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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.
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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. --
476:"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.
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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.
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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
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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
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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
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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
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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
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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 ℝ. But the definition, as given, is only for subsets of ℝ. Thus for example, property #1 is, strictly speaking, nonsense, because the
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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
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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...
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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
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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
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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
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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,
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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
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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
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In order for people to understand more easily, let's requre the example set A included in . 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 from each equivalent class, then A is not lebesgue measurable.
1887:. Since length, area and volume are different concepts, the reader deduces that the interval must have λ() = 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() is not zero...
423:). The reader must look below, to the section on "Construction of the Lebesgue measure", to find a definition which applies to subsets of ℝ. In order to prevent confusion when reading above the "Construction" section, the "Construction" should be the "Definition". --ScottEngles--
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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.
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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 x has Hausdorff dimension 2. However its Lebesgue measure is 0.01.
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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
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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:
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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!).
2103:, 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
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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.
2111:, 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.
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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.
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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!)
1494:{\displaystyle \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\}}
2632:{\displaystyle \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\}.}
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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.
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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...
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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.
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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. -
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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
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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:
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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. --
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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
1953:. 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.
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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.
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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.
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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
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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
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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.
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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.
402:(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
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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
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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):
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I think this issue deserves more explanation. It seems strange and confusing that you add the lengths of intervals that might overlap. --
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I agree. In addition, there should be some definition of Lebesgue measure in more than one dimension, before the Properties section. --
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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.
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article says "All the subsets of Rn whose dimension is smaller than n have null Lebesgue measure in Rn.". Am I missing something? --
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Which part is false? The fact that Lebesgue measure is full dimensional Hausdorff measure is a known fact (page 56 of Mattila's
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But would a fractal obtained by actually scaling itself by 3/8 and duplicating it infinitely many times have nonzero measure?
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are their images, not their paths. And, isn't the image of a space-filling curve in the plane two-dimensional? Indeed, the
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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.
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should this be clarified or is it not of much importance as a large part of the mathematics community accepts AC anyway?
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I agree. While the two concepts are related somewhat I think, I don't think their relationships is so straightforward.
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intervals do not overlap. If that intuitive explanation doesn't make sense, refer to the definition of outer measure.
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I suggest to add the following generalisation to higher dimensions, which easily follows from the definition of the
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A diagram might be even better, of course, but some text to tease out the related concepts would be quite helpful.
1852:"...the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a length, area or volume..."
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Some geometrical examples would be very helpful for non-specialists to understand what null sets are and are not.
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The introductory paragraph should be in the same language as the rest of the article (or am I missing something?)
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In the article, we should mention the measurable sets and unmeasurable sets. Since not all sets are measurable.
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A definition has been added, but it references a function l() which is not defined. That would be helpful.
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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.
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A might consist of uncountable many points, so A is not necessary to belongs to the class of Borel sets.
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the bound on the radii decreases to zero. My bad. But the statement I just made above is correct.
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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inverse f with domain K is included in C. G is lebesgue measurable, but it is not a Borel set.
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comes to the Haussdorf measure, it does not matter if one uses balls or rectangles, I think.
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Yeah I think you're right about the SVC set. I'll just delete that passage from the article.
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such λ(A) can not exist, hence A is not lebesgue measurable. More information can be found:
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1614:) is a generalization of the Lebesgue measure that is useful for measuring the subsets of
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Good point. It's not actually defined anywhere in the article. That needs to be fixed.--
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but which fits better in the present context when shaped in this way: For any
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Suppose A is measurable, then A+r (r is a rational number) is also measurable.
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What a pleasure was to read through this very well-written article! Good work!
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Added more information as requested. I think it is now more clear what
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Can any of you give an example of a non-Lebesgue measurable set?
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Is that correct? The SVC set is listed as having dimension 1 at
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of positive Lebesgue measure contains a non-measurable subset).
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I agree with the previous post. Could someone please define
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A={ai: ai} is infinite. No equivalent members of relation B.
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I feel a definition should be added to this article.
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is a sequence of intervals with open boundaries with
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Is there not a definition for the Lebesgue measure?
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1801:Geometry of sets and measures in Euclideans space
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1993:How is the length of the hypotenuse calculated?
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1238:{\displaystyle I={\text{ (or }}I=(a,b))}
1959:(And is SVC's dimension actually 1?) --
1951:List of fractals by Hausdorff dimension
1582:(3rd ed.). New York: Macmillan. p. 56.
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297:Knowledge:WikiProject Mathematics
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1879:Then, the reader learns that λ(
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376:Is it clearer now? --AxelBoldt
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291:and see a list of open tasks.
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1610:The Hausdorff measure (see
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63:New to Knowledge? Welcome!
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2181:Definition is rather dense
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1897:Are space-filling arcs 1D?
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815:Addition of a minor detail
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93:Be welcoming to newcomers
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1832:I'll add the text back.
1547:of all lower bounds). --
983:{\displaystyle l(I_{k})}
929:{\displaystyle l(I_{k})}
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323:project's priority scale
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1672:22:55, 2 May 2007 (UCT)
1657:05:02, 1 May 2007 (UTC)
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640:Bx={y: y-x is rational}
414:17:22, 4 Mar 2004 (UTC)
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1053:{\displaystyle I_{k}}
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2010:Pythagorean theorem
1857:Lebesgue measurable
1814:Hausdorff dimension
1612:Hausdorff dimension
1505:What am I missing?
398:(the rationals) in
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2277:{\displaystyle C}
1710:Hausdorff measure
1601:Hausdorff measure
1536:{\displaystyle E}
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1133:{\displaystyle E}
1113:{\displaystyle I}
1093:{\displaystyle E}
1073:{\displaystyle I}
1026:{\displaystyle E}
883:comment added by
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2015:
2014:77.124.183.77
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1580:Real Analysis
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1576:Royden, H. L.
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136:Find sources:
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123:Verifiability
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879:— Preceding
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319:Mid-priority
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244:Mid‑priority
222:WikiProjects
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32:This is the
2284:which is a
2259:German page
2244:Morningcrow
2229:Morningcrow
2193:Luke Maurer
1510:This.is.mvw
847:Definition?
822:Alok Bakshi
492:measurable.
294:Mathematics
285:mathematics
241:Mathematics
161:free images
44:not a forum
2660:Categories
2208:Erel Segal
2136:Erel Segal
1995:just-emery
1989:Hypotenuse
1563:References
793:Vitali set
489:Cantor set
440:WeCantKnow
343:Properties
2152:Smangerel
2028:Null sets
1912:, so the
1245:given by
101:if needed
84:Be polite
34:talk page
1961:Army1987
1922:Army1987
1918:null set
1889:Paolo.dL
1578:(1988).
1545:supremum
1177:interval
938:Gpayette
881:unsigned
832:CMummert
497:CMummert
412:Revolver
371:Revolver
69:get help
42:This is
40:article.
2286:product
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