Measure Theory/Measurable Functions, Length-integral, and the Need for Approximations

Integration is Hard

The main thing that I want to communicate in this lesson, is that integration in this system is hard.

Consider trying to compute something which should be simple, like $\int x^{2}\mathbf {1} _{[0,1]}$.

It is easy enough to prove that $\int x^{2}\mathbf {1} _{[0,1]}\geq 1/2$. One can do this by building a sequence of simple functions, which strongly resemble what one computes for Riemann integration.

$\psi _{n}(x)=\sum _{i=1}^{n}c_{i}\mathbf {1} _{[(i-1)/n,i/n]}$

Exercise 1. Follow the Simple Integrals

Find $\int \psi _{n}$ and then prove that the limit $\lim _{n\to \infty }\int \psi _{n}=1/2$.

Then explain why this shows that $\int x^{2}\mathbf {1} _{[0,1]}\geq 1/2$.

Resuming

The hard part is the reverse inequality, trying to show that $\int x^{2}\mathbf {1} _{[0,1]}\leq 1/2$.

In order to do this we need to show that 1/2 is an upper bound on the set ${\mathfrak {s}}$ of simple integral approximations.

But this requires considering every possible collection of measurable sets, and that space of objects is just too intractable.

Therefore, approaching integration directly is just too hard. Instead, however, we may use the ubiquitous strategy of analysis: Approximate these objects and then let the approximations go to zero.

Therefore this chapter is devoted to finding approximation theorems for sets and functions. These approximations will allow us to more easily approximate, and then precisely calculate, integrals.

Littlewood's Three Principles

J.E. Littlewood summarized the three core approximation results wonderfully when he said

There are three principles, roughly expressible in the following terms: Every (measurable) set is nearly a finite union of intervals; every [measurable] function is nearly continuous; every convergent sequence of [measurable] functions is nearly uniformly convergent.

In each case, "nearly" means that the statement is true except on a small set. A "small set" is "as small as you like", in the same way that the limit of a sequence is a number "as close as you like" to the tail of the sequence.

To demonstrate: When we say that every measurable set is nearly a finite union of open intervals, we mean the following.

Let $E\in {\mathcal {M}}$ be any measurable set and let $\varepsilon \in {\mathbb {R}}^{+}$ be any positive real number.

Then there exists a collection of open intervals $I_{1},\dots ,I_{n}$ such that

$\lambda \left(E\Delta \bigcup _{i=1}^{n}I_{i}\right)<\varepsilon$

So you can choose $\varepsilon$ "as small as you like" and then E and $\bigcup _{i=1}^{n}I_{i}$ are equal, except for $E\Delta \bigcup _{i=1}^{n}I_{i}$, which is small.