Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math.independent

Topic: Wikipedia definition of pi system is self-contradictory -- please
help me clarify.

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Paul

Posts: 407
Registered: 7/12/10
Re: Wikipedia definition of pi system is self-contradictory -- please
help me clarify.

Posted: Jul 10, 2014 4:01 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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



Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.