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Paul
Posts:
763
Registered:
7/12/10


Re: Wikipedia definition of pi system is selfcontradictory  please help me clarify.
Posted:
Jul 10, 2014 4:01 AM


On Thursday, July 10, 2014 1:23:36 AM UTC+1, quasi wrote: > Paul wrote: > > > > > >The wikipedia defines a pisystem on a set omega as being a > > >collection P of subsets of omega such that P is nonempty > > >and such that A intersect B belongs to P whenever A and B > > >belong to P. > > > > > >However, the article goes on to give another definition that > > >contradicts the previous one  namely "That is, P is a > > >nonempty family of subsets of omega that is closed under > > >finite intersections." > > > > > >These are different definitions. > > > > In this context, the phrase "closed under finite intersections" > > was clearly intended to mean "the intersection of any _nonempty_ > > finite collection of subsets of P belongs to P". > > > > Modulo the implicitly assumed "nonempty", the definitions are > > equivalent. > > > > >Does anyone know whether omega is required to belong to P > > > > No  the first definition makes that perfectly clear. > > > > quasi
Thanks, quasi. I agree with you. The answer to my question could also have been resolved by looking at some of the wikipedia examples, some of which exclude omega.
The reason I was initially confused was that I've been reading some online notes which give an extension lemma about a pisystem. I was unable to understand the proof, and was searching for reasons why. Now, I understand the problem. The author of the online notes (I don't want to publicise his error in a public forum) erroneously left out the hypothesis that mu(omega) = v(omega) where mu and v are the measures that are shown to be equivalent. His result is thus false as stated. (This is just a simple typo by the author, but enough to confuse me as a beginner).
Thank you,
Paul Epstein



