Dynkin’s lemma is a result in measure theory showing that the -algebra (http://planetmath.org/SigmaAlgebra) generated by any given -system (http://planetmath.org/PiSystem) on a set coincides with the Dynkin system generated the -system. The result can be used to prove that measures are uniquely determined by their values on -systems generating the required -algebra. For example, the Borel -algebra on is generated by the -system of open intervals for and consequently the Lebesgue measure is uniquely determined by the property that .
Note that this lemma generalizes the statement that a Dynkin system which is also a -system is a -algebra.
Let be a -system on a set . Then . That is, the smallest Dynkin system containing coincides with the -algebra generated by .
As is a -system, the set contains . We show that is also a Dynkin system.
First, for every , so is in . Second, if are in and then is in showing that . Finally. if is a sequence increasing to and then is a sequence in increasing to . As Dynkin systems are closed under limits of increasing sequences this shows that and therefore . So is indeed a Dynkin system. In particular, and for all and .
We now set which, as shown above, contains . Also, as in the argument above for , is a Dynkin system. Therefore, is contained in and it follows that for any . So is both a -system and a Dynkin system.
We can now show that is a -algebra. As it is a Dynkin system, for every and, as it is also a -system, this shows that is an algebra of sets on . Finally, choose any sequence . Then, is a sequence in increasing to which, as is Dynkin system, must be in . So, is a -algebra and must contain . Conversely, as is a Dynkin system (as it is a -algebra) containing , it must also contain . ∎
- 1 David Williams, Probability with martingales, Cambridge Mathematical Textbooks, Cambridge University Press, 1991.
|Date of creation||2013-03-22 18:33:05|
|Last modified on||2013-03-22 18:33:05|
|Last modified by||gel (22282)|
|Synonym||pi-system d-system lemma|