monotone class theorem
Monotone Class theoremFernando Sanz Gamiz
It is enough to prove that is an algebra, because an algebra which is a monotone class is obviously a -algebra.
Let . Then is clear that is a monotone class and, in fact, , for if , then since is a field, hence by minimality of ; consequently by definition of . But this shows that for any we have for any , so that and again by minimality . But what we have just proved is that is an algebra, for if we have showed that , and, of course, . ∎
One of the main applications of the Monotone Class Theorem is that of showing that certain property is satisfied by all sets in an -algebra, generally starting by the fact that the field generating the -algebra satisfies such property and that the sets that satisfies it constitutes a monotone class.
for any Borel sets and any finite . Using the Monotone Class Theorem one can show, for example, that any event in is independent of any event in . For, by independence
for every and any measurable rectangle . A second application of the theorem shows finally that the above relation holds for any and
|Title||monotone class theorem|
|Date of creation||2013-03-22 17:07:34|
|Last modified on||2013-03-22 17:07:34|
|Last modified by||fernsanz (8869)|