Measure Theory/Measuring Sets of Real Numbers

Measuring Sets of Real Numbers

Things are about to go from bad to worse.

We just learned that "the swaparoo",

$\lim _{m\to \infty }\int _{a}^{b}g_{m}(x)\ dx=\int _{a}^{b}\lim _{m\to \infty }g_{m}(x)\ dx$

is not generally valid.

We learned that simply putting conditions on the sequence of functions, although moderately successful, is a quixotic but failed project.

We also learned that Lebesgue proposed a new system of integration, which begins by partitioning the range, rather than the domain. However, this now requires defining a system of measuring sets of real numbers.

In this lesson we are about to find out that this, tragically, is also a sort of failed project!

Stay strong, though. There will be a redemption arc to the story, later.

The Length Function

To repeat what we are after, we would like to be able to take any set $A\subseteq {\mathbb {R}}$ and assign to it a number, which should intuitively capture the idea of its "measurement".

This measurement of sets of real numbers should be consistent with the idea that the measure of an interval is its length. That is to say, let $a,b\in {\mathbb {R}}$ and $a\leq b$, and I any interval with bounds at a and b. Then the length of I is the positive difference of its end-points, which we write as

$\ell (I)=b-a$

Note that some intervals are infinite in length, like $(0,\infty )$. Therefore it is convenient to introduce the extended real numbers,

${\mathbb {R}}^{*}={\mathbb {R}}\sqcup \{-\infty ,\infty \}$

That way, we can say that $\ell$ is well-defined for every interval and not just the bounded intervals.

If, in particular, we write $({\mathbb {R}}^{*})^{\geq 0}$ for the non-negative extended real numbers, then this is the range of the function $\ell$.

If we write ${\mathcal {I}}$ as the set of all intervals of real numbers, then

$\ell :{\mathcal {I}}\to ({\mathbb {R}}^{*})^{\geq 0}$

Note that $\ell ((0,1]\sqcup (2,3))$ is undefined because $(0,1]\sqcup (2,3)\notin {\mathcal {I}}$.

The Length Measure Function

It is the job of the "length measure function", which we will write as $\lambda$, to extend the concept of $\ell$. That is to say, $\lambda$ is the "generalized length" or "length measure", so that we can talk about the measure of every set.

$\lambda :{\mathcal {P}}({\mathbb {R}})\to ({\mathbb {R}}^{*})^{\geq 0}$

Still, though, we would like the length-measure function, $\lambda$, to agree with the length function $\ell$ on the intervals. That is to say, we would still like $\lambda ((a,b))=b-a$, just as with $\ell$. That is to say, we would like

$\lambda |_{\mathcal {I}}=\ell$

But also we would like to be able to "combine" intervals and measure the result. That is to say, we would like $\lambda ((0,1)\sqcup (2,3))=1+1=2$. We call this property of a measure "additivity".

More generally, if sets $A,B\subseteq {\mathbb {R}}$ are any disjoint sets, then we expect $\lambda (A\sqcup B)=\lambda (A)+\lambda (B)$.

Exercise 2. Not Disjoint Set $E=[0,2]\cup [1,3]$ and find $\lambda (E)$. What about $\ell (E)$?

Summary of What's to Come

In the next lesson, we will see a proof that this is not possible. That is to say, there is no function $\lambda$ which meets all the conditions that we have set for it.

After that, we learn how to reset our goals slightly, in order to still carry out the integration project.

The rest of the first section is dedicated to understanding the definition of the length measure function, after we have appropriately reset our requirements for it.

What results is commonly known as the Lebesgue measure, although I will persist in calling it "length measure".