View source for Talk:Measure (mathematics)

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==Need for two separate articles==

We need to have two separate articles "Measure" and "Measure Theory". A measure is clearly a small part of Measure Theory. All Wikipedia pages in other languages distinguish these two.   <!-- Template:Unsigned --><span class="autosigned" style="font-size:85%;">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Vitamindeth|Vitamindeth]] ([[User talk:Vitamindeth#top|talk]] • [[Special:Contributions/Vitamindeth|contribs]]) 16:02, 17 November 2021 (UTC)</span> <!--Autosigned by SineBot-->

== 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:<BR>
<MATH> B \supset A </MATH><BR>
<MATH> B = A \cup (B-A) </MATH><BR>
<MATH> A \cap (A-B) = \emptyset </MATH><BR>
<MATH> \mu(B) = \mu(A) + \mu(B-A) \ge \mu(A)</MATH><BR>

since the measure is valued between <math> [0 .. \infty] </math>. 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.
[[User:JamesSug|JamesSug]] 03:09, 24 October 2006 (UTC)

== "σ-finiteness can be compared in this respect to the Lindelöf property of topological spaces" ==

Is this OR? The [[σ-finite measure]] article does not mention Lindelöf spaces either. Maybe more closely related to [[σ-compact space]]s, though that would probably still need a reference. [[User:1234qwer1234qwer4|1234qwer]][[User talk:1234qwer1234qwer4|1234qwer]][[Special:Contribs/1234qwer1234qwer4|4]] 22:39, 2 May 2022 (UTC)

== Semifinite measures ==

I am writing a section on semifinite measures. The Korean Wiki page on measures (https://ko.wikipedia.org/wiki/측도) does a good job briefly discussing semifinite measures. (They also discuss localizable measures, which we could also add some material on.) Any help to expand/develop this section would be greatly appreciated.

Some references, ordered by the quality of their discussion of semifinite measures (best references first):
* {{cite book |last1=Mukherjea |first1=A |last2=Pothoven |first2=K |title=Real and Functional Analysis, Part A: Real Analysis |date=1985 |publisher=Plenum Press |edition=Second |url=https://link.springer.com/book/9781441974587}}
** The first edition was published with ''Part B: Functional Analysis'' as a single volume: {{cite book |last1=Mukherjea |first1=A |last2=Pothoven |first2=K |title=Real and Functional Analysis |date=1978 |publisher=Plenum Press |edition=First |url=https://link.springer.com/book/10.1007/978-1-4684-2331-0}}
* {{cite book |last1=Edgar |first1=Gerald A |title=Integral, Probability, and Fractal Measures |date=1998 |publisher=Springer |isbn=978-1-4419-3112-2}}
* {{cite book |last1=Fremlin |first1=D.H. |title=Measure Theory, Volume 2: Broad Foundations |date=2016 |publisher=Torres Fremlin |edition=Hardback}} Second printing.
* {{cite book |last1=Folland |first1=Gerald B |title=Real Analysis: Modern Techniques and Their Applications |date=1999 |publisher=Wiley |isbn=0-471-31716-0}}
* {{cite book |last1=Royden |first1=H.L. |last2=Fitzpatrick |first2=P.M. |author1-link=Halsey Royden |title=Real Analysis |date=2010 |publisher=Prentice Hall |location=Exercise 17.8 |page=342 |edition=Fourth}} First printing. Note that there is a later (2017) second printing. Though usually there is little difference between the first and subsequent printings, in this case the second printing not only deletes from page 53 the Exercises 36, 40, 41, and 42 of Chapter 2 but also offers a (slightly, but still substantially) different presentation of part (ii) of Exercise 17.8. (The second printing's presentation of part (ii) of Exercise 17.8 (on the Luther{{sfn|Luther|1967|loc=Theorem 1}} decomposition) agrees with usual presentations,{{sfn|Mukherjea|1985|p=90}}{{sfn|Folland|1999|p=27|loc=Exercise 1.15.a}} whereas the first printing's presentation provides a fresh perspective.)
* {{cite book |last1=Nielsen |first1=Ole A |title=An Introduction to Integration and Measure Theory |date=1997 |publisher=Wiley |isbn=0-471-59518-7}}
* {{cite book |last1=Berberian |first1=Sterling K |title=Measure and Integration |date=1965 |publisher=MacMillan}}
* {{cite book |last1=Fremlin |first1=D.H. |title=Measure Theory, Volume 4: Topological Measure Spaces |date=2013 |publisher=Torres Fremlin |edition=Second}}
* {{cite book |last1=Torchinsky |first1=Alberto |title=Problems in Real and Functional Analysis |date=2015 |publisher=American Mathematical Society |isbn=978-1-4704-2057-4}}
* {{cite book |last1=Fremlin |first1=D.H. |title=Measure Theory, Volume 5: Set-theoretic Measure Theory |date=2015 |publisher=Torres Fremlin}} Second printing.
* {{cite book |last1=Fremlin |first1=D.H. |title=Measure Theory, Volume 3: Set-theoretic Measure Theory |date=2012 |publisher=Torres Fremlin | edition=Second |at=Exercise 342X(a), p. 179}}
* {{cite journal |last1=Luther |first1=Norman Y |title=A decomposition of measures |journal=Canadian Journal of Mathematics |date=1967 |volume=20 |pages=953-959 |doi=10.4153/CJM-1968-092-0 |url=https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/decomposition-of-measures/B4356A177FCF019EA9B0A3C8AC40BB6B}}
[[User:Thatsme314|Thatsme314]] ([[User talk:Thatsme314|talk]]) 09:14, 17 June 2022 (UTC)
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== About two measure theories ==

There exists a different viewpoint for measure theory. See the paragraphs ''Wikipedia, NPOV and mathematics'' and ''Two measure theories in competition'' in my Wikipedia user page. [[User:UKe-CH|UKe-CH]] ([[User talk:UKe-CH|talk]]) 22:15, 26 November 2022 (UTC)

== Null empty set is not implied by finite measure of E ==

Additive countability does not imply null empty set if there is one element of E whose measure is finite, because the empty set is a subset of every set and is not therefore disjoint. [[Special:Contributions/2001:14BB:113:4C04:5561:8C07:F12E:ED42|2001:14BB:113:4C04:5561:8C07:F12E:ED42]] ([[User talk:2001:14BB:113:4C04:5561:8C07:F12E:ED42|talk]]) 10:52, 11 December 2022 (UTC)
:This is not what "disjoint" means. The empty set has empty intersection with any other set. [[User:1234qwer1234qwer4|1234qwer]][[User talk:1234qwer1234qwer4|1234qwer]][[Special:Contribs/1234qwer1234qwer4|4]] 12:37, 11 December 2022 (UTC)

== Archimedes ==

Hi, re the "citation needed" tag for Archimedes, her is the ref given in the Wiki Archimedes article: 
:https://web.archive.org/web/20040703122928/http://www.math.ubc.ca/~cass/archimedes/circle.html
T [[Special:Contributions/84.208.65.62|84.208.65.62]] ([[User talk:84.208.65.62|talk]]) 07:40, 20 March 2024 (UTC)
:Thank you. Now inserted. — [[User:Rgdboer|Rgdboer]] ([[User talk:Rgdboer|talk]]) 02:04, 21 March 2024 (UTC)