Talk:Lebesgue measure

message: Many of the properties of the Lebesgue measure as stated on this page are simply properties of measures in general.  Items 3 and 4 for example, and perhaps others (I don't know, since I'm just learning about the subject as I browse here!)
Wouldn't it be better to list only those properties that are specific to the Lebesgue measure, or at least to indicate which ones are?  The standard definition is already there at measure
Also, a measure is defined on its page as a function on a sigma algebra; if that's really part of the definition, then saying "the Lebesgue measurable sets therefore form a sigma algebra" seems a little redundant and/or confusing.
- Stuart
: When the article says, "the Lebesgue measurable sets form a sigma algebra", this is said first without knowing that Lebesgue measure IS a measure.  In other words, we're verifying the axioms for the particular case of Lebesgue measure on Euclidean space.  This is much the same thing that happens when we verify the group axioms for a set of elements and operation -- it's not redundant, we just have to check that it really is a group. Revolver
Is it clearer now? --AxelBoldt
Thanks.  I'm still a little unsure as to which properties listed are common to all measures, and which are special to Lebesgue measure, but that's mainly due to my near-total ignorance of the whole subject.
- Stuart
Can any of you give an example of a non-Lebesgue measurable set?
JeffBobFrank 03:37, 4 Mar 2004 (UTC)
replies: text: Sure, it's not too difficult; I'll either put it up here or on a separate article. You basically take a set of representatives of the cosets of Q (the rationals) in R (the reals), (this is where choice comes in) and consider all the translates of this set by rationals. The translates form a countable partition of R, and you use this to get a contradiction. Of course, there are some details I'm leaving out. A proof is given in Royden and a slightly more general result in papa Rudin (that every subset of R of positive Lebesgue measure contains a non-measurable subset). Revolver 17:22, 4 Mar 2004 (UTC)
thread_title:  Properties 

message: What a pleasure was to read through this very well-written article! Good work! Oleg Alexandrov (talk) 13:43, 28 October 2005 (UTC)
thread_title:  Nice article 

message: I have added one more property of lebesgue integral related with the linear transformation. I was wondering if the property of translation invariance be reatined because it's a special case of a linear transformation.
Thanks...
Alok Bakshi 06:18, 8 February 2007 (UTC)
replies: text: I don't see any reason to eliminate it.  The two list items could be merged together, but the translation case is important enough to be specifically mentioned. CMummert · talk 14:33, 8 February 2007 (UTC)
thread_title:  Addition of a minor detail 

message: Is there not a definition for the Lebesgue measure? Sancho (talk) 05:50, 14 March 2007 (UTC)
replies: text: Good point. It's not actually defined anywhere in the article. That needs to be fixed.--CSTAR 05:58, 14 March 2007 (UTC)

text: A definition has been added, but it references a function l() which is not defined.  That would be helpful.  — Preceding unsigned comment added by 198.69.66.251 (talk) 21:52, 21 January 2013 (UTC)

text: I agree with the previous post.  Could someone please define  l(I_k) ?  Thanks!!! Gpayette (talk) 01:38, 22 January 2013 (UTC)
replies: text:  Added more information as requested. I think it is now more clear what  l(I_k)  means. Jorgecarleitao (talk) 07:36, 22 January 2013 (UTC)

text: \lambda^*(E) = \operatorname{inf} \left\{\sum_{k=1}^\infty \ell(I_k) : {(I_k)_{k \in \mathbb N}} \text{ is a sequence of intervals with open boundaries with }  \bigcup_{k=1}^\infty I_k \subseteq E\right\}.
What am I missing?
--This.is.mvw (talk) 10:30, 5 December 2019 (UTC)

text: You can approximate E from above (like in the article, then you need to take the infimum of all upper bound) or from below (like in your definition, but then you need to take the supremum of all lower bounds). --Tillmo (talk) 11:03, 8 December 2022 (UTC)
thread_title:  Definition? 

message: These sentences are part of the introduction:
# "...the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a length, area or volume..."
# "Sets which can be assigned a volume are called Lebesgue measurable;" 
# "the volume or measure of the Lebesgue measurable set A is denoted by λ(A)."
Note that in the first sentence, length, area and volume are clearly defined as different concepts. 
After reading the second sentence, the reader is puzzled: it appears that for some unclear reason the editor who wrote this sentence did not consider sets which can be assigned a length or area...
Then, the reader learns that λ(A) is a volume. Since length, area and volume are different concepts, the reader deduces that the [0,1] interval must have λ([0,1]) = 0.
But a doubt remains unsolved: why in the first sentence the author also refers to length and area? Where's the mistake? 
Later on, the reader discovers that λ([0,1]) is not zero...
Paolo.dL 11:29, 4 November 2007 (UTC)
thread_title:  Ambiguity in the introduction 

message: :All countable sets are null sets, but there are sets in Rn whose dimension is smaller than n which are not null sets. Space filling arcs in R2 are examples.
We're speaking about subsets of R2, so the arcs are their images, not their paths. And, isn't the image of a space-filling curve in the plane two-dimensional? Indeed, the null set article says "All the subsets of Rn whose dimension is smaller than n have null Lebesgue measure in Rn.".
Am I missing something? --Army1987 (talk) 15:03, 8 February 2008 (UTC)
replies: text: Is that correct? The SVC set is listed as having dimension 1 at List of fractals by Hausdorff dimension. While it *looks* like the union of two copies of itself scaled by 3/8,, it actually isn't, as the holes removed in the second iteration are long 1/16 each, not 1/12, and so on. But would a fractal obtained by actually scaling itself by 3/8 and duplicating it infinitely many times have nonzero measure? [Of course not, after each iteration the measure is 6/8 of the previous one. I think this argument shows that *any* self-similar set of dimension < n has n-dimensional Lebesgue measure equal to zero. --Army1987 (talk) 11:35, 9 February 2008 (UTC)
thread_title:  Are space-filling arcs 1D? 

message: How is the length of the hypotenuse calculated? just-emery (talk) 02:43, 18 June 2009 (UTC)
replies: text: By Pythagorean theorem --77.124.183.77 (talk) 10:50, 9 July 2009 (UTC)
thread_title:  Hypotenuse 

message: This whole section could stand re-writing.
The introductory paragraph should be in the same language as the rest of the article (or am I missing something?)
Some geometrical examples would be very helpful for non-specialists to understand what null sets are and are not.
(Stevan White (talk) 18:11, 16 October 2010 (UTC)
thread_title:  Null sets 

message: In the article, we should mention the measurable sets and unmeasurable sets. Since not all sets are measurable. Jackzhp (talk) 14:53, 19 January 2011 (UTC)
thread_title: Measurability

message: I feel a definition should be added to this article. 145.99.195.193 (talk) 21:15, 3 May 2011 (UTC)
thread_title:  Still no definition 

message: I reverted today an edit by User:128.178.71.35, who added that the intervals must be disjoint in the definition of outer measure. I said that the definition was correct before his edit, and I was right, but I also said that after his modification, it was not correct anymore, for example for  E = R, and I was wrong about this. Indeed, since the intervals can be semi-open, one can manage with disjoint intervals. However I believe that it is not a good idea to impose more restrictions to the family of intervals, since it is not needed. Bdmy (talk) 11:20, 7 May 2013 (UTC)
replies: text: I think this issue deserves more explanation. It seems strange and confusing that you add the lengths of intervals that might overlap. --Erel Segal (talk) 08:46, 2 July 2014 (UTC)
thread_title:  disjoint or not disjoint? 

message: It's hard (for me, anyway) to get an intuition from just the symbols alone. If I'm not mistaken, the formula makes it the infimum of the lengths of countable open covers of E, but I had to bang my head against it for a while (and I may be wrong!).
A diagram might be even better, of course, but some text to tease out the related concepts would be quite helpful.
Luke Maurer (talk) 05:22, 9 January 2014 (UTC)
replies: text: I agree. In addition, there should be some definition of Lebesgue measure in more than one dimension, before the Properties section. --Erel Segal (talk) 08:47, 2 July 2014 (UTC)
thread_title:  Definition is rather dense 

message: The definition on this page is only for $\mathbb{R}$, which was confusing and unhelpful since I had gone here trying to remember the exact definition for $\mathbb{R}^n$, and in that, some generalized notion of volume (which corresponds to length, area, and "normal" volume in the first three dimensions) was used when I had learned it. (There are some other differences with the version I learned in school and the one here, but that's beside the point, since that's not the issue here.) I might be able to add one if I find a useful one from MathWorld or the such, but this should probably be noted -- I'm not nearly as good as math as I wish I were, so perhaps someone else might be able to handle this better (or sooner) than I would. --Morningcrow (talk) 05:31, 5 July 2015 (UTC)
replies: text: I found that the "Construction" section was more helpful in giving me the generalized case, but if that's meant to also define measure, some semblance of this should be moved up. --Morningcrow (talk) 05:39, 5 July 2015 (UTC)

text: \lambda^{\!*\!}(E) = \inf \left\{\sum_{k=1}^\infty \operatorname{vol}(C_k) : {(C_k)_{k \in \mathbb N}} \text{ is a sequence of products of open intervals with } E\subset \bigcup_{k=1}^\infty C_k\right\}.
--Tillmo (talk) 11:17, 8 December 2022 (UTC)
thread_title:  Definition doesn't work for higher dimensions