Essential supremum

In mathematics, the concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, 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.

While the exact definition is not immediately straightforward, intuitively the essential supremum of a function is the smallest value that is greater than or equal to the function values everywhere while ignoring what the function does at a set of points of measure zero. For example, if one takes the function $f(x)$ that is equal to zero everywhere except at $x=0$ where $f(0)=1$, then the supremum of the function equals one. However, its essential supremum is zero because we are allowed to ignore what the function does at the single point where $f$ is peculiar. The essential infimum is defined in a similar way.

Definition

As is often the case in measure-theoretic questions, the definition of essential supremum and infimum does not start by asking what a function ''f'' does at points ''x'' (i.e., the ''image'' of ''f''), but rather by asking for the set of points ''x'' where ''f'' equals a specific value ''y'' (i.e., the preimage of ''y'' under ''f'').

Let $f: X \to \mathbb$ be a real valued function defined on a set ''X''. The supremum of a function is characterised by the following property: $f(x) \le \sup f \le \infty$ for ''all'' $x \in X$ and if for some $a \in \mathbb \cup +\infty$ we have $f(x) \le a$ for ''all'' $x \in X$ then $\sup f \le a$.
More concretely, 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

:$f^(a, \infty) = \$

is empty. Let

:$U_f = \ \,$

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

:$\sup f=\inf U_f \,$

if the set of upper bounds $U_f$ is nonempty, and $\sup f = + \infty$ otherwise.

Now assume in addition that $(X,\Sigma,\mu)$ is a measure space and, for simplicity, assume that the function $f$ is measurable. Similar to the supremum, the essential supremum of a function is characterised by the following property: $f(x) \le \operatorname \sup f \le \infty$ for $\mu$-''almost all'' $x \in X$ and if for some $a \in \mathbb \cup +\infty$ we have $f(x) \le a$ for $\mu$-''almost all'' $x \in X$ then $\operatorname \sup f \le a$. More concretely, a number $a$ is called an ''essential upper bound'' of ''f'' if the measurable set $f^(a, \infty)$ is a set of $\mu$-measure zero, i.e., if $f(x)\le a$ for $\mu$-''almost all'' $x$ in $X$. Let

:$U^_f = \\,$

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

:$\operatorname \sup f=\inf U^_f \,$

if $U^_f \ne \varnothing$, and $\operatorname\sup f = + \infty$ otherwise.

Exactly in the same way one defines the essential infimum as the supremum of the ''essential lower bounds'', that is,

:$\operatorname \inf f=\sup \$

if the set of essential lower bounds is nonempty, and as $-\infty$ otherwise; again there is an alternative expression as $\operatorname \inf f = \sup\$ (with this being $-\infty$ if the set is empty).

Examples

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

:$f(x)= \begin 5, & \text x=1 \\ -4, & \text x = -1 \\ 2, & \text \end$

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 this function are both 2.

As another example, consider the function
:$f(x)= \begin x^3, & \text x\in \mathbb Q \\ \arctan x, & \text x\in \mathbb R\smallsetminus \mathbb Q \\ \end$
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 follows that the essential supremum is /2 while the essential infimum is −/2.

On the other hand, consider the function ''f''(''x'') = ''x''³ defined for all real ''x''. Its essential supremum is $+\infty$, and its essential infimum is $-\infty$.

Lastly, consider the function
:$f(x)= \begin 1/x, & \text x \ne 0 \\ 0, & \text x = 0. \\ \end$
Then for any $\textstyle a \in \mathbb R$, we have $\textstyle \mu(\) \geq \tfrac$ and so $\textstyle U_f^ = \varnothing$ and $\operatorname \sup f = + \infty$.

Properties

* If $\mu(X)>0$ we have $\inf f \le \operatorname \inf f \le \operatorname\sup f \le \sup f$. If $X$ has measure zero $\operatorname\sup f=-\infty$ and $\operatorname\inf f = +\infty$. Dieudonné J.: Treatise On Analysis, Vol. II. Associated Press, New York 1976. p 172f.
* $\operatorname\sup (fg) \le (\operatorname\sup f)(\operatorname\sup g)$ whenever both terms on the right are nonnegative.

See also

* ''L''^''p'' space

Notes

References

{{Measure theory

Measure theory
Integral calculus