Talk:Σ-algebra

Is exist a infinite sigma algebra on an set X such that be countable?

No, any sigma-algebra is either finite or uncountable. Prumpf 13:14, 12 Oct 2004 (UTC)

The opening remarks suggest that a sigma algebra satisfies the field axioms - is this true? If so what are the '+' and 'x' operations etc.? --SgtThroat 13:08, 10 Nov 2004 (UTC)

I don't think so. The natural operations are union and intersection, and the identities are trivial, but you hit a problem with the inverses' properties of fields. --Henrygb 01:20, 21 Jan 2005 (UTC)

Font used for denotion in math papers is not important?

The following sentence was deleted: "σ-algebras are sometimes denoted using capital letters of the Fraktur typeface".

Yes, this typeface is not used in this article, but reading math papers I found, that they are usually denoted using it. I did not know, how it is called and how should these letters be read and hoped to find this out in this article, but failed. I found the name of the typeface in other place and I thought this note will be helpful for other people. But it's considered not important...

BTW, no note, that similar constructions which are closed under finite set operations are usually called algebras (this term obviously appeared before σ-algebras). The article does not contain anything more than a definition copied from MathWorld and trivial examples. But other trivial info is irrelevant here... Cmapm 01:07, 3 Jun 2005 (UTC)

Those examples are not trivial. Some of those examples are simple, and such serve to illustrate the concent. Some other examples listed there are actuallly very important, so not trivial either.

The information you inserted is not trivial either. I said it was "not valuable". Please feel free to put it back. It is just when I read the article as a whole, I found that minor point about the font distracting from the overall concent. But there is of course room for disagreement. Oleg Alexandrov 01:43, 3 Jun 2005 (UTC)

Also, you are more than welcome to add content to this article if you feel it is incomplete. Oleg Alexandrov 01:44, 3 Jun 2005 (UTC)

question

For definition 2 of a sigma algebra, it says that for a sigma-algebra X, if E is in X, then so is the complement of E. Does this mean the complement of E in S (i.e. S-E)? Or the complement of E with some universal set?

Thanks!

It means the former. I will now try to make it more explicit in the article. Oleg Alexandrov 02:53, 23 Jun 2005 (UTC)

Name and history

Where does the name "sigma-algebra" come from? When were sigma-algebras introduced? -- Tobias Bergemann 13:39, 22 July 2005 (UTC)

Notation

I find it somewhat distracting that the notation used in this article, that in sigma-ideal, and that in measureable function, are not in concordance. I'm also unnerved that the usage of X and S in this article is reversed from the common usage I am used to seeing. That is, I'm used to seeing X be the set, and Σ be the collection of subsets, so that (X, Σ) is the sigma-algebra. Sometimes, S is used in place of Σ. Can I flip the notation used here, or will this offend sensibilities?

Also, do we have any article that defines the notation (X, Σ, μ) as a measure space? (I needed a wikilink for this in dynamical system but didn't find one). linas 13:37, 25 August 2005 (UTC)

Actually, I want to harmonize the notation in all three articles. But before doing so, we should agree on a common notation. I propose:

• X be the set
• Σ be the collection of subsets
• E and E_n are the elements of Σ (and leave alone A_n for those articles that already use that).
• μ is the measure.
• (X, Σ) and (Y, T) are the sigma algebras,
• (X, Σ, μ) and (Y, T, ν) are measure spaces.

This change will eliminate/replace the use of F, ${\mathcal {A}}$ and Ω in these three articles. Ugh. Measure (mathematics) is not even self-consistent, switching notation half-way through. linas 13:50, 25 August 2005 (UTC)

Other articles includde:

• Measure-preserving dynamical system

Agree that notation should be harmonised if possible. Your choices are good, although I would perhaps use X' and Σ' instead of Y and T. I guess then μ' is slightly problematic. Whatever :-) Dmharvey File:User dmharvey sig.png Talk 14:38, 26 August 2005 (UTC)

Agree with Linas and Dmharvey's remarks. Linas, doing all these changes will require very careful reading of all the articles and very patient changes. If you have the time, go for it. :) Oleg Alexandrov 15:29, 26 August 2005 (UTC)

Well FWIW, I've just added sigma additivity (copied over from PlantetMath: "additive" using Oleg's new conversion tool ;-) It uses ${\mathcal {A}}$, which I must say I rather like. But I agree with making things consistent and what Linas has proposed would be much better than what we now have. My least favorite part of the suggestion is the T, why not use Tau: Τ ? Paul August ☎ 18:39, August 26, 2005 (UTC)

I've gone and changed the notation in the articles to the proposed standard set forth here, excepting that Τ is used rather than T. For the record, the (Ω,F,P) notation is a standard hailing from probability theory, but its place wasn't in the measure theory articles. I've left the ${\mathcal {A}}$ in sigma additivity because there's a theoretical possibility the concept could come of use in some context other than measure theory, and besides, Paul likes it. Vivacissamamente 04:40, 30 August 2005 (UTC)

By the way, there is a discussion going on at talk:sigma additivity about whether it should be merged into measure (mathematics). I'd appreciate it if others would share their thoughts. Paul August ☎ 15:38, August 28, 2005 (UTC)

Looks like the conversion is complete, thanks to Vivacissamamente -- linas 15:06, 30 August 2005 (UTC)

I agree that there is value in notational consistency. In probability theory, it is commonplace to use Ω for the underlying space, and F or some variant of that letter for the set of all measureable sets. So I'm not so sure notational consistency should cross subject-matter boundaries. Michael Hardy 23:19, 30 August 2005 (UTC)

Except that this subject inherently crosses the boundaries. I don't know what the probability theory notation is, but a sentance should be addded to this article stating that in probability theory, the notation Ω, F is used in place of (X, Σ) but otherwise has the same meaning (or not). linas 06:16, 1 September 2005 (UTC)

Relation to field of sets

Anyone care to wikilink field of sets much earlier in the article, and expound on the difference between that and this? (The difference being that here, the number of intersections & unions is countable)? linas 15:06, 30 August 2005 (UTC)