uniqueness of measures extended from a -system
The following theorem allows measures to be uniquely defined by specifying their values on a -system (http://planetmath.org/PiSystem) instead of having to specify the measure of every possible measurable set. For example, the collection of open intervals forms a -system generating the Borel -algebra (http://planetmath.org/BorelSigmaAlgebra) and consequently the Lebesgue measure is uniquely defined by the equality .
Choose any such that and set . For any , and the requirement that agree on gives , so contains . We show that is a Dynkin system in order to apply Dynkin’s lemma. It is clear that . Suppose that are in . Then, the additivity of and gives
so and is a Dynkin system containing . By Dynkin’s lemma this shows that contains .
We have shown that for any and with . In the particular case where and are finite measures then it follows that simply by taking . More generally, choose a sequence of sets satisfying and . For any , is a pairwise disjoint sequence of sets in with and . So, and the countable additivity of and gives
|Title||uniqueness of measures extended from a -system|
|Date of creation||2013-03-22 18:33:08|
|Last modified on||2013-03-22 18:33:08|
|Last modified by||gel (22282)|