proof of existence of the essential supremum
Suppose that is a -finite measure space and is a collection of measurable functions . We show that the essential supremum of exists and furthermore, if it is nonempty then there is a sequence such that
As any -finite measure is equivalent to a probability measure (http://planetmath.org/AnySigmaFiniteMeasureIsEquivalentToAProbabilityMeasure), we may suppose without loss of generality that is a probability measure. Also, without loss of generality, suppose that is nonempty, and let consist of the collection of maximums of finite sequences of functions in . Then choose any continuous and strictly increasing . For example, we can take
As is a bounded and measurable function for all , we can set
Then choose a sequence in such that . By replacing by the maximum of if necessary, we may assume that for each . Set
Also, every is the maximum of a finite sequence of functions in . Therefore, there exists a sequence such that
Similarly, for any ,
It follows that is a nonnegative function with nonpositive integral, and so is equal to zero -almost everywhere. As is strictly increasing, and therefore -almost everywhere.
Finally, suppose that satisfies (-a.e.) for all . Then, and,
-a.e., as required.
|Title||proof of existence of the essential supremum|
|Date of creation||2013-03-22 18:39:25|
|Last modified on||2013-03-22 18:39:25|
|Last modified by||gel (22282)|