Essential supremum and essential infimum

Encyclopedia : E : ES : ESS : Essential supremum and essential infimum

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, often times one is not that interested in a property holding all the time, that is for all elements in a set, but rather almost all the time, that is, except on a set of measure zero.

Let [(X, \Sigma, \mu)] be a measure space and let [f:X \to \mathbb] 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 of measure zero, in other words, 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, we define the essential supremum [\mathrm \sup f] as follows. Let [a \in \mathbb], and define

[ M_ = \,\, ]

the subset of [X] where [f(x)] is greater than [a.] Let

[ A_ = \: \mu(M_a) = 0\},\,]

the set of real numbers for which [M_a] has measure zero. If [A_0 = \emptyset], then the essential supremum is defined to be [\infty.] Otherwise, the essential supremum of [f] is defined as

[ \mathrm \sup f=\inf A_0.\, ] or,

[ \mathrm \sup f=\inf \: \mu(\) = 0\}.\, ]

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

[ \mathrm \inf f=\sup \: \mu(\) = 0\}.\, ]

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 5, & \mbox x=1 \\ -4,& \mbox x = -1 \\ 2,& \mbox \end ]

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 and respectively, which are of measure zero. Everywhere else, the function takes the value 2. Thus, the essential supremum and the essential infimum of these functions are both 2.

As another example, consider the function

[ f(x)= \begin x^3, & \mbox x\in \mathbb Q \\ \arctan ,& \mbox x\in \mathbb R\backslash \mathbb Q \\ \end ]

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 turns out then that the essential supremum is π/2 while the essential infimum is −π/2.

Properties

[\inf f \le \mathrm \inf f \le \mathrm \sup f \le \sup f]

Essential supremum and essential infimum

Examples

Properties

See also