Talk:Σ-algebra: Difference between revisions

Content deleted Content added

Inline

Revision as of 20:58, 25 January 2006

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)

Here is another question. I know this phrasing is standard, but it is quite confusing to people new to sigma-algebras. Let A be a collection of subsets of X. We often say the following: "The sigma-algebra generated by A contains A". In fact, something like this is mentioned in this article. However, the sigma-algebra generated by A does not actually contain A...afterall, A is a collection of sets. Rather, the sigma-algebra generated by A contains all the elements of A. I know this must be obvious to many, but I found it very confusing when first encountering sigma-algebras...and I know that I was not alone.

In the examples section, it is said: "First note that there is a σ-algebra over X that contains U, namely the power set of X." Again, I think we should be perfectly clear. U is not a member of the power set of X. Rather, all memebers of U are members of the power set of X.

To make this abundantly clear, why not include very trivial examples of a sigma-algebra.

Let X = {1,2,3}

Let C = { {1}, {2} }

Then σ(C) = { {}, {1}, {2,3}, {2}, {1,3}, {1,2,3} }.

This is so very clear and obvious. Also notice, C is not a member of sigma-algebra. So we really should refrain from saying "C is in the sigma-algebra generated by C." It is sloppy even though it is standard.

I think you are confusing "containment" and "membership". A set A is said to contain a set B, iff B is a subset of A, that is iff every member of B is a member of A. So, since as you point out, every member of C is a member of σ(C), σ(C) contains C. Paul August ☎ 05:43, 12 October 2005 (UTC)[reply]

Well, that certainly would explain it! As you can see, I STILL am getting used to this. :-) Either way, it might be helpful, perhaps to people like myself, to see an example like that above. But your comment is correct and now understood by me. "containment" is NOT the same as "membership". Is there a wikipeida entry for "contains"? If so, we probably should link to it to avoid this confusion...as the mathematical usage of the word "contains" is very different from everday usage.

Name and history

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

Small sigma and delta are often used the union and intersetion are involved. They seems to be the Greek abbreviation of German words: Summe (sum) and Durchschnitt (intersection). Pura 00:10, 3 October 2005 (UTC)[reply]

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)[reply]

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)[reply]

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)[reply]

Boolean algebra

I was wondering, is a sigma-algebra also a boolean algebra. If so, should this be included in the definition? It seems that we are always using the axioms and results (demorgan) of boolean algebras. --anon

Well, any field of sets is a boolean algebra. So I guess this remark belongs in field of sets rather than the particular case of sigma-algebra. Oleg Alexandrov (talk) 21:46, 15 January 2006 (UTC)[reply]

Families?

"In mathematics, a σ-algebra ... over a set X is a family Σ of subsets of X that is closed under countable set operations..."

Is "family" meant here in the sense of family (mathematics), or is it just a loose way of saying "set"? If family (mathematics) is meant, then a link should be added. If set is meant, why not just say "set"? Dbtfz 06:04, 19 January 2006 (UTC)[reply]

Meaning of terms

I've just read through the current article (having had no knowledge of sigma algebras), and was confused by some terms. It is unclear whether my confusion arises from addressable weaknesses in the article or from lack of prerequisite knowledge on my part.

The problem terms were "countable set operations" (first para), and "(countable) sequence" and "(countable) union" (both in Property 3). I know what a countable set is, and what a set operation is, but the rest of the article leave me unable to guess at the combination.

My guess at the meanings is confounded by an example earlier in this talk page:

Let X = {1,2,3}

Let C = { {1}, {2} }

Then σ(C) = { {}, {1}, {2,3}, {2}, {1,3}, {1,2,3} }.

.. since I had assumed P3 would require {1} U {2} = {1,2} to be included (and hence also its complement {3}). Hv 20:26, 25 January 2006 (UTC)[reply]

As far as I can see, that example was written by a confused person and is wrong. For exactly the reasons you state. Now, as for your confusion... well, I'm not sure what you don't get. You can take the union of two sets, the union of three sets, the union of alef-0 sets, or the union of beth-2 sets. The first three are countable set operations (in fact countable unions), while the last is not. -lethe ^talk 20:33, 25 January 2006 (UTC)

Notice how in the definition, the word "countable" is in parentheses. This is because the notation E₁, E₂, … already implies a countable set of sets. The subscripts give a bijection to N. The union of a countable number of sets is a countable set operation. So is the intersection of a countable number of sets. This is what is meant by the phrase "countable set operation". -lethe ^talk 20:36, 25 January 2006 (UTC)

If the example was wrong that clears up most of my confusion (though I'm surprised nobody pointed it out at the time). I'd be interested to see an example of a subset that does not need to be included in the σ-algebra because it can only be generated as the union of an uncountable number of the subsets that are included. Hv 20:40, 25 January 2006 (UTC)[reply]

What is it you want to see an example of? If the set can only be generated by uncountable operations, then it does have to be explicitly included, since the axioms of a σ-algebra won't get you to uncountable unions. Unless you mean you want an example of a set that isn't in the algebra. I can surely give you an example. Let C be the set of all singleton subsets of R. Then σ(C) is the set of all countable sets of real numbers. Any uncountable set, for example (0,1), will not be in σ(C), even though it is generated by union of elements of C, because the union is uncountable. -lethe ^talk 20:48, 25 January 2006 (UTC)

@@ Line 117: / Line 117: @@
 ::Notice how in the definition, the word "countable" is in parentheses.  This is because the notation ''E''<sub>1</sub>, ''E''<sub>2</sub>, … already implies a countable set of sets.  The subscripts give a bijection to '''N'''.  The union of a countable number of sets is a countable set operation.  So is the intersection of a countable number of sets.  This is what is meant by the phrase "countable set operation". -[[User:Lethe/sig|lethe]] <sup>[[User talk:Lethe/sig|talk]]</sup> 20:36, 25 January 2006 (UTC)
 :::If the example was wrong that clears up most of my confusion (though I'm surprised nobody pointed it out at the time). I'd be interested to see an example of a subset that does not need to be included in the σ-algebra because it can only be generated as the union of an uncountable number of the subsets that ''are'' included. [[User:Hv|Hv]] 20:40, 25 January 2006 (UTC)
-What is it you want to see an example of?  If the set can ''only'' be generated by uncountable operations, then it ''does'' have to be explicitly included, since the axioms of a σ-algebra won't get you to uncountable unions.  Unless you mean you want an example of a set that isn't in the algebra.  I can surely give you an example.  Let ''C'' be the set of all [[singleton]]s in '''R'''.  Then σ(''C'') is the set of all countable sets of real numbers.  Any uncountable set, for example (0,1), will not be in σ(''C''), even though it is generated by union of elements of ''C'', because the union is uncountable.  -[[User:Lethe/sig|lethe]] <sup>[[User talk:Lethe/sig|talk]]</sup> 20:48, 25 January 2006 (UTC)
+What is it you want to see an example of?  If the set can ''only'' be generated by uncountable operations, then it ''does'' have to be explicitly included, since the axioms of a σ-algebra won't get you to uncountable unions.  Unless you mean you want an example of a set that isn't in the algebra.  I can surely give you an example.  Let ''C'' be the set of all [[singleton]] subsets of '''R'''.  Then σ(''C'') is the set of all countable sets of real numbers.  Any uncountable set, for example (0,1), will not be in σ(''C''), even though it is generated by union of elements of ''C'', because the union is uncountable.  -[[User:Lethe/sig|lethe]] <sup>[[User talk:Lethe/sig|talk]]</sup> 20:48, 25 January 2006 (UTC)