Talk:Measure (mathematics)

Measure (mathematics) was a good article, but it was removed from the list as it no longer met the good article criteria at the time. There are suggestions below for improving the article. If you can improve it, please do; it may then be renominated.
Review: September 8, 2006.

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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. — Preceding unsigned comment added by Vitamindeth (talk • contribs) 16:02, 17 November 2021 (UTC)[reply]

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)[reply]

"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 spaces, though that would probably still need a reference. 1234qwer 1234qwer 4 22:39, 2 May 2022 (UTC)[reply]

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):

Mukherjea, A; Pothoven, K (1985). Real and Functional Analysis, Part A: Real Analysis (Second ed.). Plenum Press.
- The first edition was published with Part B: Functional Analysis as a single volume: Mukherjea, A; Pothoven, K (1978). Real and Functional Analysis (First ed.). Plenum Press.
Edgar, Gerald A (1998). Integral, Probability, and Fractal Measures. Springer. ISBN 978-1-4419-3112-2.
Fremlin, D.H. (2016). Measure Theory, Volume 2: Broad Foundations (Hardback ed.). Torres Fremlin. Second printing.
Folland, Gerald B (1999). Real Analysis: Modern Techniques and Their Applications. Wiley. ISBN 0-471-31716-0.
Royden, H.L.; Fitzpatrick, P.M. (2010). Real Analysis (Fourth ed.). Exercise 17.8: Prentice Hall. p. 342.{{cite book}}: CS1 maint: location (link) 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^[1] decomposition) agrees with usual presentations,^[2]^[3] whereas the first printing's presentation provides a fresh perspective.)
Nielsen, Ole A (1997). An Introduction to Integration and Measure Theory. Wiley. ISBN 0-471-59518-7.
Berberian, Sterling K (1965). Measure and Integration. MacMillan.
Fremlin, D.H. (2013). Measure Theory, Volume 4: Topological Measure Spaces (Second ed.). Torres Fremlin.
Torchinsky, Alberto (2015). Problems in Real and Functional Analysis. American Mathematical Society. ISBN 978-1-4704-2057-4.
Fremlin, D.H. (2015). Measure Theory, Volume 5: Set-theoretic Measure Theory. Torres Fremlin. Second printing.
Fremlin, D.H. (2012). Measure Theory, Volume 3: Set-theoretic Measure Theory (Second ed.). Torres Fremlin. Exercise 342X(a), p. 179.
Luther, Norman Y (1967). "A decomposition of measures". Canadian Journal of Mathematics. 20: 953–959. doi:10.4153/CJM-1968-092-0.

Thatsme314 (talk) 09:14, 17 June 2022 (UTC)[reply]

References

^ Luther 1967, Theorem 1.
^ Mukherjea 1985, p. 90.
^ Folland 1999, p. 27, Exercise 1.15.a.

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. UKe-CH (talk) 22:15, 26 November 2022 (UTC)[reply]

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. 2001:14BB:113:4C04:5561:8C07:F12E:ED42 (talk) 10:52, 11 December 2022 (UTC)[reply]

This is not what "disjoint" means. The empty set has empty intersection with any other set. 1234qwer 1234qwer 4 12:37, 11 December 2022 (UTC)[reply]

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 84.208.65.62 (talk) 07:40, 20 March 2024 (UTC)[reply]

Thank you. Now inserted. — Rgdboer (talk) 02:04, 21 March 2024 (UTC)[reply]

[FOOTNOTELuther1967Theorem_1-1] Luther 1967, Theorem 1.

[FOOTNOTEMukherjea198590-2] Mukherjea 1985, p. 90.

[FOOTNOTEFolland199927Exercise_1.15.a-3] Folland 1999, p. 27, Exercise 1.15.a.

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