Essential Supremum and Essential Infimum

Essential Supremum And Essential Infimum

In mathematics, 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 one often deals with statements which are not valid everywhere, that is for all elements in a set, but rather almost everywhere, that is, except on a set of measure zero.

Let (X, Σ, μ) be a measure space, and let f : X → R 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

is empty. In contrast, a is called an essential upper bound if the set

is contained in a set of measure zero, that is to say, 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, the essential supremum of f, ess sup f, is defined by

if the set of essential upper bounds is not empty, and ess sup f = +∞ otherwise.

Exactly in the same way one defines the essential infimum as the largest essential lower bound, that is,

if the set of essential lower bounds is not empty, and as −∞ otherwise.