if A0= and esssupf=- if A0=.

Essential supremum of a collection of functions

Let (Ω,,μ) be a measure space, and 𝒮 be a collectionMathworldPlanetmath of measurable functions f:Ω¯. The Borel σ-algebra on ¯ is used.

If 𝒮 is countableMathworldPlanetmath then we can define the pointwise supremum of the functions in 𝒮, which will itself be measurable. However, if 𝒮 is uncountable then this is often not useful, and does not even have to be measurable. Instead, the essential supremum can be used.

The essential supremum of 𝒮, written as esssup𝒮, if it exists, is a measurable function f:Ω¯ satisfying the following.

  • fg, μ-almost everywhere (, for any g𝒮.

  • if g:Ω¯ is measurable and gh (μ-a.e.) for every h𝒮, then gf (μ-a.e.).

Similarly, the essential infimum, essinf𝒮 is defined by replacing the inequalitiesMathworldPlanetmath’ by ‘’ in the above definition.

Note that if f is the essential supremum and g:Ω¯ is equal to f μ-almost everywhere, then g is also an essential supremum. Conversely, if f,g are both essential supremums then, from the above definition, fg and gf, so f=g (μ-a.e.). So, the essential supremum (and the essential infimum), if it exists, is only defined almost everywhere.

It can be shown that, for a σ-finite measure μ, the essential supremum and essential infimum always exist ( Furthermore, they are always equal to the supremum or infimumMathworldPlanetmath of some countable subset of 𝒮.

Title essential supremum
Canonical name EssentialSupremum
Date of creation 2013-03-22 12:21:29
Last modified on 2013-03-22 12:21:29
Owner gel (22282)
Last modified by gel (22282)
Numerical id 9
Author gel (22282)
Entry type Definition
Classification msc 28C20
Synonym ess-sup
Synonym ess sup
Related topic Supremum
Related topic LpSpace
Related topic ExistenceOfTheEssentialSupremum
Defines essential infimum
Defines ess-inf
Defines ess inf