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Properties

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)

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)

The examples and properties which make reference to higher dimension cartesian products of ℝ, really require that the definition of Lebesgue measure be extended to ℝⁿ. But the definition, as given, is only for subsets of ℝ. Thus for example, property #1 is, strictly speaking, nonsense, because the given definition doesn't apply to it (to A). The reader must look below, to the section on "Construction of the Lebesgue measure", to find a definition which applies to subsets of ℝⁿ. In order to prevent confusion when reading above the "Construction" section, the "Construction" should be the "Definition". --ScottEngles--50.134.248.28 (talk) 16:26, 19 February 2014 (UTC)[reply]

Sorry if I'm wrong (I am very new to this subject), but the second-to-last sentence of the sub-section before "Intuition" in the "Definition" section describes non-measurable sets. The next sentence claims their existence. I thought this was only true assuming AC, though? Am I confused? If not, should this be clarified or is it not of much importance as a large part of the mathematics community accepts AC anyway?WeCantKnow (talk) 05:16, 29 May 2019 (UTC)[reply]

Nice article

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

Borel vs. Lebesgue

"there are many more Lebesgue-measurable sets than there are Borel measurable sets". cardinal of the class of borel sets (=c=2^N0) < cardinal of the class of Lebesgue measurable set(=2^(2^N0)) = cardinal of the powerset of R(=2^(2^N0), but the class of all lebesgue messurable sets is still a strict subset of the power set of R. It is just hard to imagine. Can someone put two examples here at the same time please. Thanks. Jackzhp 17:22, 4 October 2006 (UTC)[reply]

The existence follows just from cardinality concerns. The cardinality of the class of Borel sets is the same as the cardinality of the real numbers, but the cardinality of the class of all subsets of the Cantor set is the same as the cardinality of the power set of the real numbers. Now every subset of the Cantor set is Lebesgue measurable of measure zero, because the Cantor set has measure zero. So most subsets of the Cantor set are Lebesgue measurable but not Borel measurable.

I would guess that a particular example of such a set could be constructed, but just like with non-Lebesgue-measurable sets it will not be canonical. CMummert 17:40, 4 October 2006 (UTC)[reply]

Please feel free to change the following text.

a lebesgue measurable, non Borel set on R

Cantor function f is a one to one map, which can map the cantor set C into a Borel set with measure 1. Hence its inverse function exists. Since every lebesgue measurable set with a positive measure contains a non lebesgue measurable set. Denote the non lebesgue measurable subset of f(C) as K, then the range G of the inverse f with domain K is included in C. G is lebesgue measurable, but it is not a Borel set.

f: C <-> f(C)

λ(C)=0, C is null; λ(f(C))=1

K\subseteq f(C)\,\!

K is a non lebesgue measurable set.

f: G <-> K

G\subseteq C\,\!

λ(G)=0, G is negligible, null.

G is not Borel set. If it is a Borel set, K is a Borel set since f is one to one map. Since K is non lebesgue measurable, so K is not a Borel set, neither is G.

Feel free to change all this.

a non lebesgue measurable set on R

In order for people to understand more easily, let's requre the example set A included in [0,1]. In order to construct the set A, define an equivalent class respect to x: Bx={y: y-x is rational}, then we select a member belong to [0,1] from each equivalent class, then A is not lebesgue measurable.

Bx={y: y-x is rational}

A\subseteq [0,1]\,\!

A={ai: ai} is infinite. No equivalent members of relation B.

Proof of A is not lebesgue measurable

Suppose A is measurable, then A+r (r is a rational number) is also measurable.

[0,1]\subseteq \cup (A+rn)\subseteq [0,2]\,\!

rn is any rational number, rn ∈[0,1]

λ([0,1])=1 <= λ(U(A+rn))=sum λ(A) << λ([0,2])=2

such λ(A) can not exist, hence A is not lebesgue measurable. More information can be found: Vitali set

Proof of A is lebesgue measurable

This is much easy, I can show it here. Although I know it is wrong, I don't know the reason why it is wrong.

Any x belongs to R, its lebesgue measure is zero. All singleton belong to the class of Borel set, so does the union of them, hence A belongs to the class of Borel sets. Lebesgue measure of A equals the sum of the lebesgue measure of all points belong to A. Hence the sum is zero.

A might consist of uncountable many points, so A is not necessary to belongs to the class of Borel sets.

Addition of a minor detail

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)[reply]

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)[reply]

Definition?

Is there not a definition for the Lebesgue measure? Sancho (talk) 05:50, 14 March 2007 (UTC)[reply]

Good point. It's not actually defined anywhere in the article. That needs to be fixed.--CSTAR 05:58, 14 March 2007 (UTC)[reply]

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)[reply]

I agree with the previous post. Could someone please define

l(I_{k})

? Thanks!!! Gpayette (talk) 01:38, 22 January 2013 (UTC)[reply]

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)[reply]

---

I am confused about the definition, enough that I'm *questioning* if it is wrong (but still presuming that I am wrong or missing something):

It says that $E$ is a subset of the infinite union of all $I_{k}$ , but shouldn't it be the other way around? Otherwise, how is $I$ dependent upon $E$ ? Am I missing something? It seems like $I$ should be the infinite set of infinitesimal ranges that sum up to $E$ .

In other words, it seems that it should be:

Given a subset $E\subseteq \mathbb {R}$ , with the length of interval $I=[a,b]{\text{ (or }}I=(a,b))$ given by $\ell (I)=b-a$ , the Lebesgue outer measure ^[1] $\lambda ^{*}(E)$ is defined as

\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)[reply]

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)[reply]

References

^ Royden, H. L. (1988). Real Analysis (3rd ed.). New York: Macmillan. p. 56. ISBN 0-02-404151-3.

Hausdorff measure

I don't know what Hausdorff measure is, but I do know what Hausdorff dimension is, and it is very difficult to construe it as a generalization of the Lebesgue measure. It is a completely different concept. As a result, I am removing the following text:

The Hausdorff measure (see Hausdorff dimension) is a generalization of the Lebesgue measure that is useful for measuring the subsets of Rⁿ of lower dimensions than n, like submanifolds, for example, surfaces or curves in R³ and fractal sets.

Loisel 04:51, 1 May 2007 (UTC)[reply]

I agree. While the two concepts are related somewhat I think, I don't think their relationships is so straightforward. Oleg Alexandrov (talk) 05:02, 1 May 2007 (UTC)[reply]

Actually, they are related in how they are defined. Lebesgue measure of a set is defined by taking the smallest (or infimum really) value that one can obtain by summing the volumes of balls whose union covers your set. The sum of these is just going to be the sum of each radius to the power n (since the volume of these balls is just cr^n for some constant depending on the dimension), n the dimension of the space where the set lives, times some constant. Hausdorff measure is defined in exactly the same way only now we vary the exponent, that is, we now sum this radii to the power of some real number s. So it allows us to study the measure theoretic properties of sets that may be too small for Lebesgue measure to pick up (e.g Cantor set, etc). So we can make the relationship between the measures straightforward in some sense, or at least make the extension from Lebesgue to Hausdorff measure intuitive.

I do agree, however, that the statement Loisel quoted above, doesn't make much sense, since the Hausdorff dimension is defined in terms of Hausdorff measure, not the other way around; that is, the Hausdorff dimension of a set is the infimum of the number s for which the s-dimensional Hausdorff measure is zero, or equivalently, the supremum of the numbers s such that the s dimensional Hausdorff measure is nonzero. So the concept of dimensions and measure are directly related. I'll take a look at the article when I get the chance. Jazzam 22:55, 2 May 2007 (UCT)

I thought that the Lebesgue measure was defined in terms of unions of rectangles, not balls, so that one gets sets without overlaps. I think the definition with unions of balls would give the same result, but things would be harder to prove since balls have small overlaps no matter what. When it comes to the Haussdorf measure, it does not matter if one uses balls or rectangles, I think. Oleg Alexandrov (talk) 03:34, 3 May 2007 (UTC)[reply]

Balls or rectangles don't matter, since the Lebesgue measure is regular. Lebesgue measure and Hausdorff dimension are two completely different concepts. The Lebesgue measure "covers a set with balls" and adds up the volumes. The Hausdorff measure "covers a set with balls" and sees what power d of the radius must be taken so that the sum of r_i^d is bounded as r_i goes to zero. Completely different. The Hausdorff dimension is constant 2 for all compact sets with interior in the plane, so it bears absolutely no relationship at all to a measure.

Loisel 20:08, 3 May 2007 (UTC)[reply]

Dimension and measure are different, clearly, that is true. I was arguing that Lebesgue and Hausdorff measures are related. In particular, Lebesgue measure is just n-dimensional Hausdorff measure (times a constant), so Lebesgue measure is really just a special case of Hausdorff measure. Hausdorff dimension of a set is defined in terms of the Hausdorff measure, but it itlsef is not a measure. I think that it would be safe to replace the statement deleted by Loisel above by something like:

Lebesgue measure is a special case of Hausdorff measure. In particular, Lebesgue measure on

\mathbb {R} ^{n}

is just a constant times the n-dimensional Hausdorff measure on

\mathbb {R} ^{n}

.

Also, the statement I made earlier about the definition of Hausdorff measure isn't exactly correct. It isn't the infimum of the sum of radii of etc... but when taking the measure of a set, one must fix an upper bound on the possible radii, then take the infimum, then take the limit of these values as the bound on the radii decreases to zero. My bad. But the statement I just made above is correct.

JAzzam 3:15, 4 May 2007 (UTC)

What you just wrote is false. Loisel 15:53, 4 May 2007 (UTC)[reply]

Which part is false? The fact that Lebesgue measure is full dimensional Hausdorff measure is a known fact (page 56 of Mattila's Geometry of sets and measures in Euclideans space, although it is a nontrivial result). I did notice, however, that in my last paragraph I defined spherical hausdorff measure instead of the usual one (that is, the infimum is taken over spheres instead of arbitrary convex sets), if that's what you mean. But it is still true that the usual n-dimensional Hausdorff measure is Lebesgue measure.

Jazzam 22:02, 4 May 2007

OK, this is my final comment in this thread because I am wasting my time. My first reply to you had the explanation, but let me do it again. Look again at the Hausdorff dimension article. Scroll down to "Hausdorff dimension and topological dimension." This says that any compact set with nonempty interior in the plane has Hausdorff dimension 2. In particular, the square [0,0.1]x[0,0.1] has Hausdorff dimension 2. However its Lebesgue measure is 0.01.

Loisel 23:17, 4 May 2007 (UTC)[reply]

Sorry, I just reread. You're talking about Hausdorff measure, not dimension. Yes, that's true, the Hausdorff measure on a manifold is the measure induced by the Lebesgue measure. I guess we had a misunderstanding.

I'll add the text back.

Loisel 00:25, 5 May 2007 (UTC)[reply]

Ambiguity in the introduction

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)[reply]

Are space-filling arcs 1D?

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)[reply]

That was nonsense. I've changed it to something that's true, although I'm not so sure it's that germane to the article. Loisel (talk) 16:29, 8 February 2008 (UTC)[reply]

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)] (And is SVC's dimension actually 1?) --Army1987 (talk) 20:19, 8 February 2008 (UTC)[reply]

Yeah I think you're right about the SVC set. I'll just delete that passage from the article. Loisel (talk) 04:57, 11 February 2008 (UTC)[reply]

Hypotenuse

How is the length of the hypotenuse calculated? just-emery (talk) 02:43, 18 June 2009 (UTC)[reply]

By Pythagorean theorem --77.124.183.77 (talk) 10:50, 9 July 2009 (UTC)[reply]

Null sets

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))[reply]

Measurability

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)[reply]

Still no definition

I feel a definition should be added to this article. 145.99.195.193 (talk) 21:15, 3 May 2011 (UTC)[reply]

disjoint or not disjoint?

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) Bdmy (talk) 11:22, 7 May 2013 (UTC)[reply]

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)[reply]

There is an inherent problem in assuming that the intervals be open and disjoint: any connected set E will fail to be covered by any collection of such intervals, which suggests that the Lebesgue outer measure of a connected set is vacuously zero (and R is connected, so this is silly). The problem with adding lengths of overlapping intervals is quite minor, given that, upon taking infima, the amount of overlap must vanish; a reader that can understand the definition through infima will not take issue with the question of overlap that you've mentioned. Thus, I think the disjointness condition is out, and if there is any concern about this overlap question, it can be elaborated upon briefly following the definition. Smangerel (talk) 23:27, 26 December 2014 (UTC)smangerel, 6:27 PM EST, Dec. 26th, 2014[reply]

The covering sets absolutely do not need to be disjoint. Look up any definition of outer measure. In the case of Lebesgue measure, overlapping intervals are not considered. The infimum taken over the sum of lengths (defined on any collection of covering intervals) will ensure that the covering intervals do not overlap. If that intuitive explanation doesn't make sense, refer to the definition of outer measure. Mkroberson0208 (talk) 14:12, 1 September 2015 (UTC)[reply]

Definition is rather dense

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)[reply]

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)[reply]

Definition doesn't work for higher dimensions

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)[reply]

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)[reply]

I suggest to add the following generalisation to higher dimensions, which easily follows from the definition of the German page but which fits better in the present context when shaped in this way: For any rectangular cuboid $C$ which is a product $C=I_{1}\times \cdots \times I_{n}$ of open intervals, let $\operatorname {vol} (C)=\ell (I_{1})\times \cdots \times \ell (I_{n})$ denote its volume. For any subset $E\subseteq \mathbb {R^{n}}$ ,

\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)[reply]

[1] Royden, H. L. (1988). Real Analysis (3rd ed.). New York: Macmillan. p. 56. ISBN 0-02-404151-3.

[1]