Essential supremum

In mathematics, the concepts of essential supremum and essential infimum are related to the notions of supremum and infimum, 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, i.e., except on a set of measure zero.

Let f : X → R be a real valued function defined on a set X. A real number a is called an upper bound for f if f(x) ≤ a for all x in X, i.e., if the set

is empty. Let

be the set of upper bounds of f. Then the supremum of f is defined by

if the set of upper bounds ${\displaystyle U_{f}}$ is nonempty, and  sup f = +∞ otherwise.

Now assume in addition that (X, Σ, μ) is a measure space and, for simplicity, assume that the function f is measurable. A number a is called an essential upper bound of f if the measurable set f⁻¹(a, ∞) is a set of measure zero, i.e., if f(x) ≤ a for almost all x in X. Let

be the set of essential upper bounds. Then the essential supremum is defined similarly as

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