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Concept
Essential infimum and essential supremum
Summary
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 because we are allowed to ignore what the function does at the single point where is peculiar. The essential infimum is defined in a similar way.
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 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 of if the measurable set is a set of -measure zero, That is, if for -almost all in Let
be the set of essential upper bounds.
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