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I'm a long way from an expert on this stuff, but I thought that a Borel-field (generated from the family of open subsets of X) is a particular (and very common) example of a sigma-field, but that not all sigma-fields are Borel-fields. Can somebody confirm or deny this? If I am correct, then the article ought not suggest in the first sentence that they are equivalent. —Preceding unsigned comment added by 158.143.65.19 ( talk) 10:34, 5 October 2010 (UTC)
Is exist a infinite sigma algebra on an set X such that be countable?
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)
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)
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!
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.
Where does the name "sigma-algebra" come from? When were sigma-algebras introduced? -- Tobias Bergemann 13:39, 22 July 2005 (UTC)
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)
I find it somewhat distracting that the notation used in this article, that in sigma-ideal, and that in measurable 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:
This change will eliminate/replace the use of F, 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:
Looks like the conversion is complete, thanks to Vivacissamamente -- linas 15:06, 30 August 2005 (UTC)
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)
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
"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)
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:
.. 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)
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 and their complements. Any uncountable set with uncountable complement, 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)
Thanks Lethe, I think it is a bit clearer now. The only other thing that seems missing is a mention of what σ-algebras are useful for, and in particular what the restriction to countable operations buys us - does the restriction allow more powerful general theorems to be proved, or is it primarily to ensure that things like Borel algebras are well-defined? And is there another type of algebra defined identically except without the restriction to countable operations? Hv 12:06, 28 January 2006 (UTC)
I am confused by the third requirment of a sigma-algebra. Does this just say that the union of countably many member sets is also a member? In any case, this needs to be rewritten to make it more transparent on a first reading. -- Njerseyguy 17:13, 27 February 2006 (UTC)
The definition would be less confusing if the first requirement "Sigma is non-empty" were replaced with "X belongs to Sigma", which is the way I have always seen it done (Wolfram Mathworld does it this way, for example). The fact that X belongs to Sigma is fairly important and using the definition here, the fact that X belongs to Sigma is nto 100% obvious at first glance and must be proven.
Gsspradlin (
talk) 21:49, 6 November 2013 (UTC)
I just consulted Folland and, unfortunately for me, he uses "Sigma is non-empty" as one of his rules. I have a copy of Halsey+Royden's Real Analysis (it appears to be an "international edition" of a fourth edition) in which they include "Sigma contains the empty set" as one of the rules. Oddly, a little later, there is a redundant definition of "sigma-algebra of subsets of R (the real numbers)" that assumes instead that R belongs to Sigma. I think that replacing "Sigma is nonempty" with either "the empty set belongs to Sigma" or "X belongs to Sigma" would be an improvement, for the reasons I give above. Gsspradlin ( talk) 23:51, 6 November 2013 (UTC)
I completely agree with Gsspradlin. At first glance, it might be a good idea to take a minimal or general axiom which "just" says that Sigma contains some set, but actually this is not useful in practice. A similar criticism applies to the subgroup criterion (subgroups should not just assumed to be non-empty, but rather they should contain the neutral element). The requirement "Sigma is non-empty" is also problematic from the perspective of constructive mathematics and from the perspective of general algebra. Regarding sources, Rudin's book "Real and complex analysis" requires that X is contained in Sigma. In my humble opinion, it would be even more natural to require that both X and the empty set are contained in Sigma. In the article, the proof that the definition of a sigma-algebra implies that X is contained, should either be omitted or replaced by a remark that for proving the containment of X and the empty set in Sigma it is enough to prove that Sigma is non-empty. Also the proof on the intersection of sigma-algebras should be adjusted; notice that the current proof already uses X as the test set. -- Martin Brandenburg. -- 85.181.227.12 ( talk) 08:34, 7 April 2015 (UTC)
The first requirement includes the statement "...X is considered to be the universal set in the following context...". I think universal set (a set containing among other things itself according to Universal_set) should be replaced by something else, perhaps "set universe" as in the article Universe_(mathematics). Joel Sjögren ( talk) 22:49, 24 May 2016 (UTC)
I'm not going to make an edit myself, but that lead sentence with "countable set operations" (and others like it) causes problems for people that are not "in the know" (such as myself). Is this article for specialists? I don't think so, personally. Specialists will likely be reading textbooks (of course, if you're not a specialist, why the hell are you reading this!). Personally, I read plenty of textbooks in other areas and find that wikipedia is far better at getting me some basic information on specialized topics then going to, say, a measure theory textbook.
Anyway, we have these words: "that is closed under countable set operations".
They describe the fact that: "if I apply a countable number of set operations to an element of the set, then the result is also an element of the set"
It seems that something like this would remove the ambiguity: "that is closed under the application of a countable number of set operations". Likely, this would require breaking that sentence into two parts.
Regards, Mark
"The empty set is in Σ": is this really needed in the definition? Closure under complementation and countable unions implies that both the empty set and X are in Σ since:
A in Σ and A^c in Σ imply their union (which is X) is in Σ which implies that X^c (which is the empty set) is in Σ.
Indeed closure under complementation and finite unions is enough to prove that the empty set and X are in sigma. Pramana 19:59, 15 June 2006 (UTC)
I don't believe any authors allow for empty sigma algebras. Are you sure? Oleg Alexandrov ( talk) 03:03, 16 June 2006 (UTC)
Perhaps we can consider the following points: 1. Halmos and Royden dont use this axiom (stated above). 2. The main use of sigma algebras is in measure and integration for which we actually need a "rich" collection of subsets of X. I am currently unaware of any "deep" results because of including this axiom in the definition. I agree that this is not a major issue, but i thought definitions should be "lean" and Halmos and Royden are good enough for me. Pramana 06:09, 16 June 2006 (UTC)
I prefer the current version too. Pramana 12:48, 16 June 2006 (UTC). And if we decide to stick with the current version, we need to correct "from 2. and 3. it follows.......". Pramana 12:58, 16 June 2006 (UTC)
Surely the fact that Σ is closed under countable unions implies that the union of zero sets is in Σ, which in turn implies that the empty set must be in Σ. So there's no need for a separate axiom demanding that Σ is nonempty, it follows from 3. Bat020 14:53, 28 March 2007 (UTC)
I remarked elsewhere that it would be less confusing to replace the condition that the sigma-algebra be nonempty with the condition that the entire set belong to the sigma-algebra (arguing by authority, this is what Rudin does), or perhaps that the empty set belong to the sigma-algebra (as Doob and S. Lang do). The way it stands, it is not 100% obvious at first glance that the empty set and the whole set belong to the sigma-algebra, and it requires a little work to show it, which seems pointless when you could just make this one of the requirements. I am assuming that no one likes empty sigma-algebras (someone commented above that Royden does, but I have never heard of anyone else doing this). Gsspradlin ( talk) 22:04, 6 November 2013 (UTC)
@Bat020 : what I propose is not an extra assumption that the empty set or the whole set belong to Sigma, but replacing the requirement that Sigma be nonempty with either (i) the empty set belongs to Sigma or (ii) the whole set belongs to Sigma. They are all equivalent, of course, but I think it's probably important that the empty set and the whole set belong to Sigma, and in the current definition, those facts are not immediately obvious and need to be proven. Rudin requires that the whole set belong to Sigma, and Royden requires that the empty set belongs to Sigma. Unfortunately for me, I admit that Folland's definition uses "Sigma is nonempty". I have not seen a definition anywhere that allows Sigma itself to be empty, and I would guess that very few people are interested in this possibility (but with mathematicians, you cannot rule it out). — Preceding unsigned comment added by Gsspradlin ( talk • contribs) 23:59, 6 November 2013 (UTC)
The only reason I can imagine that mathematicians could want the empty set to be a σ-algebra would be to have a null σ-algebra. But {∅, X} already is the null σ-algebra as it contains zero information in terms of identifying proper subsets of X; X and ∅ being identifiable already by default. I also agree, especially for the typical reader, that there is little motivation for making non-empty part of the definition when saying X ∈ Σ is equivalent and more to the point. It also makes it quickly clear that {∅, X} is not only a σ-algebra but also the smallest σ-algebra. Daren Cline ( talk) 15:22, 8 April 2015 (UTC)
The examples states: "The collection of subsets of X which are countable or whose complements are countable is a σ-algebra, which is distinct from the powerset of X if and only if X is uncountable." If X is N, then this is false - there are subsets of N which are countable and whose complements is also countable. ( 67.102.227.19 19:50, 25 September 2006 (UTC))
Can someone go over and give a bit more context to Sigma homomorphism? It now links back here but my measure theory is way to rusty to give the proper context. -- Chrispounds 03:50, 9 October 2006 (UTC)
I'm confused by the statement that the power set of X is always a sigma-algebra. If the power set of the reals is the set of all subsets of reals, then I assume it contains the Vitali sets. I thought the aim was to avoid such sets because they're not measurable. LachlanA 03:41, 31 October 2006 (UTC)
In the section on generated sigma-algebras, one of the steps involves taking the intersection of multiple sigma algebras. Does this mean that the intersection of two sigma-algebras is always another sigma-algebra? What about countable and uncountable intersections? Calumny 17:38, 27 August 2007 (UTC)
Could one write something about sigma-fields? Here it states that those are somewhat interchangable... could this be clarified? —Preceding unsigned comment added by 83.6.96.216 ( talk) 17:07, 31 August 2008 (UTC)
The text currently asserts that
but nothing is said about what would constitute an "explicit" description. I would argue that, just taking the example of the Borel sets of the reals for concreteness, the following description is explicit: the Borel sets are the interpretations of the Borel codes, where a Borel code is a gadget of the sort described at infinity-Borel set, but where the ordinal height of the tree thereby generated is countable.
The burden is on those who would claim that there is no explicit description to say what is meant by that and prove that none exists. Therefore I am re-removing the text. -- Trovatore ( talk) 18:01, 20 November 2008 (UTC)
What differs Sigma-algebra from Sigma-ring? Only first axiom of whole set to be member of sigma-algebra or something more? —Preceding unsigned comment added by 212.87.13.75 ( talk) 02:58, 6 December 2008 (UTC) Ok, I see something: complement vs. relative complement. One could write something about differences and when one concept is used and when other (as with delta-ring and sigma-ring). 212.87.13.75 ( talk) 03:01, 6 December 2008 (UTC)
There should be a definition or at least a link to "sigma-ring" Gsspradlin ( talk) 21:55, 6 November 2013 (UTC)
Is it the case that any sigma-algebra over X can be understood as the power set of X' where X' is X with elements lumped together (ie, with some sort of equivalence relation)?
To put it another way, does any sigma-algebra over X define an equivalence relation over X, by virtue of it having lumps that are not split into subsets by any intersection operation in the algebra? —Preceding unsigned comment added by Paul Murray ( talk • contribs) 00:31, 13 January 2009 (UTC)
Why countable? Does the theory break in some way if uncountable unions are allowed? 207.241.239.70 ( talk) 04:02, 8 March 2009 (UTC)
No thanks to wikipedia or any other resources online, I only recently figured out how these connect to probability (ie that a probability space is a set, along with a sigma algebra and a probability measure). A particularly important (and probably obvious to most of the editors but not to a beginning undergrad looking through wikipedia's math articles) insight was that the probability spaces commonly encountered in undergrad probability have as a sigma algebra nothing more than the power set of the sample space. —Preceding unsigned comment added by 128.174.230.125 ( talk) 15:43, 22 May 2009 (UTC)
This page has had the "Too technical" template added to it. Since I saw no discussion on this page about the reasons for the template's addition, and since, in my opinion, the article is at about the right technical level for the likely or intended audience, I removed the template. But my edit was reverted, so I'm attempting to start a discussion here. In particular I'd like to know other editor's opinions on this article, the reasons why this article is thought too technical, and for suggestions for how this article might be made usefully less technical. Paul August ☎ 15:12, 20 January 2010 (UTC)
I've added a new section called "Motivation" (I did this before noticing any the above suggestions or remarks, and haven't thought yet how to roll any of these in.) Does this help at all? Paul August ☎ 17:04, 20 January 2010 (UTC)
I rearranged the introduction a little to put a slightly more intuitive part ahead of the formal definition. Feel free to revert if you think it's not agreeable. Ray Talk 18:07, 20 January 2010 (UTC)
Shouldn't there be some link to Algebra of sets at the beginning of the article since it belongs to this type of algebra? -- kupirijo ( talk) 12:00, 5 October 2010 (UTC)
Names with words should always refer to the same concept. Anytime notation and/or naming has been abused (which is to be avoided), it has to be made clear. As an example, in the paragraph that starts with "Elements of the σ-algebra are called measurable sets..." (section Sigma-algebra#Definition_and_properties), it is stated that "A function between two measurable spaces is called measurable function..." I guess that a measurable function f should be either f:X→X' or f:Σ→Σ', but not f:(X,Σ)→(X',Σ') as stated in the text. A related point: in section Talk:Sigma-algebra#Notation (X,Σ) is called the sigma-algebra, while in the main article it is Σ the sigma-algebra and (X,Σ) is the measurable space. Names and underlying concepts should be unified. —Preceding unsigned comment added by Sanchin s ( talk • contribs) 02:43, 7 December 2010 (UTC)
The explanation of measurable sets could do with a bit of clarification and wikification. For example, exactly which operations should we expect the privileged family of sets to be closed under. There should be links for every technical term, down to the level of "subset", including "operation" and "closure". —Preceding unsigned comment added by 74.176.113.247 ( talk) 23:20, 29 December 2010 (UTC)
I found the following to be vague in this section:
"If the subsets of X in Σ correspond to numbers in elementary algebra, then the two set operations union (symbol ∪) and intersection (∩) correspond to addition and multiplication. The collection of sets Σ is completed to include countably infinite operations."
Should I take it to mean that there is only on possible Homomorphism between a sigma-algebra and elementary algebra or instead is the intent to say that a sigma algebra is analogous to elementary algebra? Further, if it is analogous to elementary algebra then in it would be helpful to know in what ways is it analogous? S243a ( talk) 19:01, 24 October 2013 (UTC)
I don't see mentioned in the article, but I seem to recall that sigma algebras are both pi-systems and d-systems. If that's true, can we get that added to the article at an appropriate place? 70.247.162.18 ( talk) 00:02, 3 November 2013 (UTC)
There is a section entitled "Relation to sigma-ring", but there is no definition of what a sigma-ring is or even a link to a definition. I think the term "sigma-ring" is much less well-known than sigma-algebra, even to mathematicians, and there should be a link or definition, or the section should simply be omitted. Gsspradlin ( talk) 21:53, 6 November 2013 (UTC)
The Borel σ-algebra cannot be constructed in a very real sense, unless you want to think of it as a limit of ever-increasing collections of sets. To do that, you start with (for example) the open sets, then include all countable intersections of those sets, then the next step is to include all countable unions of the collection form the first step, then countable intersections from the second step, and so on. "ad infinitum" is a perfectly apt description of this process.
One common misconception about Borel sets is that they can all be represented in some fashion (for example, constructed in terms of unions of intervals). Unfortunately, this is nowhere close to the truth.
It is really the word "construction" that I object to. Perhaps there is another way to explain it without (at least at this point in the article) referring to "transfinite" operations. Daren Cline 19:06, 4 March 2014 (UTC) — Preceding unsigned comment added by Darencline ( talk • contribs)
I understand your notation, though analysts and probabilists have used other notation. For example, see Borel set, which would be an apt link here.
And I agree that "ad infinitum" may be a little glib. I was partly assuming that since the article is introductory at this point it just needs to be accurate, if still imprecise. (This isn't even what has provoked me most to think about contributing.) But I was also mis-remembering the theorem. Still, I don't see that is not closed under complements. I do see why it is not closed under countable unions.
Frankly, I tend to think that any discussion of a construction of a σ-algebra can lead to misconceptions. And so I like to just say to students that there is no representation for an arbitrary Borel set; we can only describe the collection in terms of classes of sets that generate the Borel σ-algebra. It also helps to bring home the point that proofs have to be done on the simpler classes and then "bootstrapped" to the Borel σ-algebra. (Which is why I also agree with the comment above that this article should refer to pi-system, lambda-system and the pi-lambda theorem, and their relevance to theory.)
My real concern here is accessibility of this article to graduate students (for example, in statistics) who know very little about analysis, let alone transfinite induction, but nevertheless do have to learn and understand σ-algebras and measure theory in order to do probability. Jumping from a finite example to constructing the Borel class tells them right away that this article isn't likely to help them much.
Indeed, the article is pretty much void of probability applications. So perhaps I can remedy that and then modify the introduction enough to show that aspect. I would also like to see more examples near the top of the article.Daren Cline 00:10, 5 March 2014 (UTC) — Preceding unsigned comment added by Darencline ( talk • contribs)
As mentioned above, the article currently gives the basics of the definition and few examples. There is some effort to discuss the use of σ-algebras for measure theory, but not much is said about probability. I would like to incorporate (or see incorporated) a number of points, including the following.
-- Daren Cline ( talk) 00:47, 6 March 2014 (UTC)
Thanks. I can use help with links to other pages. I don't intend for this to be lengthy; just to show more uses for σ-algebras. Daren Cline ( talk) 14:25, 6 March 2014 (UTC)
I added two concepts to the Motivations section. The first, limit of sets, is supposed to indicate one simple reason why closure under countable unions and intersections is necessary. However, in order to do this I felt it necessary to define set limits. Maybe that's straying a bit off topic, but it is also true that the definition is not usually known very well, and can be at odds with one's intuition based on limits of points. (Is there an article for this specific topic?)
The second, use of sub σ-algebras, is extremely important to probability and this just presents the idea which will be expanded on in a section specific to probability uses of σ-algebras. -- Daren Cline ( talk) 14:33, 12 March 2014 (UTC)
"A useful property is the following: for two functions f and g defined on X, σ(f) ⊂ σ(g) if and only if there exists a measurable function h such that f(x) = h(g(x))." — First, this is often called the Doob-Dynkin lemma. Second, I doubt: does it hold as written, or rather, modulo null sets? Moreover, working with σ-algebras on a probability space, I (and many others) always include all null sets in every σ-algebra (by assumption or by construction). Then we have a one-to-one correspondence between σ-algebras and operators of conditional expectation. Otherwise we have too many different σ-algebras. Boris Tsirelson ( talk) 07:20, 13 March 2014 (UTC)
No, really, I should not doubt: the claim holds as written. However, for now it is not clear which kind of functions are considered. For an arbitrary (Y,B) the claim definitely fails. For Rn with the Borel σ-algebra it holds. Boris Tsirelson ( talk) 08:46, 13 March 2014 (UTC)
In the examples section currently is the statement that the power set is also known as the discrete σ-algebra. I've been wondering about this as it was new to me. In fact, I think the σ-algebra generated by the discrete topology is something quite different: the collection of countable subsets and their complements. Does anyone have thoughts on this? Should the wording be changed? -- Daren Cline ( talk) 20:09, 28 March 2014 (UTC)
Yes, that's what I was thinking of: xn can only converge to x if there exists N such that xn = x for all n ≥ N, right? I guess I've forgotten the term for that – will have to look it up. (correction: that kind of convergence does imply singleton sets, and thus every set, is open.) --
Daren Cline (
talk) 16:53, 29 March 2014 (UTC)
Of course, but I would expect it to be assumed. -- Daren Cline ( talk) 16:53, 29 March 2014 (UTC)
I'm pretty sure that F() appearing in that integral is the probability density function (PDF), not the cumulative density function (CDF). (We integrate PDFs to get probabilities; we take differences of CDFs to get probabilities.) Could someone who *knows* the correct answer either correct the integral or add a note about how this unusual use of the CDF is correct? -- 216.137.30.166 ( talk) 00:59, 6 September 2015 (UTC)
@ 216.137.30.166: There is no such thing as a "cumulative DENSITY function". The words "cumulative" and "density" contradict each other. The letters "CDF" stand for "cumulative distribution function." Michael Hardy ( talk) 18:37, 23 February 2019 (UTC)
The consensus is to merge separable sigma algebra into sigma algebra. Cunard ( talk) 00:08, 1 August 2016 (UTC)
hi guys,
want to put out a feeler for the appetite to merge separable sigma algebra into sigma algebra. i feel the former could be slotted in under a subsection of the latter.
hopefully this is not too radical? the page for separable sigma algebra is quite bare and i think merging into sigma algebra will improve its exposure.
thoughts? 174.3.155.181 ( talk) 21:40, 29 June 2016 (UTC)
The contents of the separable sigma algebra page were merged into Σ-algebra on June 2016. For the contribution history and old versions of the redirected page, please see its history; for the discussion at that location, see its talk page. |
The last sentence in this section says "If the measure space is separable, it can be shown that the corresponding metric space is, too." But in the previous paragraph, "separability" of a measure space was defined to be separability of the corresponding metric space. Perhaps there's some other (equivalent) way of defining it, and this got lost when the separable sigma algebra page was merged into the sigma algebra page? — Preceding unsigned comment added by 73.114.17.117 ( talk) 15:45, 4 October 2016 (UTC)
It is stated that:
Shouldn't this π-system be defined using the union of instead? — Preceding unsigned comment added by Sopasakis p ( talk • contribs) 11:52, 13 August 2017 (UTC)
In "σ-algebra generated by an arbitrary family", it says the generated algebra is made from the elements of F via a countable number of complement, union, and intersection operations. But intersection can be defined in terms of union and complement: A intersection B = (A' union B')' via De Morgan's laws. Plus, the sigma algebra itself is defined as being closed under compliment and union, so why add intersection to this section? Vityou123 ( talk) 08:26, 14 September 2023 (UTC)
I am confident that the most recent edit (see the diff listing) was made with good intentions, but the grammar is faulty, and some topics, "such as" ... one topic that may be relevant here ... that is, the issue of
have already been discussed -- at length -- on this very "Talk:" page.
This section of this "Talk:" page is not intended to address the above mentioned (in a "bullet" item) "issue". Instead, this section is intended mainly to call attention to some grammatical errors in the first sentence of this article, after the (currently) most recent edit.
Also, the addition of "contain X itself" ... [if not reverted] should perhaps be moved to a separate sentence. This "addition" strikes me as somehow failing to be 'worded' correctly, in order to maintain consistency with the ['requirement'] items already in that list ... which are [consistently] adjectival phrases. (Would it be inconsistent only if we fixed "is" to say "that is"? That doesn't help, as leaving it saying "is" instead of "that is" ... would be incorrect.) I think it -- (the phrase "contain X itself") -- should be removed (that is, I think the addition of it should be reverted); but if it is not, then ... moving it to a separate sentence would afford an opportunity to mention the fact that this situation -- (the fact that Σ must "contain X itself") -- while TRUE, is already 'implied' (or "required") by some other rules in the definition.
Personally, I would be fine with just reverting the most recent edit ... the one which created the " Latest revision as of 17:31, 25 February 2024" version of the article.
However, I was hesitant to do so (without discussing it here first) partly because there might not have been enough space in the edit comment to express the ideas included here ... in order to make it clear that
Thanks for listening.
If there are no objections, then ... I intend to revert the most recent ["17:31, 25 February 2024"]
.
Mike Schwartz (
talk) 18:57, 27 February 2024 (UTC)
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersections.
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I'm a long way from an expert on this stuff, but I thought that a Borel-field (generated from the family of open subsets of X) is a particular (and very common) example of a sigma-field, but that not all sigma-fields are Borel-fields. Can somebody confirm or deny this? If I am correct, then the article ought not suggest in the first sentence that they are equivalent. —Preceding unsigned comment added by 158.143.65.19 ( talk) 10:34, 5 October 2010 (UTC)
Is exist a infinite sigma algebra on an set X such that be countable?
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)
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)
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!
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.
Where does the name "sigma-algebra" come from? When were sigma-algebras introduced? -- Tobias Bergemann 13:39, 22 July 2005 (UTC)
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)
I find it somewhat distracting that the notation used in this article, that in sigma-ideal, and that in measurable 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:
This change will eliminate/replace the use of F, 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:
Looks like the conversion is complete, thanks to Vivacissamamente -- linas 15:06, 30 August 2005 (UTC)
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)
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
"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)
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:
.. 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)
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 and their complements. Any uncountable set with uncountable complement, 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)
Thanks Lethe, I think it is a bit clearer now. The only other thing that seems missing is a mention of what σ-algebras are useful for, and in particular what the restriction to countable operations buys us - does the restriction allow more powerful general theorems to be proved, or is it primarily to ensure that things like Borel algebras are well-defined? And is there another type of algebra defined identically except without the restriction to countable operations? Hv 12:06, 28 January 2006 (UTC)
I am confused by the third requirment of a sigma-algebra. Does this just say that the union of countably many member sets is also a member? In any case, this needs to be rewritten to make it more transparent on a first reading. -- Njerseyguy 17:13, 27 February 2006 (UTC)
The definition would be less confusing if the first requirement "Sigma is non-empty" were replaced with "X belongs to Sigma", which is the way I have always seen it done (Wolfram Mathworld does it this way, for example). The fact that X belongs to Sigma is fairly important and using the definition here, the fact that X belongs to Sigma is nto 100% obvious at first glance and must be proven.
Gsspradlin (
talk) 21:49, 6 November 2013 (UTC)
I just consulted Folland and, unfortunately for me, he uses "Sigma is non-empty" as one of his rules. I have a copy of Halsey+Royden's Real Analysis (it appears to be an "international edition" of a fourth edition) in which they include "Sigma contains the empty set" as one of the rules. Oddly, a little later, there is a redundant definition of "sigma-algebra of subsets of R (the real numbers)" that assumes instead that R belongs to Sigma. I think that replacing "Sigma is nonempty" with either "the empty set belongs to Sigma" or "X belongs to Sigma" would be an improvement, for the reasons I give above. Gsspradlin ( talk) 23:51, 6 November 2013 (UTC)
I completely agree with Gsspradlin. At first glance, it might be a good idea to take a minimal or general axiom which "just" says that Sigma contains some set, but actually this is not useful in practice. A similar criticism applies to the subgroup criterion (subgroups should not just assumed to be non-empty, but rather they should contain the neutral element). The requirement "Sigma is non-empty" is also problematic from the perspective of constructive mathematics and from the perspective of general algebra. Regarding sources, Rudin's book "Real and complex analysis" requires that X is contained in Sigma. In my humble opinion, it would be even more natural to require that both X and the empty set are contained in Sigma. In the article, the proof that the definition of a sigma-algebra implies that X is contained, should either be omitted or replaced by a remark that for proving the containment of X and the empty set in Sigma it is enough to prove that Sigma is non-empty. Also the proof on the intersection of sigma-algebras should be adjusted; notice that the current proof already uses X as the test set. -- Martin Brandenburg. -- 85.181.227.12 ( talk) 08:34, 7 April 2015 (UTC)
The first requirement includes the statement "...X is considered to be the universal set in the following context...". I think universal set (a set containing among other things itself according to Universal_set) should be replaced by something else, perhaps "set universe" as in the article Universe_(mathematics). Joel Sjögren ( talk) 22:49, 24 May 2016 (UTC)
I'm not going to make an edit myself, but that lead sentence with "countable set operations" (and others like it) causes problems for people that are not "in the know" (such as myself). Is this article for specialists? I don't think so, personally. Specialists will likely be reading textbooks (of course, if you're not a specialist, why the hell are you reading this!). Personally, I read plenty of textbooks in other areas and find that wikipedia is far better at getting me some basic information on specialized topics then going to, say, a measure theory textbook.
Anyway, we have these words: "that is closed under countable set operations".
They describe the fact that: "if I apply a countable number of set operations to an element of the set, then the result is also an element of the set"
It seems that something like this would remove the ambiguity: "that is closed under the application of a countable number of set operations". Likely, this would require breaking that sentence into two parts.
Regards, Mark
"The empty set is in Σ": is this really needed in the definition? Closure under complementation and countable unions implies that both the empty set and X are in Σ since:
A in Σ and A^c in Σ imply their union (which is X) is in Σ which implies that X^c (which is the empty set) is in Σ.
Indeed closure under complementation and finite unions is enough to prove that the empty set and X are in sigma. Pramana 19:59, 15 June 2006 (UTC)
I don't believe any authors allow for empty sigma algebras. Are you sure? Oleg Alexandrov ( talk) 03:03, 16 June 2006 (UTC)
Perhaps we can consider the following points: 1. Halmos and Royden dont use this axiom (stated above). 2. The main use of sigma algebras is in measure and integration for which we actually need a "rich" collection of subsets of X. I am currently unaware of any "deep" results because of including this axiom in the definition. I agree that this is not a major issue, but i thought definitions should be "lean" and Halmos and Royden are good enough for me. Pramana 06:09, 16 June 2006 (UTC)
I prefer the current version too. Pramana 12:48, 16 June 2006 (UTC). And if we decide to stick with the current version, we need to correct "from 2. and 3. it follows.......". Pramana 12:58, 16 June 2006 (UTC)
Surely the fact that Σ is closed under countable unions implies that the union of zero sets is in Σ, which in turn implies that the empty set must be in Σ. So there's no need for a separate axiom demanding that Σ is nonempty, it follows from 3. Bat020 14:53, 28 March 2007 (UTC)
I remarked elsewhere that it would be less confusing to replace the condition that the sigma-algebra be nonempty with the condition that the entire set belong to the sigma-algebra (arguing by authority, this is what Rudin does), or perhaps that the empty set belong to the sigma-algebra (as Doob and S. Lang do). The way it stands, it is not 100% obvious at first glance that the empty set and the whole set belong to the sigma-algebra, and it requires a little work to show it, which seems pointless when you could just make this one of the requirements. I am assuming that no one likes empty sigma-algebras (someone commented above that Royden does, but I have never heard of anyone else doing this). Gsspradlin ( talk) 22:04, 6 November 2013 (UTC)
@Bat020 : what I propose is not an extra assumption that the empty set or the whole set belong to Sigma, but replacing the requirement that Sigma be nonempty with either (i) the empty set belongs to Sigma or (ii) the whole set belongs to Sigma. They are all equivalent, of course, but I think it's probably important that the empty set and the whole set belong to Sigma, and in the current definition, those facts are not immediately obvious and need to be proven. Rudin requires that the whole set belong to Sigma, and Royden requires that the empty set belongs to Sigma. Unfortunately for me, I admit that Folland's definition uses "Sigma is nonempty". I have not seen a definition anywhere that allows Sigma itself to be empty, and I would guess that very few people are interested in this possibility (but with mathematicians, you cannot rule it out). — Preceding unsigned comment added by Gsspradlin ( talk • contribs) 23:59, 6 November 2013 (UTC)
The only reason I can imagine that mathematicians could want the empty set to be a σ-algebra would be to have a null σ-algebra. But {∅, X} already is the null σ-algebra as it contains zero information in terms of identifying proper subsets of X; X and ∅ being identifiable already by default. I also agree, especially for the typical reader, that there is little motivation for making non-empty part of the definition when saying X ∈ Σ is equivalent and more to the point. It also makes it quickly clear that {∅, X} is not only a σ-algebra but also the smallest σ-algebra. Daren Cline ( talk) 15:22, 8 April 2015 (UTC)
The examples states: "The collection of subsets of X which are countable or whose complements are countable is a σ-algebra, which is distinct from the powerset of X if and only if X is uncountable." If X is N, then this is false - there are subsets of N which are countable and whose complements is also countable. ( 67.102.227.19 19:50, 25 September 2006 (UTC))
Can someone go over and give a bit more context to Sigma homomorphism? It now links back here but my measure theory is way to rusty to give the proper context. -- Chrispounds 03:50, 9 October 2006 (UTC)
I'm confused by the statement that the power set of X is always a sigma-algebra. If the power set of the reals is the set of all subsets of reals, then I assume it contains the Vitali sets. I thought the aim was to avoid such sets because they're not measurable. LachlanA 03:41, 31 October 2006 (UTC)
In the section on generated sigma-algebras, one of the steps involves taking the intersection of multiple sigma algebras. Does this mean that the intersection of two sigma-algebras is always another sigma-algebra? What about countable and uncountable intersections? Calumny 17:38, 27 August 2007 (UTC)
Could one write something about sigma-fields? Here it states that those are somewhat interchangable... could this be clarified? —Preceding unsigned comment added by 83.6.96.216 ( talk) 17:07, 31 August 2008 (UTC)
The text currently asserts that
but nothing is said about what would constitute an "explicit" description. I would argue that, just taking the example of the Borel sets of the reals for concreteness, the following description is explicit: the Borel sets are the interpretations of the Borel codes, where a Borel code is a gadget of the sort described at infinity-Borel set, but where the ordinal height of the tree thereby generated is countable.
The burden is on those who would claim that there is no explicit description to say what is meant by that and prove that none exists. Therefore I am re-removing the text. -- Trovatore ( talk) 18:01, 20 November 2008 (UTC)
What differs Sigma-algebra from Sigma-ring? Only first axiom of whole set to be member of sigma-algebra or something more? —Preceding unsigned comment added by 212.87.13.75 ( talk) 02:58, 6 December 2008 (UTC) Ok, I see something: complement vs. relative complement. One could write something about differences and when one concept is used and when other (as with delta-ring and sigma-ring). 212.87.13.75 ( talk) 03:01, 6 December 2008 (UTC)
There should be a definition or at least a link to "sigma-ring" Gsspradlin ( talk) 21:55, 6 November 2013 (UTC)
Is it the case that any sigma-algebra over X can be understood as the power set of X' where X' is X with elements lumped together (ie, with some sort of equivalence relation)?
To put it another way, does any sigma-algebra over X define an equivalence relation over X, by virtue of it having lumps that are not split into subsets by any intersection operation in the algebra? —Preceding unsigned comment added by Paul Murray ( talk • contribs) 00:31, 13 January 2009 (UTC)
Why countable? Does the theory break in some way if uncountable unions are allowed? 207.241.239.70 ( talk) 04:02, 8 March 2009 (UTC)
No thanks to wikipedia or any other resources online, I only recently figured out how these connect to probability (ie that a probability space is a set, along with a sigma algebra and a probability measure). A particularly important (and probably obvious to most of the editors but not to a beginning undergrad looking through wikipedia's math articles) insight was that the probability spaces commonly encountered in undergrad probability have as a sigma algebra nothing more than the power set of the sample space. —Preceding unsigned comment added by 128.174.230.125 ( talk) 15:43, 22 May 2009 (UTC)
This page has had the "Too technical" template added to it. Since I saw no discussion on this page about the reasons for the template's addition, and since, in my opinion, the article is at about the right technical level for the likely or intended audience, I removed the template. But my edit was reverted, so I'm attempting to start a discussion here. In particular I'd like to know other editor's opinions on this article, the reasons why this article is thought too technical, and for suggestions for how this article might be made usefully less technical. Paul August ☎ 15:12, 20 January 2010 (UTC)
I've added a new section called "Motivation" (I did this before noticing any the above suggestions or remarks, and haven't thought yet how to roll any of these in.) Does this help at all? Paul August ☎ 17:04, 20 January 2010 (UTC)
I rearranged the introduction a little to put a slightly more intuitive part ahead of the formal definition. Feel free to revert if you think it's not agreeable. Ray Talk 18:07, 20 January 2010 (UTC)
Shouldn't there be some link to Algebra of sets at the beginning of the article since it belongs to this type of algebra? -- kupirijo ( talk) 12:00, 5 October 2010 (UTC)
Names with words should always refer to the same concept. Anytime notation and/or naming has been abused (which is to be avoided), it has to be made clear. As an example, in the paragraph that starts with "Elements of the σ-algebra are called measurable sets..." (section Sigma-algebra#Definition_and_properties), it is stated that "A function between two measurable spaces is called measurable function..." I guess that a measurable function f should be either f:X→X' or f:Σ→Σ', but not f:(X,Σ)→(X',Σ') as stated in the text. A related point: in section Talk:Sigma-algebra#Notation (X,Σ) is called the sigma-algebra, while in the main article it is Σ the sigma-algebra and (X,Σ) is the measurable space. Names and underlying concepts should be unified. —Preceding unsigned comment added by Sanchin s ( talk • contribs) 02:43, 7 December 2010 (UTC)
The explanation of measurable sets could do with a bit of clarification and wikification. For example, exactly which operations should we expect the privileged family of sets to be closed under. There should be links for every technical term, down to the level of "subset", including "operation" and "closure". —Preceding unsigned comment added by 74.176.113.247 ( talk) 23:20, 29 December 2010 (UTC)
I found the following to be vague in this section:
"If the subsets of X in Σ correspond to numbers in elementary algebra, then the two set operations union (symbol ∪) and intersection (∩) correspond to addition and multiplication. The collection of sets Σ is completed to include countably infinite operations."
Should I take it to mean that there is only on possible Homomorphism between a sigma-algebra and elementary algebra or instead is the intent to say that a sigma algebra is analogous to elementary algebra? Further, if it is analogous to elementary algebra then in it would be helpful to know in what ways is it analogous? S243a ( talk) 19:01, 24 October 2013 (UTC)
I don't see mentioned in the article, but I seem to recall that sigma algebras are both pi-systems and d-systems. If that's true, can we get that added to the article at an appropriate place? 70.247.162.18 ( talk) 00:02, 3 November 2013 (UTC)
There is a section entitled "Relation to sigma-ring", but there is no definition of what a sigma-ring is or even a link to a definition. I think the term "sigma-ring" is much less well-known than sigma-algebra, even to mathematicians, and there should be a link or definition, or the section should simply be omitted. Gsspradlin ( talk) 21:53, 6 November 2013 (UTC)
The Borel σ-algebra cannot be constructed in a very real sense, unless you want to think of it as a limit of ever-increasing collections of sets. To do that, you start with (for example) the open sets, then include all countable intersections of those sets, then the next step is to include all countable unions of the collection form the first step, then countable intersections from the second step, and so on. "ad infinitum" is a perfectly apt description of this process.
One common misconception about Borel sets is that they can all be represented in some fashion (for example, constructed in terms of unions of intervals). Unfortunately, this is nowhere close to the truth.
It is really the word "construction" that I object to. Perhaps there is another way to explain it without (at least at this point in the article) referring to "transfinite" operations. Daren Cline 19:06, 4 March 2014 (UTC) — Preceding unsigned comment added by Darencline ( talk • contribs)
I understand your notation, though analysts and probabilists have used other notation. For example, see Borel set, which would be an apt link here.
And I agree that "ad infinitum" may be a little glib. I was partly assuming that since the article is introductory at this point it just needs to be accurate, if still imprecise. (This isn't even what has provoked me most to think about contributing.) But I was also mis-remembering the theorem. Still, I don't see that is not closed under complements. I do see why it is not closed under countable unions.
Frankly, I tend to think that any discussion of a construction of a σ-algebra can lead to misconceptions. And so I like to just say to students that there is no representation for an arbitrary Borel set; we can only describe the collection in terms of classes of sets that generate the Borel σ-algebra. It also helps to bring home the point that proofs have to be done on the simpler classes and then "bootstrapped" to the Borel σ-algebra. (Which is why I also agree with the comment above that this article should refer to pi-system, lambda-system and the pi-lambda theorem, and their relevance to theory.)
My real concern here is accessibility of this article to graduate students (for example, in statistics) who know very little about analysis, let alone transfinite induction, but nevertheless do have to learn and understand σ-algebras and measure theory in order to do probability. Jumping from a finite example to constructing the Borel class tells them right away that this article isn't likely to help them much.
Indeed, the article is pretty much void of probability applications. So perhaps I can remedy that and then modify the introduction enough to show that aspect. I would also like to see more examples near the top of the article.Daren Cline 00:10, 5 March 2014 (UTC) — Preceding unsigned comment added by Darencline ( talk • contribs)
As mentioned above, the article currently gives the basics of the definition and few examples. There is some effort to discuss the use of σ-algebras for measure theory, but not much is said about probability. I would like to incorporate (or see incorporated) a number of points, including the following.
-- Daren Cline ( talk) 00:47, 6 March 2014 (UTC)
Thanks. I can use help with links to other pages. I don't intend for this to be lengthy; just to show more uses for σ-algebras. Daren Cline ( talk) 14:25, 6 March 2014 (UTC)
I added two concepts to the Motivations section. The first, limit of sets, is supposed to indicate one simple reason why closure under countable unions and intersections is necessary. However, in order to do this I felt it necessary to define set limits. Maybe that's straying a bit off topic, but it is also true that the definition is not usually known very well, and can be at odds with one's intuition based on limits of points. (Is there an article for this specific topic?)
The second, use of sub σ-algebras, is extremely important to probability and this just presents the idea which will be expanded on in a section specific to probability uses of σ-algebras. -- Daren Cline ( talk) 14:33, 12 March 2014 (UTC)
"A useful property is the following: for two functions f and g defined on X, σ(f) ⊂ σ(g) if and only if there exists a measurable function h such that f(x) = h(g(x))." — First, this is often called the Doob-Dynkin lemma. Second, I doubt: does it hold as written, or rather, modulo null sets? Moreover, working with σ-algebras on a probability space, I (and many others) always include all null sets in every σ-algebra (by assumption or by construction). Then we have a one-to-one correspondence between σ-algebras and operators of conditional expectation. Otherwise we have too many different σ-algebras. Boris Tsirelson ( talk) 07:20, 13 March 2014 (UTC)
No, really, I should not doubt: the claim holds as written. However, for now it is not clear which kind of functions are considered. For an arbitrary (Y,B) the claim definitely fails. For Rn with the Borel σ-algebra it holds. Boris Tsirelson ( talk) 08:46, 13 March 2014 (UTC)
In the examples section currently is the statement that the power set is also known as the discrete σ-algebra. I've been wondering about this as it was new to me. In fact, I think the σ-algebra generated by the discrete topology is something quite different: the collection of countable subsets and their complements. Does anyone have thoughts on this? Should the wording be changed? -- Daren Cline ( talk) 20:09, 28 March 2014 (UTC)
Yes, that's what I was thinking of: xn can only converge to x if there exists N such that xn = x for all n ≥ N, right? I guess I've forgotten the term for that – will have to look it up. (correction: that kind of convergence does imply singleton sets, and thus every set, is open.) --
Daren Cline (
talk) 16:53, 29 March 2014 (UTC)
Of course, but I would expect it to be assumed. -- Daren Cline ( talk) 16:53, 29 March 2014 (UTC)
I'm pretty sure that F() appearing in that integral is the probability density function (PDF), not the cumulative density function (CDF). (We integrate PDFs to get probabilities; we take differences of CDFs to get probabilities.) Could someone who *knows* the correct answer either correct the integral or add a note about how this unusual use of the CDF is correct? -- 216.137.30.166 ( talk) 00:59, 6 September 2015 (UTC)
@ 216.137.30.166: There is no such thing as a "cumulative DENSITY function". The words "cumulative" and "density" contradict each other. The letters "CDF" stand for "cumulative distribution function." Michael Hardy ( talk) 18:37, 23 February 2019 (UTC)
The consensus is to merge separable sigma algebra into sigma algebra. Cunard ( talk) 00:08, 1 August 2016 (UTC)
hi guys,
want to put out a feeler for the appetite to merge separable sigma algebra into sigma algebra. i feel the former could be slotted in under a subsection of the latter.
hopefully this is not too radical? the page for separable sigma algebra is quite bare and i think merging into sigma algebra will improve its exposure.
thoughts? 174.3.155.181 ( talk) 21:40, 29 June 2016 (UTC)
The contents of the separable sigma algebra page were merged into Σ-algebra on June 2016. For the contribution history and old versions of the redirected page, please see its history; for the discussion at that location, see its talk page. |
The last sentence in this section says "If the measure space is separable, it can be shown that the corresponding metric space is, too." But in the previous paragraph, "separability" of a measure space was defined to be separability of the corresponding metric space. Perhaps there's some other (equivalent) way of defining it, and this got lost when the separable sigma algebra page was merged into the sigma algebra page? — Preceding unsigned comment added by 73.114.17.117 ( talk) 15:45, 4 October 2016 (UTC)
It is stated that:
Shouldn't this π-system be defined using the union of instead? — Preceding unsigned comment added by Sopasakis p ( talk • contribs) 11:52, 13 August 2017 (UTC)
In "σ-algebra generated by an arbitrary family", it says the generated algebra is made from the elements of F via a countable number of complement, union, and intersection operations. But intersection can be defined in terms of union and complement: A intersection B = (A' union B')' via De Morgan's laws. Plus, the sigma algebra itself is defined as being closed under compliment and union, so why add intersection to this section? Vityou123 ( talk) 08:26, 14 September 2023 (UTC)
I am confident that the most recent edit (see the diff listing) was made with good intentions, but the grammar is faulty, and some topics, "such as" ... one topic that may be relevant here ... that is, the issue of
have already been discussed -- at length -- on this very "Talk:" page.
This section of this "Talk:" page is not intended to address the above mentioned (in a "bullet" item) "issue". Instead, this section is intended mainly to call attention to some grammatical errors in the first sentence of this article, after the (currently) most recent edit.
Also, the addition of "contain X itself" ... [if not reverted] should perhaps be moved to a separate sentence. This "addition" strikes me as somehow failing to be 'worded' correctly, in order to maintain consistency with the ['requirement'] items already in that list ... which are [consistently] adjectival phrases. (Would it be inconsistent only if we fixed "is" to say "that is"? That doesn't help, as leaving it saying "is" instead of "that is" ... would be incorrect.) I think it -- (the phrase "contain X itself") -- should be removed (that is, I think the addition of it should be reverted); but if it is not, then ... moving it to a separate sentence would afford an opportunity to mention the fact that this situation -- (the fact that Σ must "contain X itself") -- while TRUE, is already 'implied' (or "required") by some other rules in the definition.
Personally, I would be fine with just reverting the most recent edit ... the one which created the " Latest revision as of 17:31, 25 February 2024" version of the article.
However, I was hesitant to do so (without discussing it here first) partly because there might not have been enough space in the edit comment to express the ideas included here ... in order to make it clear that
Thanks for listening.
If there are no objections, then ... I intend to revert the most recent ["17:31, 25 February 2024"]
.
Mike Schwartz (
talk) 18:57, 27 February 2024 (UTC)
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersections.