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Lebesgue Measure

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 concpet 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.

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 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.

Note that an iterative process is not in fact required: we can define the set A without any reference to iterations, see below A definition without iterations.

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 cannot be more 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 . 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 . And so, Lebesgue measure must reject the possibility that the set A includes the entire interval between 0 and 1.

Analyzing the Lebesgue result

Let us suppose that Lebesgue theory is correct, and that there are points not in the set A; clearly any such points cannot be rational, and so they must be irrational; we will call the 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 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. Furthermore, from the above it follows that there cannot be any degenerate interval in the set A, and that every interval in the set B is degenerate.

It is easy to show that, even if there are irrationals in the set B, they could not 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. Hence there exists a set that consists of all the complete intervals of A together with their endpoints - the union of the set A and those associated endpoints. 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 the endpoints of the complete intervals of A. There are various ways of demonstrating the absurdity of this bizarre notion:

For that to be the case, there would have to be right endpoints of complete intervals of A that are not left endpoints of any complete interval of A; otherwise the entire interval 0 to 1 would be included by the intervals of A and these endpoints, and there could be no other points remaining. Given any such right endpoint R1 that is not also a left endpoint of a complete interval of A, that point R1 (an irrational) is a left endpoint of a non-degenerate interval of the real number line of arbitrary width. Every rational in such an interval is the midpoint of a non-degenerate interval. Now, if the point R1 is not the left endpoint of a complete interval of A, then consider any other complete interval of A such that its left endpoint L1 (an irrational) is greater than R1. So now, we have two irrationals R1 and L1, and where L1 is greater than R1. Hence there must be at least one rational between those two points. And since that is the case, then there must be another complete interval of A between those two points. As indicated above, there exists a set that consists of all the complete intervals of A between R1 and L1 together with their endpoints. We now suppose that nevertheless, between R1 and L1 there remains some point L2 that is greater than R1, such that every number r that is greater than R1 and less than L2 is a point of the set B, i.e:

For all r, R1 < r < L2, r is an element of the set B. But this is impossible, since R1 and L2 are irrational, and hence if there could exist such a right endpoint R1 and a left endpoint L2 that are not coincident, there would be infinitely many rational numbers between R1 and L2. But that is impossible since every rational is in A, and every enumerated rational and its associated interval is already accounted for.

Hence there cannot be any points in the set B other than endpoints of complete intervals of A.

And for another reason as to why there could not be any points in B other than endpoints of complete intervals of A:

Since every interval in the set A has a rational endpoint, and since each such endpoint is itself the midpoint of another interval, the only way in which an irrational might remain is by there being a sequence of intervals of decreasing size, where the size of the intervals is decreasing in such a way that as the intervals become closer and closer to the irrational point, the irrational point is never reached by any one of the sequence of intervals. Hence if there could be irrationals in the set B, then every such irrational must be associated with an infinite sequence of rational numbers that consists of all right endpoints (of the enumerated intervals) within a complete interval on the left side of the irrational, and an infinite sequence of all left endpoints within a complete interval on the right side of the irrational. Furthermore, since such a sequence of endpoints can only define one such irrational, each such irrational in B must have its own unique associated infinite sequence of rationals, and which has no points in common with any other such associated sequence of rationals (apart from perhaps a single point that is a common initial point of both the left and right sequences of a complete interval of A). Hence every irrational that might be in the set B must be an endpoint of a complete interval of A where the endpoint is the limit of a sequence of rationals within that complete interval.

Contradiction

This brings us to the crux of the matter. There is a severe inconsistency in Lebesgue measure theory, since it asserts that the endpoints of a set of non-degenerate intervals, each of which has zero measure, has a total measure greater than that of the intervals between those endpoints - and which are all of 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 measure of at least 8⁄ 9 , while the intervals of set A, which includes every rational (and also many irrationals) cannot have a length greater 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 without actually addressing the actual contradiction, see, for example Fallacy by deflection.

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 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. 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:

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. 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 A and B from the overall length between a and b.

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:

(i) calculate a limiting condition for the total length of the intervals, without including any consideration of the relationships between the endpoints of the intervals

or

(ii) 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.

In case (i), you get a value of number theory: a numerical value of 1⁄ 9 .

In case (ii), you get a value of set theory: the set of points between 0 and 1.

These are two completely different types of values. To assume that the value (i) must imply the other case (ii) is absurd, and indicates a complete failure to understand limitlessness. Also note that the measure of each interval remaining after each iteration is less than 1⁄ 2 , 1⁄ 4 , 1⁄ 8 , … and which has the limiting value of 0.

You can also see a formal paper on some of the problems of calculating the measure of some 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:

R 1 − L 1 + R 2 − L 2 + R 3 − L 3 + …

where L 1 and R 1 are the left and right endpoints of the first interval, L 2 and R 2 are the left and right endpoints of the second interval, L 3 and R 3 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:

1⁄ 10 + 1⁄ 100 + 1⁄ 1000 + 1⁄ 10000 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 .

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). But there are infinitely many ways in which we can order the addition of the intervals, and in fact, as previously noted, we can define the set A in a way that does not specify any order of addition of lengths at all, see below A definition without iterations.

Furthermore, the two endpoints of any given interval do not need to appear consecutively in any such ordering. For example, if L 1 , R 1 , L 2 , R 2 , L 3 , R 3 , L 4 , R 4 are in ascending order, then the total length of a finite number of such intervals (L 1 , R 1 ), (L 2 , R 2 ), (L 3 , R 3 ), (L 4 , R 4 ) can be given by R 4 − L 1 + R 3 − L 2 + R 1 − L 2 + R 2 − L 3 + R 3 − L 4 .

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: Fallacy by deflection for an argument that the measure of the set A must be 1⁄ 9 .

A definition without iterations It should be noted that while the above iterative definition of the set A is a fairly informal definition, we can formally define it without any reference to iteration by: r ∈ A ⇔ {∃n ∈ N, n > 0, ∧ [q(n) – 1/(2·10n) < r < q(n) + 1/(2·10n)]} where q(n) is a function that lists the rational numbers between 0 and 1, defined for n > 0. Any number that satisfies the definition must be in the set A. And, as stated above, for any given q(n) each left and right endpoint is defined as not included in the interval for that q(n). Hence if there could be any number in the interval 0 to 1 and not in the set A, it would have to be in a closed interval whose endpoints are rational numbers – which is impossible - since every rational is the midpoint of some interval.

A specific listing of rational numbers Some people have suggested that they can circumvent the contradiction by using enumerations (see also One-to-one correspondences and Listing the rationals) of the rationals that are defined in terms of various conditional requirements, which render the enumeration and the sequence of intervals interdependent. Rather than trying to construct a set of rules as to which enumerations are applicable, all that is required is one specific enumeration. We can define that the set A is to be given by one specific enumeration using the pattern of rationals: 1⁄ 2 1⁄ 3 1⁄ 4 1⁄ 5 1⁄ 6 … 2⁄ 3 2⁄ 4 2⁄ 5 2⁄ 6 … 3⁄ 4 3⁄ 5 3⁄ 6 … 4⁄ 5 4⁄ 6 … 5⁄ 6 …

We go through this pattern, leaving out any duplicates, which gives the first terms of the enumeration as 1⁄ 2, 1⁄ 3, 2⁄ 3, 1⁄ 4, 3⁄ 4 ,1⁄ 5, 2⁄ 5, 3⁄ 5, 4⁄ 5, 1⁄ 6, 5⁄ 6 , … Given this enumeration, either there are no points in the interval 0 to 1 that are not in the set A, or if there could be, then as demonstrated above, there could not be ‘more’ of such points than in the set A, since each such point would be associated with a set of infinitely many points in A - this renders Lebesgue measure contradictory. 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.