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Essential infimum and essential supremum

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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 if we apply the Lebesgue-Borel measure and are allowed to 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.