Essential infimum and 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, that is, 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 that is equal to zero everywhere except at where then the supremum of the function equals one. However, its essential supremum is zero since (under the Lebesgue measure) one can ignore what the function does at the single point where is peculiar. The essential infimum is defined in a similar way.

Definition

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

Let   be a real valued function defined on a set   The supremum of a function   is characterized by the following property:   for all   and if for some   we have   for all   then   More concretely, a real number   is called an upper bound for   if   for all   that is, if the set   is empty. Let   be the set of upper bounds of   and define the infimum of the empty set by   Then the supremum of   is   if the set of upper bounds   is nonempty, and   otherwise.

Now assume in addition that   is a measure space and, for simplicity, assume that the function   is measurable. Similar to the supremum, the essential supremum of a function is characterised by the following property:   for  -almost all   and if for some   we have   for  -almost all   then   More concretely, a number   is called an essential upper bound of   if the measurable set   is a set of  -measure zero,[a] That is, if   for  -almost all   in   Let   be the set of essential upper bounds. Then the essential supremum is defined similarly as   if   and   otherwise.

Exactly in the same way one defines the essential infimum as the supremum of the essential lower bounds, that is,   if the set of essential lower bounds is nonempty, and as   otherwise; again there is an alternative expression as   (with this being   if the set is empty).

Examples

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

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   where   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   It follows that the essential supremum is   while the essential infimum is  

On the other hand, consider the function   defined for all real   Its essential supremum is   and its essential infimum is  

Lastly, consider the function   Then for any     and so   and  

Properties

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If   then   and otherwise, if   has measure zero then [1]  

If the essential supremums of two functions   and   are both nonnegative, then  

Given a measure space   the space   consisting of all of measurable functions that are bounded almost everywhere is a seminormed space whose seminorm   is the essential supremum of a function's absolute value when  [nb 1]

See also

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Notes

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  1. ^ For nonmeasurable functions the definition has to be modified by assuming that   is contained in a set of measure zero. Alternatively, one can assume that the measure is complete.
  1. ^ If   then  

References

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  1. ^ Dieudonné J.: Treatise On Analysis, Vol. II. Associated Press, New York 1976. p 172f.

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