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Essential suprema and infima

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Essential suprema and infima

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 that 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

\{x\in X: f(x)>a\}

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

\{x\in X: f(x)>a\}

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

\mathrm{ess } \sup f=\inf \{a \in \mathbb{R}: \mu(\{x: f(x) > a\}) = 0\}\,

if the set \{a \in \mathbb{R}: \mu(\{x: f(x) > a\}) = 0\} 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,

\mathrm{ess } \inf f=\sup \{b \in \mathbb{R}: \mu(\{x: f(x) < b\}) = 0\}\,

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

Examples

On the real line consider the Lebesgue measure and its corresponding σ-algebra Σ. Define a function f by the formula

f(x)= \begin{cases} 5, & \text{if } x=1 \\
                           -4, & \text{if }  x = -1 \\
                           2, & \text{ otherwise. }
\end{cases} 

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 this function are both 2.

As another example, consider the function

f(x)= \begin{cases} x^3, & \text{if } x\in \mathbb Q \\
                           \arctan{x} ,& \text{if } x\in \mathbb R\backslash \mathbb Q \\
\end{cases} 

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 follows that the essential supremum is π/2 while the essential infimum is −π/2.

Lastly, consider the function f(x) = x3 defined for all real x. Its essential supremum is +∞, and its essential infimum is −∞.

Properties

  • \inf f \le \liminf f \le \mathrm{ess } \inf f \le \mathrm{ess }\sup f \le \limsup f \le \sup f
  • \mathrm{ess }\sup (fg) \le (\mathrm{ess }\sup f)(\mathrm{ess }\sup g) whenever both terms on the right are nonnegative.

This article incorporates material from Essential supremum on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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