Lebesgue Measure

Page last updated 18 Nov 2024

Henri Lebesgue

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

Given this definition, there are only three possibilities. Either:

1. The entire closed interval between 0 and 1 is covered by the intervals, or
2. 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), or
3. The definition is irredeemably inherently contradictory, and is not a valid definition of two complementary sets of points.

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

If it is 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:

Some people, when faced with this unpalatable contradiction, attempt to evade the contradiction by asserting that there must be other points in the set B as well as those single points that are between two intervals of A. While it is true that in certain particular cases there can be singularity endpoints of one set where it is impossible to define a specific interval of the complementary set which has that point as an endpoint, but even if that were the case here, there could not be sufficiently many such points to make any difference to the measure of the set B. This is covered in more detail below in The converging sequence plea. Despite my requests to these people to provide a logical argument to back up their claims, they simply resort to hand-waving rather than any reasoned argument.

Contradiction

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.

Further absurdity

While the case above is over a unit interval, it is a simple matter to scale everything by any given multiple n as a natural number. We let the new overall interval be n times the previous case, so it is now [0, n]. In the new sequence of intervals, each interval is now n times the previous size, hence the sequence is now ⁿ⁄₁₀ , ⁿ⁄₁₀₀ , ⁿ⁄₁₀₀₀ …. For the new enumeration, each number in the enumeration is obtained by multiplying the corresponding number of the previous enumeration by n.

Hence everything is simply scaled by n. Hence, according to Lebesgue measure, the measure of the points in A cannot be more than ⁿ⁄₉ , and the measure of all points not in A must be at least ⁸ⁿ⁄₉ , in other words, the measure of all points not in A has been multiplied by n. But the scaling simply means that any single irrational points that were not in the original set A have only been repositioned to a new position (the new position is given by multiplying the previous position by n). By the rules of set theory the set of points of the interval between 0 and 1 is equinumerous with any interval between 0 and m, for any natural number m. Yet the claim is that, according to Lebesgue measure, these points now somehow constitute a measure that is n times the size of the original measure of at least ⁸⁄₉ (but less than 1).

For example if n =1,000,000, these points, simply because they have moved to a different location, according to Lebesgue measure now constitute a measure that is at least ^8,000,000⁄₉ instead of the previous value of between ⁸⁄₉ and 1. So where does the extra measure of 888,888 units come from? Do the points somehow magically become a million times wider? And simply by choosing a different n, we can change the Lebesgue “measure” of this set of single isolated points by as much as we want.

This nonsensical result is a stark demonstration of the inherent absurdity of Lebesgue measure.

Welcome to a fantasy land where hocus-pocus takes precedence over logic.

Attempts to evade the contradictions

Some people appear to have some difficulty accepting that Lebesgue theory results in a contradiction. As shown above, the only possibility is that the set A contains only non-degenerate intervals, and the other set B contains only single point degenerate intervals, and there are no points common to both.

Misunderstandings of this topic almost invariably involve an avoidance of any consideration 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 PDF On Smith-Volterra-Cantor sets and their measure which explains how all contradictions can be avoided by a logical analysis of the definitions involved.

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 !

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.)

1. For any set of single degenerate points that is denumerable, the Lebesgue measure of that set is zero.
2. 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.)
3. 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 an attempt to avoid contradictions when Lebesgue theory is used along with set theory and the axiom of choice, then it has to be claimed that 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. See also The Axiom of Choice.)

The axioms of Lebesgue theory of measure are based around the requirement that if, for a given interval, any set A is defined for that interval, then the sum of the Lebesgue measures of the that set and its complement (the set of elements not in A, see complement) 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.

Limits and Limitlessness

If one does not believe that the definition of sets A and B are inherently and irredeemably contradictory, then one must provide a means of arriving at a satisfactory result. The problems in the Lebesgue reading 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, some people appear to fail to take into account the fact that, whatever finite number n you choose, and then consider that n rationals have been assigned as midpoints of intervals, there are, not a billion times as many rationals not so assigned, nor a billion to the power of a billion times as many rationals not so assigned, nor a googleplex times as many rationals not so assigned, nor Graham’s number times as many rationals not so assigned, there will be infinitely many times that quantity of rationals still to be assigned an interval with its midpoint as that rational. Regardless of however large your n is, there will always be infinitely many times that quantity of rationals still to be assigned an interval with its midpoint as that rational. In other words, you never “run out” of rationals onto which a non-degenerate interval can be attached. That is why there can be two different limits that can be associated with the definition - you can either:

1. calculate a limiting condition for the total length of intervals of decreasing size, without including any consideration of the relationships between the endpoints of the intervals
    or
2. 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 limit of the summation of lengths of 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) indicates a complete failure to understand limitlessness. Also note that the measure of the remaining intervals are decreasing, and there 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 PDF On Smith-Volterra-Cantor sets and their measure. For more on set theory, see the pages that give an overview of set theory, starting at Overview of set theory: Part 1: Different types of set theories.

An Alternative basis for Calculating Measure

If we reject the notion that a set of points can have a measure even if both that set and its complement do not define any non-degenerate intervals, then we can have a valid basis for the determination of the measure of a set, by specifying which sets have properties that contribute to a valid determination of measure, as follows:

A definition of a set of points A is valid for the calculation of a measure if and only if every point of the set A is either a point of a non-degenerate interval of that set or else it is the endpoint of the open end of an interval of the complement of the set A.

Note that a point that is an endpoint that is the open endpoint of an interval necessarily implies that that interval is non-degenerate, since:

A degenerate hypothetical interval (x, x) that is open at both ends is contradictory, since x is defined to be both the left endpoint of the interval and not in the interval, and also the right endpoint of the interval and not in the interval. Hence there can be no points in this hypothetical interval, and such an interval does not exist.

Similarly, the hypothetical degenerate semi-open (Footnote: An interval that has one endpoint that is in the interval and one endpoint that is not in the interval; also can be called semi-closed.) interval (x, x] is contradictory, since on the one hand x is defined to not be within the interval (since it must be the left endpoint of an interval where the left endpoint is not within that interval), while on the other hand it is defined to be within the interval (since it must be the right endpoint of an interval where the right endpoint is within that interval). Hence such an interval does not exist. The hypothetical degenerate semi-open interval [x, x) is similarly contradictory, and such an interval does not exist.

Note also that a sequence of nested intervals that are all semi-open do not have a valid limit state since, as shown above, the hypothetical interval (x, x] or [x, x) is contradictory, and such an interval does not exist.

If we take as an example, the set of all rationals, or the set of all irrationals, then since every point is a degenerate closed interval, no point can be the open endpoint of an interval of the complementary set, since all the points of the complementary set are all also degenerate closed intervals.

Different orders of summation

In the assertion that the set A has a measure of ¹⁄₉ it is assumed that it is 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:

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 . We can also note that if we start at zero, the sequence of midpoints is ¹⁄₂₀, ²¹⁄₂₀₀, ²²¹⁄₂₀₀₀ … and the limiting value of these midpoint values is also ¹⁄₉. (Footnote: 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 such simple limiting summation value. Unlike the case above where the midpoints of the sequence converge neatly towards a single point, any enumeration of the rationals over a given interval must always zig-zag across that interval, and can never converge to a single point. For any finite n in the recursive process there always remain infinitely many rationals that have not been covered by any interval, it is also the fact that those remaining rationals never fall into a tidy sequence that converges towards a single point.

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 you can take a specific case such as the limit of the adding the interval 0 to ¹⁄₁₀ , then the interval ¹⁄₁₀ to ¹¹⁄₁₀₀ , then ¹¹⁄₁₀₀ to ¹¹¹⁄₁₀₀₀ , and so on, where the intervals follow on directly adjacent to each other and converge to one single limiting point, and apply that same assumption to a completely different scenario. The history of mathematics is not short of errors that resulted from the assumption that a general case follows from a specific case. And, in terms of set membership, 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, in the application of a limit, there is no particular order of incrementation of intervals. That is, for any point that is not in A, that point must be obtained by a limit, and for multiple such points, it is not the case that a limit is “reached” for one point, and then, subsequently, a limit is “reached” for another point, and then another. No, in the application of a limit state, there is no particular order of “reaching” the limit points, and hence there is no particular order of summation and subtraction of the endpoints, and one cannot assume that one can apply the simplistic case to this scenario. When applying the limit, that application of a limit does not of itself entail any enumeration of the individual limit points. The assertion that there can be a non-contradictory result where the total length of the set A can be no more than 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.⁠ (Footnote: A reader suggested that there is more than one way of examining measure in order to see the full picture, using the analogy of blindfolded men examining an elephant. But the measure of a simple interval between two points is, very simply, the difference of their values. There’s no nuance involved. And if Platonism were correct then, any set of points must either have a total measure that “exists” independently of any method of calculation, and which corresponds to such a simple interval, or else there is no measure of that set that is an intrinsic property of measure for the set and which “exists” independently of any method of calculation.)

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. The set of rational points, as in section Analyzing the Lebesgue result above that constitute the lowest enumeration numbers of the complete intervals of A is a denumerable set, and hence the associated irrational endpoints of the complete intervals also constitute a denumerable set. I have pointed out that there is association between the points of the complement of A and a set of certain rationals of A, as detailed in section Analyzing the Lebesgue result above (the lowest enumeration numbers of the complete intervals of A). That set of rationals is a denumerable set, and that implies that the associated irrational endpoints of the complete intervals also constitute a denumerable set. However, the indication is that it may be impossible to define an enumeration function that gives the set of all those associations from the definition of the set A. But as I indicate above, that does not imply that a definition of that enumeration function is impossible. I have since updated this 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.