Page last updated 19 Oct 2021
Lebesgue measure is a theory that arose from the concept of a “real number line”. Mathematicians began to contemplate what it meant to refer to distances between points on such a line in case of sets of points that had rather involved definitions, and came up with the concept of a “real number line”.
The “real number line”
Mathematicians noticed that if you thought of a line of real numbers like a physical line stretched between two points, then given any real number, you could have a corresponding point on your “real number line”. And, being Platonists, they assumed that such a “real number line” actually exists as a mathematical object, and is composed of an accumulation of points.
This was a fundamental error. The reality is that the notion of a real number line is a notion that is inherently a fractal, where no matter how close one zooms in, the line always looks the same. It may be a simple one-dimensional fractal, but a fractal it is, and that means that there never is a situation where the fractality ends and - behold - you then have a solid line where you cannot fit in any more points.
Because of this, there cannot be an actual sequence of all the real numbers between any two values (such as 0 and 1) where every number is set in order according to its value, since for any real number, there is no ‘next’ number. Similarly, there cannot be an actual sequence of points that somehow make an actual line. Moreover, by the very definition of a point, a point has no length or width, so that it is impossible for a collection of points to constitute a line.
But if you recognize that when you define a line where one end corresponds to 0 and the other end corresponds to 1, you are only defining a concept, not describing any actual thing, and that, since there is no limit to how many real numbers you can have between 0 and 1, then similarly, there is no limit to the number of points you can define on this line. But you never actually reach the state where the line is ‘filled’ with points. (Footnote: As Wittgenstein remarked: “But the curve is not composed of points, it is a law that points obey, or again, a law according to which points can be constructed.” as in Philosophical Grammar, University of California Press, 2005.)
This is in direct opposition to the Platonist stance which insists that all the points on the line ‘exist’ simultaneously, thus constituting an entire continuous “real number line”. For more on Platonism see Platonism, The Myths of Platonism, Platonism’s Logical Blunder, Numbers, chairs and unicorns and the posts Moderate Platonism and Descartes’ Platonism.
An example of Lebesgue measure theory
First, a couple of definitions:
An open interval is an interval that does not include the endpoints that define that interval (for example the open interval whose endpoints are 1⁄3 and 1⁄2 is the set of all points between 1⁄3 and 1⁄2 but not including the points 1⁄3 and 1⁄2).
A closed interval is an interval whose endpoints are included in the interval.
Now, let’s consider a definition of a set A of ever decreasing intervals that is defined like this:
We start with the closed interval between 0 and 1. Now take a suitable listing (Footnote: See One-to-one correspondences and Listing the rationals.) of the rational numbers between 0 and 1 (for details see below A specific listing of rational numbers). Then, going through this list of rational numbers, for the first rational we define an associated open interval 1⁄10 wide with that rational at the midpoint of the interval; our set now includes all the numbers in that interval (not including the endpoints). For the next number, define an associated open interval 1⁄100 wide with that rational at the midpoint of the interval; we add those numbers to our set. For the next number, define an associated open interval 1⁄1000 wide with that rational at the midpoint of the interval; we add those numbers to our set. And so on, with each subsequent open interval being 1⁄10 of the length of the previous interval. (Footnote: Note that an iterative process is not in fact required: we can define the set A without any reference to iterations, see Formal set definitions.)
Given this definition, there are only two possibilities. Either:
- The entire closed interval between 0 and 1 is covered by the intervals, or
- There are some irrational points between 0 to 1 that are not covered by any interval (obviously, by the definition, there cannot be any rational points that are not covered by some interval).
Now, according to conventional mathematics using the axioms of Lebesgue measure theory, the total length of the intervals of the set A must be less than 1⁄9. This number is calculated by using a calculation that gives the limiting value of the sum of 1⁄10 + 1⁄100 + 1⁄1000 + 1⁄10000 + … which gives the value of 1⁄9. (Footnote: See Geometric Series for how this is calculated.) Note that this is a maximum value, since if some intervals overlap, the limiting sum will be less than 1⁄9.
Which means that since the length of the original interval (from 0 to 1) is 1, then the remaining length, according to the axioms of Lebesgue measure theory, must be at least 8⁄9. And so, according to this theory, there must be sufficiently many points remaining that can account for this value 8⁄9.
Analyzing the Lebesgue result
Given that there are points not in the set A, then it is clear that any such points cannot be rational, and so they must be irrational; we will call this set of such irrationals the set B.
Complete Interval: Note that in the following we call an interval of A a “complete interval ” if it is not a sub-interval of any interval of A (apart from itself).
Now, if there are any points in B, then, according to the definition, there must be complete intervals of A that have left and right endpoints (the points that are the lower and upper bounds of the complete interval ). These endpoints cannot be rational, since every rational is the midpoint of some enumerated interval, hence any such endpoints must be irrationals. (Footnote: This is an indication that the notion that such a set which contains irrationals is well-defined by the strict application of the above definition of open intervals cannot be correct.) Furthermore, from the definition of the set A it follows that there cannot be any degenerate interval (Footnote: A degenerate interval is a single point.) in the set A, and that every interval in the set B is degenerate. (Footnote: Clearly, there could not be any intervals in the set B that contain more than a single point, since for any two real numbers, there are infinitely many rationals between them.)
It is easy to show that the irrationals in the set B cannot possibly give rise to a total width greater than the total width of the set A, as follows:
Every rational has an associated n by the enumeration listing of the rationals. And every rational is in some complete interval of A. For any given rational, either it is the rational with the lowest associated n of all the rationals in that complete interval or it is not. If it is, then there is a unique association of that natural number n with the left irrational endpoint of that complete interval of A. Since every rational is enumerated, every complete interval of A is included by such an association, and hence every left endpoint of a complete interval is accounted for by such an association. (Footnote: In the same way, there is also a unique correspondence of the lowest associated n of all the rationals in that complete interval and the right endpoint of that interval.) Some people assume that this means that the endpoints of A are denumerable, but to be denumerable means that there must be an enumeration function that defines that enumeration, so that for a given natural number n, the function uniquely gives the appropriate endpoint. In this case however, it is impossible to define a function that uniquely defines the association of every complete interval of A with every left endpoint, and hence the association is not denumerable. (Footnote: Perhaps someone might want to make the claim that there if there is a set of such associations then there must be an independent “existing” but “indefinable” function that gives those associations, and also an independent “existing” but “indefinable” enumeration function that enumerates the endpoints. This is the conventional Platonist stance which also asserts that different infinitely large sets can have different “sizes”, even though all infinite sets are limitlessly large, in this case this leads to the notion that the set of endpoints is of the same “size” as the set of natural numbers, that they have the same cardinality - but this serves only to demonstrate the fallacy of assuming for infinite sets that the existence or non-existence of a mapping function indicates the size of an infinite set, see also Proof that there are more real numbers than numbers? The notion that there “exist” “indefinable” enumeration functions only serves to compound the contradictions inherent in Lebesgue measure, since by that notion every possible set that is an enumeration of some “indefinable” real numbers also “exists” - and every such “enumerated” set, by conventional measure theory, must have zero measure. But since there are no “indefinable” real numbers that could not be in some such “indefinably enumerated” set, then where could the measure of a non-denumerable set of “indefinable” real numbers come from? It would be absurd if the measure of a single point depends on whether it is in a denumerable or a non-denumerable set.)
Clearly, the endpoints of the complete intervals of A, all of which are of zero size, cannot constitute a greater measure than those intervals for which they are endpoints, since all those intervals are of non-zero size. But according to the axioms of Lebesgue measure theory, the total measure of all the complete intervals of A must be less than 1⁄9, and that the remaining measure of at least 8⁄9 is accounted for by points that are not in the set A.
Some people, when faced with this unpalatable contradiction, make the bizarre attempt to get around the contradiction by claiming that there must be other points in the set B as well as those single points that are between two intervals of A. I have asked these people to prove me wrong by providing a logical argument to back up their claims but they have been unable to do so.
This brings us to the crux of the matter. There is a contradiction inherent in Lebesgue measure theory, since it asserts that the endpoints of the intervals of A, where each such endpoint has zero measure, has a total measure greater than that of the intervals between those endpoints - and infinitely many of these intervals have non-zero measure. That is an absurdity.
Yes, according to the axioms of Lebesgue measure theory, that set of single irrational points (each of which has precisely zero length) has a total length greater than 8⁄9, while the intervals of set A, which includes every rational (and also many irrationals) must have a total length less than 1⁄9.
Conventional mathematics claims that although there are infinitely many intervals between these irrational points, these irrational points constitute a “bigger” infinity than that of the intervals - that there are somehow ‘more’ of these points than the intervals between them ! And that somehow (although exactly how is never divulged) because there is a ‘bigger’ infinity of these single points, they have a total measure of at least 8⁄9 even though each such single point has a measure of precisely zero. But, as we have seen above, the infinity of these irrational points cannot possibly be of a “bigger” infinity than that of the rational numbers.
Welcome to fantasy land.
Attempts to evade the contradiction
Some people appear to have some difficulty accepting that their beloved Lebesgue theory results in a contradiction, and try to devise various arguments against it while apparently being oblivious to the inherent contradictions of those arguments. These attempts almost invariably involve a misunderstanding of the application of limit states to summations involving infinitely many items.
The page Understanding sets of decreasing intervals explains why such definitions of ever-decreasing intervals are inherently contradictory because by such definitions, without any limit included in the definition, the endpoints of any union of overlapping intervals of the set A cannot be defined as either rational or irrational, since for any endpoint of an interval given by the definition of A there is always another interval that extends that interval past that rational endpoint. The fallacies involved in attempts to circumvent these contradictions are analyzed in detail on the page Understanding Limits and Infinity and also in the paper On Smith-Volterra-Cantor sets and their measure which explains how all contradictions can be avoided by a logical analysis of the definitions involved.
As is so often the case, Platonists refuse to let contradictions get in the way of their beloved and bizarre notions. As an hilarious example of how Platonists mange to congratulate themselves on pretending that there isn’t a contradiction involved, see the web-page An apparent inconsistency of Lebesgue measure. As Wilfrid Hodges (Footnote: Wilfrid Hodges, An Editor Recalls Some Hopeless Papers, The Bulletin of Symbolic Logic, Vol 4, No. 1, March 1998.) has remarked (with reference to flawed attempts to attack the diagonal argument): ‘to attack an argument, you must find something wrong in it. Several authors believed that you can avoid [that] by simply doing something else.’ But that is precisely what the protagonists on the web-page do - they attempt to avoid the contradiction by doing something else other than finding something wrong with the contradictory statement. But while on the one hand, it is hilarious, it is also pathetic and sad that they are unable to see that the presence of a contradiction is telling them that there is something fundamentally wrong with their mathematical foundations. They are so sure that there is nothing wrong with the axioms of Lebesgue measure theory that they cannot contemplate the possibility that it might be an inconsistent theory.
Now for a few details of Lebesgue’s theory of measure.
The rules of Lebesgue’s theory
Lebesgue’s theory of measure is a theory that has to be bolted on to conventional number theory, by inventing axioms that are not inherent in fundamental number theory. (Footnote: Note that Lebesgue measure theory has never had any confirmation of any efficacy in relation to any real world application - unlike the conventional usage of numbers, which have been used time and time again in real world applications.) The reason for this necessity for bolting on is that in conventional number theory, for any two different numbers, there is a numerical value that is simply the difference between those two numbers, while the difference between a number and itself is precisely zero. But when you have the concept of a “real number line”, the notion of an interval now corresponds to the notion of the difference between two numbers. And what people refer to as a single point on the real number line corresponds to a single number; this isn’t really an interval, but sometimes it is referred to as a degenerate interval - in which case the measure of such a degenerate interval is precisely zero (the difference between a number and itself).
A measure, in its very simplest form, is simply the difference between two real numbers. And one expects that more complex measures would be dependent on multiples of such basic measures. But Lebesgue measure manages to assume that a collection of single zeros (each consisting of the difference between a number and itself) can somehow constitute a measure that is greater than zero.
Yes, really ! I’m not kidding.
The key assertions in Lebesgue theory are essentially: (Footnote: These are, of course, somewhat simplified here, but the essential facets of the theory are given by this.)
- For any set of single degenerate points that is denumerable, the Lebesgue measure of that set is zero.
- For a set of non-overlapping intervals, but only provided the intervals are denumerable, the Lebesgue measure is the sum of the lengths of all of the intervals. (Footnote: It also assumes that there is always a simple summation of the lengths of infinitely many ever-decreasing intervals, which is incorrect, see below Different orders of summation. )
- For a set of numbers between two numbers a and b that is not made up of either of the two above types, the Lebesgue measure cannot be deduced directly, but is given by subtracting the total of Lebesgue measures of the sets of type (i) and (ii) from the overall length between a and b. (Footnote: Note: In order to avoid contradictions when Lebesgue theory is used along with set theory and the axiom of choice, then this means there must be sets of points that don’t have any measure - not a zero measure, nor some finite measure, nor an infinite measure - just no measure at all. Which means that, when using set theory and the axiom of choice, there are sets of points for which the axioms of the Lebesgue theory of measure cannot give a measure for those sets. )
The axioms of Lebesgue theory of measure are based around the requirement that if an interval is split into two sets of points, then the sum of the Lebesgue measures of the two sets must always sum up to the total length of the interval. Now, while it might be nice to have that requirement satisfied, the Lebesgue method of doing so comes at a high price. The downsides are many. One major downside is that it is never explained how a collection of infinitely many zeros (the measures of single points separated by non-degenerate intervals) can be a finite non-zero value.
But the principal downside is that it leads to a direct contradiction - as in the case described above of ever decreasing intervals.
The problems arise because of a failure to acknowledge that some definitions involve limitlessness, such as the recursive algorithm defined above that never terminates. Now, although a definition involves limitlessness, what you can do is to applying a limiting condition. But you must be careful. If there is a choice of limiting conditions that can be applied, then you must be sure to choose the limiting condition that corresponds to whatever aspect of the limitlessness that you are attempting to calculate a limiting value for. In the case of the ever decreasing intervals as described above, you can either:
- calculate a limiting condition for the total length of the intervals, without including any consideration of the relationships between the endpoints of the intervals
- calculate a limiting condition for the totality of points that are in the set of points given by all defined intervals, without including any consideration of the actual lengths of the intervals. This is done by applying a limit to every union of overlapping intervals of A, which results in irrational values for the endpoints of the complete intervals of A.
In case (a), you get a value of number theory: a numerical value of 1⁄9.
In case (b), you get a value of set theory: a set of points between 0 and 1.
These are two completely different types of values. To assume that the value (a) must imply the other case (b) is absurd, and indicates a complete failure to understand limitlessness. Also note that the measure of the remaining intervals are decreasing, and the is no finite width at which this stops, so that the limiting value is zero width.
You can also see a formal paper on how to eliminate contradictions in the calculation of the total measure of sets that are defined in terms of limitlessness, see On Smith-Volterra-Cantor sets and their measure.
Different orders of summation
In the assertion that the set A has a measure of 1⁄9 it is assumed that it a very simple matter - that one simply adds up an interval of length 1⁄10 then add another interval of length 1⁄100 and so on - and that you can extrapolate that to infinity. But the definition of each of those lengths is dependent on the interval it is associated with - each length is defined by the left endpoint and the right endpoint for each case. So while it is simply asserted that the total length obtained by adding up infinitely many decreasing fractions, this conceals the fact that the calculation that is actually being defined is:
R1 − L1 + R2 − L2 + R3 − L3 + …
where L1 and R1 are the left and right endpoints of the first interval, L2 and R2 are the left and right endpoints of the second interval, L3 and R3 are the left and right endpoints of the third interval, and so on.
For a special case where each subsequent interval is added so as to ‘touch’ the previous one (is adjacent to) the previous one - the endpoints coincide - then we can have:
|0.10 − 0.00||+||0.11 − 0.10||+||0.111 − 0.110||+||0.1111 − 0.1110|
For a finite sum, the left endpoint of one interval coincides with the right endpoint of the previous interval, and so the corresponding endpoint numbers cancel out - in the above 0.10, 0.11, and 0.111 cancel out, leaving 0.1111 as the correct summation. If the process continues infinitely, the limiting value is 0.111… which is equal to 1⁄9. Note that for a finite sum of intervals, the two endpoints of any given interval do not need to appear consecutively in any such ordering. For example, if L1, R1, L2, R2, L3, R3, L4, R4 are in ascending order, then the total length of a finite number of such intervals (L1, R1), (L2, R2), (L3, R3), (L4, R4) can be given by R4 − L1 + R3 − L2 + R1 − L 2 + R2 − L3 + R3 − L4.
But for the case of intervals that are not adjacent, and where the process continues infinitely, there is not necessarily any simple such summation. It is well known that for an infinite series that has both positive and negative terms, the limiting sum is dependent both on the values and the order in which they appear in the series (see Sums of infinitely many fractions: 1). The Lebesgue measure value of 1⁄9 arises from the naive assumption that the calculation of the limit of a simple sequence of sums of decreasing intervals with enumerable rational endpoints and where the resultant set has denumerable rational endpoints, must also apply to a set that results in intervals that have non-denumerable irrational endpoints. As previously noted, we can define the set A in a way that does not specify any order of addition of intervals at all, see Formal set definitions.
The simplistic summation of infinitely many interval lengths overlooks the crucially important fact that there can be any order of summation and subtraction of the endpoints, which can result in different limiting values. The naive assumption that one can always calculate the size of such a set by simply adding the lengths is completely erroneous. Since there are infinitely many different possible orderings, ignoring the fact that different orderings can result in different limiting values is absurd. The assertion that the total length of the set A must be 1⁄9 is an absurdity that should be obliterated from mathematics.
One correct calculation of measure?
If Platonism is correct, then the measure of any set of points must be an intrinsic property of the set - rather than being merely a human invention that is used for certain purposes. And so, if Platonism is correct, then there can only be one correct calculation of the measure of any set of points. Clearly, Lebesgue measure cannot be the correct Platonist theory of measure, since it leads directly to a blatant contradiction. There is no logical reason to suppose that Lebesgue theory is a theory that reflects some Platonist measure that exists independently of the human mind. It follows that there is no reason to promote Lebesgue measure theory as the ‘correct’ theory of measure.
Also see Understanding Limits and Infinity which demonstrates the logical fallacies in some arguments regarding the set A.
Note: Mention in a sci.math forum
I came across a sci.math forum A Counter-argument against Lebesgue Measure Theory? that makes some erroneous claims about my site. In case anyone might think that there is some substance in what is discussed there, I thought I should point out a few facts about what is written there. Below relates to what was there on 01 Oct 2021.
Two of the posters claim that I assert that the points of the complement of A constitute a denumerable set. This is completely incorrect. I did not claim that there is an such an enumeration function, but I pointed out that there is association between the points of the complement of A and certain rationals of A, but I also pointed out that it is impossible to define an enumeration function that gives the set of all those associations. I have since updated the page to deal with that misconception.
The posters there also glibly assert that all intervals of the set A must have rational endpoints - and simply ignore the consequent contradiction. As I point out on this site, for any set whose intervals are open and non-degenerate, an endpoint of any interval of it is, by definition, not a point of that set, but a point of its complement. And for the case in question this generates a contradiction, since all endpoints of the intervals of the set A are (unless the definition includes a limit) defined to be rational, while all points of the complement of A must also, by the same definition, be irrational. And it cannot be the case that the points of the complement of A are not the endpoints of the intervals of A, since that would imply that, since the points are different, then there must be a non-degenerate interval between them, and that non-degenerate interval would then have to be in the complement of A, which is impossible.
One poster claims that while he could demonstrate by a counter-example that my argument is wrong, he declines to do so, so we only have his word for it - and yet this same person complains that I engage in “wishy-washy wordy arguments and unsupported claims”, apparently completely unaware of the irony. In addition, he also concludes that I am a crank, not because he has demonstrated any error in my writing, but because he thinks that non-crank mathematicians do not use certain terms of the English language. Logic is clearly not his strong point. I suspect that his claimed counter-example might be the same as that claimed in a comment to this site, see my response to this claim, where I point out the fallacy therein.
When these people observe that I get a result that differs to their results, they simply assume that that must mean that their result is the correct one and that I am wrong. It never occurs to them that perhaps it might indicate yet another contradiction/paradox to add to the list of all the contradictions/paradoxes that are already in their system, and they never stop to consider whether all those contradictions/paradoxes might actually indicate something fundamentally wrong with their system.
A specific listing of rational numbers
An example of an enumeration of the rationals can be given by using the pattern of rationals:
This gives a list that begins as 1⁄2, 1⁄3, 2⁄3, 1⁄4, 3⁄4, 2⁄4, 1⁄5, 2⁄5, 3⁄5, 4⁄5,1⁄6, 2⁄6, 3⁄6, 4⁄6, 5⁄6,… Note that this gives duplicates like 2⁄4, 3⁄21, but the important fact is that the enumeration includes every rational between 0 and 1 (not including 0 or 1). The definition of the list can made so as to remove the duplicates id required.
Clearly, there can be infinitely many similar enumerations based on similar definitions. An example is given below - note that this enumeration follows a pattern that for each subsequent denominator, the values run from the lowest to the highest value of the numerator. For every subsequent denominator, this gives a pattern of rationals across the interval 0 to 1. This patterning continues infinitely as the terms progress. The enumeration can be represented by an algorithm as follows, where the calculation of the nth rational does not require the calculation of all of the rationals prior to n in the enumeration:
- Let t = 0
- If √1 + 8(n + t) is a Natural number, then:
- m = (−1 + √1 + 8(n + t) )/2 and the nth rational is (m − t)/(m + 1)
- Otherwise let t = t + 1 and repeat from step 2.
This can easily be made into a program as has been done here:
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