Lebesgue Measure

Lebesgue measure is a measure of the length of a set of points, also called the $n$-dimensional "volume" of the set. So, intuitively:

• 1-dimensional volume is the length of a line segment,
• 2-dimensional volume is the area of a surface,
• 3-dimensional volume is the volume that we know of
• and so on.

These measures are defined over the space of real numbers $\mathbb{R}$, as opposed to, say, the space of natural numbers $\mathbb{N}$.

The basics

• Lebesgue measure of a set of real numbers between $[a, b]$ is $b-a$. This is consistent with our intuitive understanding of the length of a line segment.
• Lebesgue measures are finitely additive. So the measure of a union of the set of points $[a, b]$ and $[c, d]$ is $(b-a) + (d-c)$. I.e. $\text{measure}([a,b]\bigcup[c,d])=(b-a)+(d-c)$. This is again consistent with our intuitive understanding of the sum of two line segments.
• In fact $$\text{measure}(\bigcup_{i=1}^{\infty} A_i)=\sum_{i=1}^{\infty}\text{measure}(A_i)$$ The above applies to any $n$-dimensional measure (length, area, volume, etc.) This is consistent with our intuitive understanding of the sum of multiple line segments, areas, volumes, etc.

Those were the basics. So far so good. We now look at Lebesgue measure for:

• countable sets (set of natural numbers, for example),
• uncountable sets (set of real numbers) and
• some seeming bizarre sets for which a Lebesgue measure cannot be defined.

Lebesgue measure for countable sets

• Lebesgue measure of integer points on the real line is $0$. This is because points are just locations on a real line and intuitively, integers are few and far between on a real line.
• Lebesgue measure of a countable set is $0$. This is a stronger statement than the above. It is interesting, since it implies that a set containing countably infinite number of points is also $0$. This is not surprising though, since between any two distinct integer points on the real line lie uncountably many real points. So intuitively, a countable set of points is just not large enough to have a non-zero Lebesgue measure!
• One implication of the above is that rational numbers have a measure of $0$. This is in spite of the fact that rationals are dense - i.e. between any 2 rational numbers, there are countably infinite number of rationals! So, countable infinity is no match for Lebesgue measure.

Lebesgue measure for uncountable sets

• As hinted earlier, Lebesgue measure for an uncountable number of contiguous points is non-zero. So for the set of points $[a, b]$, this is $b-a$.
• Removing a countable number of points from an uncountable set does not change its measure. E.g the measure of the half open set $(a, b]$ (which excludes just the point $a$) or the full open set $(a, b)$, which excludes both points $a$ and $b$ is $b-a$.
• One implication of the above is that if we were to remove, say, all rational numbers from the set $[a, b]$, its measure is still $b-a$. So we removed all countably infinite number of rationals, and yet the measure doesn't budge. Cool!
• Lebesgue measure of an uncountable set need not always be non-zero. Georg Cantor gave the construction of an uncountable set, called the Cantor set, which has uncountably infinite points, and yet has a measure of $0$, as shown in the next section.

Cantor Set - an uncountable set with 0 measure

1. Start with the set of points $\mathbb{R}[0, 1]$.
2. Divide it into 3 parts - $[0, 1/3]$, $(1/3, 2/3)$, $[2/3, 1]$ and remove the middle part $(1/3, 2/3)$ from the set, as depicted in the diagram below. (Note: the first line in the diagram represents $\mathbb{R}[0,1]$ and those below it represent the construction of the Cantor set as detailed in this and subsequent steps.)

3. Of the remaining parts, divide each into 3 parts, as shown in the diagram above. E.g.
1. Divide $[0, 1/3]$ into $[0, 1/9]$, $(1/9, 2/9)$, $[2/9, 1/3]$ and again remove the middle part $(1/9, 2/9)$ from the set.
2. Similarly, divide $[2/3, 1]$ into $[2/3, 7/9]$, $(7/9, 8/9)$ and $[8/9, 1]$ and remove the middle part, $(7/9, 8/9)$ from the set.
4. Keep repeating the above step, as shown in the diagram above.

• The segments that we end up removing have a combined measure of $$\frac{1}{3}+\frac{2}{9}+\frac{4}{27}+\frac{8}{81}\cdots=\frac{1}{3}\sum_{i=0}^{\infty}\bigg(\frac{2}{3}\bigg)^i=1$$
• This means that the number of points left has a Lebesgue measure of 0, even though the number of points left is uncountable!
• This is bizarre indeed. However, there is something even more bizarre. There are some sets that are not Lebesgue measurable! I.e. we cannot define a measure for them. A Vitali set is one such non-measurable set.

Vitali Set - a non-measurable set
Construction of this set relies on the Axiom of Choice and is slightly involved.

1. Definition: We define 2 points, $a$ and $b$, as equivalent if $a-b$ is a rational number. We denote this as $a\sim b$. Note that $a$ and $b$ themselves, don't have to be rationals for them to be equivalent. Examples of equivalence are:
1. $\frac{1}{2}\sim \frac{1}{3}$, since $\frac{1}{2}-\frac{1}{3}=\frac{1}{6}$, a rational number.
2. $\pi\sim\pi+\frac{1}{2}$, since $\pi-(\pi+\frac{1}{2})=-\frac{1}{2}$, a rational number.
2. We partition the set of real numbers $\mathbb{R}[0, 1]$ into equivalence classes using the above definition. All points within an equivalence class are equivalent to each other, of course.

3. Each equivalence class has countably many points, one corresponding to each rational number.
4. There are uncountably many such equivalence classes, one corresponding to each irrational number (that is not separated from another irrational number by a rational distance).
5. By axiom of choice, we can construct a set containing one point from each of these equivalence classes, which we call $V$, the Vitali set.
6. Note that since elements of $V$ are drawn from $\mathbb{R}[0,1]$, $\text{measure}(V)\leq 1$.
7. Definition: We define a translation of $V=\{v_0,v_1,v_2,\cdots\}$ called $V_q=\{v_0+q,v_1+q,v_2+q,\cdots\}$ where $q\in\mathbb{Q}[0,1]$, the set of rationals between $[0,1]$. So for each rational number $q$, we have a corresponding uncountable set $V_q$ as shown in the diagram above.
8. Note that $\text{measure}(V)=\text{measure}(V_q)\quad\forall q\in\mathbb{Q}[0,1]$, since each $V_q$ is merely a translation of $V$.
9. Also note that $$\bigcup_{q\in\mathbb{Q}[0,1]}V_q=\mathbb{R}[0,2]$$ i.e. we just partitioned $\mathbb{R}[0,2]$ into countably many partitions, $V_q, q\in\mathbb{Q}[0,1]$, with each partition containing an uncountable number of points.
10. With points (8) and (9) in mind, what is the Lebesgue measure of $V$?
1. If it is $0$, from (8), it means that each of $V_q$ has a measure of $0$. It implies, from (9), that $\text{measure}(\mathbb{R}[0,2])=0$ which is false, since $\text{measure}(\mathbb{R}[0,2])=2$.
2. If it is greater than $0$, say $a$, then from (8), each of $V_q$ has a measure of $a$. Hence, from (9), since there are infinitely many $q$s, $\text{measure}(\mathbb{R}[0,2])=\infty$ which is false, since $\text{measure}(\mathbb{R}[0,2])=2$.
11. Hence, $V$ is not Lebesgue measurable!

In closing
We saw that there are some sets with Lebesgue measure $0$ and some with non-zero. We also saw that there are some uncountable sets with measure $0$ and some sets with an undefined measure.

References

• Wapner, Leonard, The Pea and the Sun, 2005.
• The Axiom of Choice (Wikipedia)
• Cantor Set (Wikipedia)
• Vitali Set (Wikipedia)

Hyperwebster - an uncountable dictionary

Introduction

There are an infinite number of points on the real line. In fact, there are more points on any segment of the real line, no matter how small, than all of natural numbers combined. This was proved by Cantor using the diagonal argument.

To get an idea of the number of points on a real line, Dr. Ian Stewart gave the following construction of an infinite dictionary, the "hyperwebster".

Construction of the Hyperwebster

An enterprising publishing company decides to print a book containing all the words that can possibly created from the English alphabet A-Z. Since it is really a collection of letters of the alphabet in any order, of any length, we will see:

• some non-sensical words like "XWPBQI" and "NKPMZ",
• some real words like "SUN" and "MOON" and
• even some composite words like "SUNMOON" and "MOONSUN".

Since not all such words have meaning, the publishing company decides to include only the words in the book, without their associated meaning, if any. 

So the "Hyperwebster" book looks like this:

A, AA, AAA, ..., AB, ABA, ABAA, ..., AC, ..., AZ, AZA, ...

B, BA, BAA, ..., BB, BBA, BBAA, ..., BC, ..., BZ, BZA, ...

C, CA, CAA, ..., CB, CBA, CBAA, ..., CC, ..., CZ, CZA, ...

Z, ZA, ZAA, ..., ZB, ZBA, ZBAA, ..., ZC, ..., ZZ, ZZA, ...

The staff at the publishing company realizes that it can partition the words into 26 volumes, one for each letter of the alphabet. So the Hyperwebster now looks like the following:

Volume A: A, AA, AAA, ..., AB, ABA, ABAA, ..., AC, ..., AZ, AZA, ...

Volume B: B, BA, BAA, ..., BB, BBA, BBAA, ..., BC, ..., BZ, BZA, ...

Volume C: C, CA, CAA, ..., CB, CBA, CBAA, ..., CC, ..., CZ, CZA, ...

Volume Z: Z, ZA, ZAA, ..., ZB, ZBA, ZBAA, ..., ZC, ..., ZZ, ZZA, ...

Next, the staff realizes that all words in volume A start with the letter A, all words in volume B start with the letter B and so on. This means that the first letter in each word can be inferred from its volume and hence, the first letter can be dropped. Excellent! The publishing company just saved some ink by not printing an infinite number of letters.

The new volumes now look like this:

Volume A: A, AA, AAA, ..., B, BA, BAA, ..., C, ..., Z, ZA, ...

Volume B: A, AA, AAA, ..., B, BA, BAA, ..., C, ..., Z, ZA, ...

Volume C: A, AA, AAA, ..., B, BA, BAA, ..., C, ..., Z, ZA, ...

Volume Z: A, AA, AAA, ..., B, BA, BAA, ..., C, ..., Z, ZA, ...

The staff realizes that each volume now looks identical, except for the name of the volume. Why would anyone buy 26 identical copies of the same content? So, the decision is made to publish a single volume called "Hyperwebster", which looks like the following:

New hyperwebster:

A, AA, AAA, ..., B, BA, BAA, ..., C, ..., Z, ZA, ...

This turns out to be identical to the original hyperwebster that they started out with.

Original hyperwebster:

A, AA, AAA, ..., AB, ABA, ABAA, ..., AC, ..., AZ, AZA, ...

B, BA, BAA, ..., BB, BBA, BBAA, ..., BC, ..., BZ, BZA, ...

C, CA, CAA, ..., CB, CBA, CBAA, ..., CC, ..., CZ, CZA, ...

Z, ZA, ZAA, ..., ZB, ZBA, ZBAA, ..., ZC, ..., ZZ, ZZA, ...

The staff realizes that:

1. the original volume can be partitioned into 26 different volumes,
2. the first letter in each volume can be dropped, making each volume identical,
3. and each volume now is really identical to the original volume
4. and steps 1-3 can be applied ad infinitum.

The publishing company wisely abandons publishing the hyperwebster, even though each execution of steps 1-4 represent an infinite amount of savings!

Moral of the story

The content of the hyperwebster is equivalent to points on a real line (replace A-Z above with 0-9 and observe that it generates all real numbers). Any subset of the real line can be chopped up into infinitely many parts, each of which has the same number of points as the original. Each of the parts in turn can be chopped up into infinitely many subparts, each having the same number of points as the original, ad infinitum. Yeah, that's a lot of points! Continuum hypothesis states that the number of such points is aleph-1.

References

• Leonard M. Wapner, "The Pea and the Sun", 2005.