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help me clarify.

Replies: 3   Last Post: Jul 10, 2014 4:01 AM

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 Paul Posts: 780 Registered: 7/12/10
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 pi-system on a set omega as being a
>
> >collection P of subsets of omega such that P is non-empty
>
> >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
>
> >non-empty 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 pi-system. 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

Date Subject Author
7/9/14 Paul
7/9/14 quasi
7/9/14 William Elliot
7/10/14 Paul