Talk:Measure (mathematics)

Edit history was combined with that of Mathematical measure on 2003 Mar 11; the move that broke the history was made on 2002 Oct 26.

Notation

This article appears to switch notation about half-way through, from X to S; please see discussion at Talk:Sigma-algebra#Notation. linas 14:11, 25 August 2005 (UTC)

Non-measureablity

The discussion pages on measure theory -- site to site -- has formed a loop with no information!

Hey, everyone, let's post an example of a non-measurabe set! (unless I missed where it was discussed on Wikipedia...)

Sorry, forgot to sign in -- MathStatWoman anonymous post on 22 dec 2005

See Banach-Tarski paradox, and also Smith-Volterra-Cantor set Vitali set and Non-measurable set; I'll add these to the article. -- linas 21:19, 22 December 2005 (UTC)

Non-measurable sets?

I think the non-measureable set term in the section of counterexamples should be emphasize to Lebesgue non-measurable sets ... e.g. Vitali set is not Lebesgue measurable because two properties of translational invariance and countably additivity are inconsistency. The Vitali set may be measurable for other measures that does not requite both two properties .... Jung dalglish 19:12, 3 February 2006 (UTC)

I'm not sure that would be correct. "Measurable space" (cf "measure space" which is the triple) is a term referring to a couple (X,Sigma) and thus not referring a particular measure (or any measure at all). In the same way, a measurable set is just a set member of some measurable space (element of the sigma algebra). Thus, "measurable" refers to the way the set is constructed, not the way it is measured. This is because the way it is constructed avoids inconsistency problems later on of course when applying a measure. I think the introductory text is good at pointing that out but the definition makes it as though measurable set is a member of Sigma in a measure space, when you only need a measurable space to refer to a measurable set. I'm going to clarify the definition.70.81.15.136 (talk) 11:56, 10 March 2009 (UTC)

Measure and countably additive measure

The section Formal definitions currently starts with the following sentence:

Formally, a countably additive measure μ is a function defined on a σ-algebra Σ over a set X with values in the extended interval [0, ∞] such that the following properties are satisfied:

The section then continues to define countably additive measures. Do mathematicians always mean countably additive measures when they refer to measures without further qualifications? If yes, this should be stated early on. As the article stands now, the unqualified notion of a measure is never formally defined. (At my first reading of the section I was always expecting a sentence starting with "A measure is then constructed from a countably additive measure by..." or similar.) —Tobias Bergemann 16:11, 16 January 2006 (UTC)

I believe anytime one says "measure" one means a "countable additive measure". Functions which are not countably additive are simple called "additive functions". I reworded the article to make that clear. Oleg Alexandrov (talk) 21:32, 16 January 2006 (UTC)

splitting off for stubs

Right now, the material at σ-finite measure seems to be a verbatim copy of the material in the corresponding section of this article. I think splitting off sections is a good idea only when an article gets too long. Thus when some analyst comes here and writes us a book on σ-finiteness, we should split it off, but until that day, we should keep all our articles intact. Therefore I propose changing it to a redirect. I request your comments. -lethe ^{talk +} 21:09, 11 March 2006 (UTC)

There at least two important facts about σ-finite measures which merit mention in such an article: Fubini's theorem and Radon Nykodym require something like this property in the hypothesis. (I know they are true more generally -- localizable measure spaces). One should also mention another example of non σ-finite measures: Hausdorff measure of dimension r on spaces of Hausdorrf dimension > r.--CSTAR 21:26, 11 March 2006 (UTC)

It was me who forked off the sigma-finite measure. I find it much easier to refer to its own sigma-finite measure article, than to refer to it as #Sigma-finite measure. CSTAR, thanks for expanding it.

I just felt that the concept is important enough, and linked enough, that it can be its own article. Oleg Alexandrov (talk) 03:50, 12 March 2006 (UTC)

Alright, I guess I can get on board now. -lethe ^{talk +} 23:45, 12 March 2006 (UTC)

Toward GA

It is rough for math neophytes to look at formulas first thing off, could it be possible to give more information on these formula... if possible. For the rest, it seems good. Lincher 15:20, 2 June 2006 (UTC)

I also think this article is really good. I added a small line to the second equation to hopefully explain it in cleartext: "the measure of the union of all E is equal to the sum of the measures of all E". The language used is less clear than the equation though, but I think that is fine. Set notation is relatively easy and intuitive actually, so I don't see so much problems with this article. Let's approve it for GA.❝Sverdrup❞ 16:18, 5 June 2006 (UTC)

I've now listed this on Wikipedia:Good articles/Review, my main concern is lack of history on the topic: when was the concept introduced and by who. Who were the major players in the development of the theory? --Salix alba (talk) 10:43, 3 September 2006 (UTC)

Proof of Monotonicity?

The properties are listed with out a hint as to why they are true? That the measure is monotonic doesn't seem obvious ( although one should expect it, because of the analogue of "area" that is drawn.) Still at least some kind of motivational argument, if not a proof should help the reader see the importance of this particular property. Later properties and applications of measure depend heavily on this property. To be able to see the big picture, I would ask why is this true?

We could give the quick proof:
$B\supset A$
$B=A\cup (B-A)$
$A\cap (A-B)=\emptyset$
$\mu (B)=\mu (A)+\mu (B-A)\geq \mu (A)$

since the measure is valued between $[0..\infty ]$. The analogue of "area" then seems to follow naturally, as we see that the measure of the super set is indeed bigger by the "diffrence" between the two sets, and that property naturally extends from the sum of disjoint sets.

I think there some quailty to this article, and usefull information in it. A litte more elobration might make it a little clearer. JamesSug 03:09, 24 October 2006 (UTC)

Definition.

Should a measure be defined on a semi-ring, then using Carathéodory's extension theorem show that the measure can be extended? --unsigned

That is possible but too much trouble. It is better to keep the definiton simple. And as you said yourself, you don't really gain in generality if you start with a semiring. Oleg Alexandrov (talk) 03:41, 25 October 2006 (UTC)

It might be worth giving the two possible definitions, as it could be handy if an article on the construction of lebesque measures from first motivation is ever written. But then mention that the definition on a semi-ring is normally not used because of Carathéodory's extension theorem, and so using the 'tighter' definition for the rest of the article...Given it was suggested more history be included/motivations I think that it only seems sensible to include that definition. --TM-77 11:35, 28 October 2006 (UTC)

Motivation and history is of course good, but this to me seems like a technical discussion not many readers would appreciate. Perhas it could become a remark somwhere at the bottom. Oleg Alexandrov (talk) 16:31, 28 October 2006 (UTC)

Heading Change

I changed the section heading for Counterexamples to Non-Measurable Sets and added a 'main article' link to that article (Which, by the way, I'm not a fan of but...) Zero sharp 21:46, 3 November 2006 (UTC)

Should be Non-measurable sets as far as I can tell. :) Oleg Alexandrov (talk) 07:42, 4 November 2006 (UTC)

Opening

The Opening says:

In mathematics, a measure is a function that assigns a number, e.g., a "size", "volume", or "probability", to subsets of a given set. The concept has developed in connection with a desire to carry out integration over arbitrary sets rather than on an interval as traditionally done, and is important in mathematical analysis and probability theory.

1. In the intro Measure should be bolded

2. How does measure act as a function to asign a number without a standard to relate to?

A. If we say we measure 1.25, then 1.25 what?

B. How do we define the sets and subsets? What about the multiples of the set?

C. Do we need a way to define what's apples and whats oranges to be able to measure?

D. Can a set or subset define other sets in another dimension?

E. Can we agree that in antiquity measures of length defined areas, and volumes and that now we include time as well?

F. Is there a way that the measure can be considered scalar or proportionate to something without a standard being defined? Rktect 00:50, 3 August 2007 (UTC)

The introduction is meant to be informal, and I think it does the job well. It gives the idea that each set is mapped to a number. Further details and a more formal definition is supplied below in the text. Oleg Alexandrov (talk) 02:43, 3 August 2007 (UTC)

Recent edits

I believe the recent changes to the intro made it too complicated and confusing. The first several sentences of an article are very important, and it is important that they are clear and to the point, rather than comprehensive. There is too much repetition now, see sentences 1 and 3. Comments? Oleg Alexandrov (talk) 02:56, 6 August 2007 (UTC)

I don't entirely disagree, but I think its wrong to say that measure applies only to spatial dimensions, so with that in mind I would support any good revision to make the opening sentence shorter.Rktect 12:06, 6 August 2007 (UTC)

I agree with Oleg. Saying that the concept of a measure is a generalization of length/area/volume is by no means the same as saying that these things are the only thing a measure can be used to model. The purpose of the first sentence is to give the reader an initial informal idea of how one can think of this, not to provide a comprehensive list of things it can be used for. –Henning Makholm 12:40, 6 August 2007 (UTC)

Practical applications

Can someone please add a section on the practical applications of measure theory? My interviews with a broad range of scientists suggests that measure theory has a zero measure of contribution to any sort of applied science. —Preceding unsigned comment added by 24.226.30.183 (talk) 05:16, 18 January 2008 (UTC)

Measure theory is applied in probability theory and mathematical analysis. It may not have a lot of direct applications itself, but it is a tool that underlines these other subjects that have plenty of applications. Oleg Alexandrov (talk) 18:51, 19 January 2008 (UTC)

Nonsense

This article is pretty much nonsense at the moment. Being a measure theorist, I am surprised how this article got to GA in the first place (but now it is demoted). The point is that many of the important topics in measure theory are non included not to mention that the lede is not up to par (politely speaking). I will start a rewrite now. PST

To User:Point-set topologist. You write: "The space X is said to be a measurable space, if the measure of X is finite". Really? Where did you see this terminology? As far as I know, a measurable space is, by definition, a set equipped with a sigma-algebra (that is, with no measure at all). If equipped also with a measure, it becomes a measure space, be the total measure finite or not. But maybe different terminologies are in use. I wait for a reference from you. Boris Tsirelson (talk) 10:01, 13 January 2009 (UTC)

"A pair (S,Σ), where S is a set and Σ is a σ-algebra on S, is called a measurable space." Page 16 of: David Williams, "Probability with Martingales", Cambridge 1991. Boris Tsirelson (talk) 10:06, 13 January 2009 (UTC)

"Let (S,Σ) be a measurable space, so that Σ is a σ-algebra on S. A map μ:Σ→[0,infinity] is called a measure on (S,Σ) if μ is countably additive. The triple (S,Σ,μ) is then called a measure space." Page 18 of the same book. Afterwards Williams defines, when a measure "(or indeed the measure space (S,Σ,μ))" is called finite, or σ-finite. Boris Tsirelson (talk) 10:13, 13 January 2009 (UTC)

Hi Boris,

A space is of course measurable if it satisfies the properties as you say. I think that some authors require the conditions to be a bit stronger. I don't have access to Rudin's book at the moment (real analysis) but I think Rudin defines it the way I do in chapter 8 (in the beginning of a particular section: sorry for making you have to seach, I just don't have the book at the moment but I know it is at the beginning of the section where he introduces measurable functions). PST--Point-set topologist (talk) 10:55, 13 January 2009 (UTC)

An online viewing confirms that on p. 8 of Rudin's Real and Complex Analysis, Rudin gives the same definition of measurable space as does Boris Tsirelson above: a set endowed with a sigma-algebra. PST may be confusing "measurable space" with "measure space". In any event, it is certainly not part of the definition of the latter that the total mass must be finite, as you (Tsirelson) well know. Plclark (talk) 13:09, 13 January 2009 (UTC)

It's not page 8 (chapter 8) and I am referring to Rudin's real analysis book. I am pretty sure that his treatment of measures starts by defining a sigma-ring on a set together with a non-negative additive set function. He then defines the space to be measurable if X is in the sigma ring and in addition, has finite measure (I am pretty sure that my ref is correct but I could be wrong; please have a look at the correct reference if you have access to it). PST--Point-set topologist (talk) 13:21, 13 January 2009 (UTC)

Sigh. There is no textbook by Walter Rudin with the title "Real Analysis". The closest title is "Real and Complex Analysis", which is what I discussed above. He also has a book called "Principles of Mathematical Analysis", and in fact in Chapter 11, Section 11.12 (page 310) of this book the definition of a measurable space that he gives includes the notion of a measure, and differs from that of a "measure space" only in that, for him, a "measure space" can be defined with respect to a sigma-ring which is not a sigma-algebra. (Most students and experts would agree that the discussion of measure theory in "Real and Complex Analysis" is more thorough, careful and definitive than that of "Principles of Mathematical Analysis".) Finally, let me say again that you can find this bibliographic information online if you search for it. Plclark (talk) 18:17, 13 January 2009 (UTC)

Why just Rudin? Here is R.M. Dudley, "Real analysis and probability", page 86: "A measurable space is a pair (X,S) where X is a set and S is a σ-algebra of subsets of X." Here is A.S. Kechris (not a probabilist), "Classical descriptive set theory", page 66: "A measurable (or Borel) space is a pair (X,S), where X is a set and S is a σ-algebra on S." Boris Tsirelson (talk) 18:46, 13 January 2009 (UTC)

And here is D.W. Stroock (also not a probabilist), "Probability theory, an analytic view", page 1: "Let (Ω,F,P) be a probability space (i.e., Ω is a nonempty set, F is a σ-algebra over Ω, and P is a measure on the measurable space (Ω,F) having total mass 1)". Boris Tsirelson (talk) 18:54, 13 January 2009 (UTC)

Paul Halmos' trusty "Measure Theory", also uses the same "measurable space" terminology (X,S) p 73. Other references which use the same measurable space terminology Paul Meyer's "Probability and Potentials". Note that in some contexts measurable spaces are also called "Borel spaces" (see for instance Mackey's papers or the appendix to Dixmier "Les C*-algèbres et leurs représentations"). However, this "Borel space" terminology is not much used now. -CSTAR (talk) 19:24, 13 January 2009 (UTC)

Removed it. Maybe I had seen the definition in some textbook on probability (most likely) but as I am not sure, I will retract my claims that the definition is true. --Point-set topologist (talk) 20:55, 13 January 2009 (UTC)

Nice. Happy editing. Boris Tsirelson (talk) 21:21, 13 January 2009 (UTC)

Why μ(Ø) = 0 appears as part of the definition?! It follows from additivity, doesn't it? Maybe it should be moved down as note? Vasili Galka (talk) 17:32, 24 June 2009 (UTC)

Follows from additivity, really? What if μ(Ø) is infinite? Boris Tsirelson (talk) 19:16, 18 August 2009 (UTC)

$\mu (\varnothing )=\mu {\Bigl (}\bigcup _{i\in \varnothing }E_{i}{\Bigr )}=\sum _{i\in \varnothing }\mu (E_{i})=0.$ --Zundark (talk) 21:30, 18 August 2009 (UTC)

I see. OK, but then, for not being too much Bourbaki-ish, we should warn the reader that the "countable collection" may be empty (if we really want to drop "μ(Ø) = 0"). Boris Tsirelson (talk) 06:33, 19 August 2009 (UTC)

The person who removed "μ(Ø) = 0" should observe that the section is now incomprehensible as a result, as the text refers to the second and third of only two listed conditions. This needs substantial revision. —Preceding unsigned comment added by 76.202.59.223 (talk) 19:24, 16 September 2009 (UTC)

Indeed. I restore the three conditions and provide more comments. This is easier to understand and harder to misunderstand. I think so; but I know that some people hate three axioms when two are enough; feel free to revert to the two axioms if you wish, but please ensure that the text remains coherent and not misleading. Boris Tsirelson (talk) 20:41, 16 September 2009 (UTC)

I took me about 45 seconds to "parse" your latest addition, although admittedly once parsed, the semantics of the passage came a little more quickly. I realize you added these two or three sentences in a conciliatory gesture to the author of the previous edit, but is such conciliation always needed? --CSTAR (talk) 20:59, 16 September 2009 (UTC)

That is, my English is cumbersome? Maybe. I am sorry, if so. Anyway, if you are able to do better, just do it! Boris Tsirelson (talk) 21:03, 16 September 2009 (UTC)

Ack! No, no. it has nothing to do with English, yours or anybody else's. Why bother to subsume condition (2) as part of (3)?--CSTAR (talk) 21:07, 16 September 2009 (UTC)

Does it lack inline citations?

User:Finell on 21 July 2009 required inline citations. However, "Editors making a challenge should have reason to believe the material is contentious, false, or otherwise inappropriate", according to Wikipedia:When to cite#Challenging another user's edits. Is it the case here? Could Finell be more specific, pointing to a problem? Boris Tsirelson (talk) 19:08, 18 August 2009 (UTC)