Talk:Measure (mathematics)

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Mathematics rating: B Class High Priority  Field: Analysis
Measure (mathematics) was a good article, but it has been removed from the list. There are suggestions below for improving the article to meet the good article criteria. Once these are addressed, the article can be renominated. Editors may also seek a reassessment of the decision if they believe there was a mistake.

Delisted version: September 8, 2006


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.

Contents

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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) \ge \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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)