Essential supremum of a function
Let be a measure space and let be a Borel measurable function from to the extended real numbers . The essential supremum of is the smallest number for which only exceeds on a set of measure zero. This allows us to generalize the maximum of a function in a useful way.
More formally, we define as follows. Let , and define
Essential supremum of a collection of functions
If is countable then we can define the pointwise supremum of the functions in , which will itself be measurable. However, if is uncountable then this is often not useful, and does not even have to be measurable. Instead, the essential supremum can be used.
The essential supremum of , written as , if it exists, is a measurable function satisfying the following.
, -almost everywhere (http://planetmath.org/AlmostSurely), for any .
if is measurable and (-a.e.) for every , then (-a.e.).
Similarly, the essential infimum, is defined by replacing the inequalities ‘’ by ‘’ in the above definition.
Note that if is the essential supremum and is equal to -almost everywhere, then is also an essential supremum. Conversely, if are both essential supremums then, from the above definition, and , so (-a.e.). So, the essential supremum (and the essential infimum), if it exists, is only defined almost everywhere.
It can be shown that, for a -finite measure , the essential supremum and essential infimum always exist (http://planetmath.org/ExistenceOfTheEssentialSupremum). Furthermore, they are always equal to the supremum or infimum of some countable subset of .
|Date of creation||2013-03-22 12:21:29|
|Last modified on||2013-03-22 12:21:29|
|Last modified by||gel (22282)|