Littlewood’s Three Principles (1/3)

When I first studied measure theory, I felt like I could never quite wrap my head around what a measurable set or function looked like. After all, the Borel Heirarchy is huge, and what the hell does a F_{\sigma\delta\sigma\sigma\delta\sigma} set look like anyway? In his 1944 text “Lectures on the Theory of Functions”, John Littlewood, a professor of mathematics at Cambridge, sought to alleviate similar confusions.

Littlewood’s three principles provide useful intuition in studying measure theory, and I’d like to spend the next three posts (Monday, Wednesday, and Friday of this week) to exposit them.

“The extent of knowledge [of real analysis] required is nothing like as great as is sometimes supposed. There are three principles, roughly expressible in the following terms:

  1. Every set is nearly a finite sum of intervals.
  2. Every function is nearly continuous.
  3. Every convergent sequence is nearly uniformly convergent.”

- John Littlewood

Of course, the sets and functions referred to above are assumed to be measurable. This post, and the two following it, will make these statements precise.

Let’s agree that for the rest of this post, that \mu is a Lebesgue-Stieltjes measure on \mathbb{R}. The domain of \mu is denoted \mathcal{M}_\mu. We’ll begin with two short lemma, whose proofs are omitted:

Lemma 1. If U\subseteq \mathbb{R} is open and non-empty, then U is a countable union of disjoint open intervals.

The proof of this actually makes for a good exercise. Hint: Pick a point in U and find an interval around it (why can you do this?). Try to extend that interval as much as you can inside of U. Deduce that U is a disjoint union of intervals. How can you index the collection of intervals to show that there are countably many?

Lemma 2. If E\in M_{\mu}, then

\mu(E)=\inf\{\mu(U): U\supset E\text{ and }U \text{is open}\}

The proof of this can be found in Folland (Page 34). The conclusion is actually stronger than this, but this is all I need to proceed. A word of caution: it’s rather late, so don’t hold me to my epsilons.

Theorem (Littlewood’s First Principle) Every finite measurable set is nearly a finite sum of intervals. That is, if E\in M_{\mu}, and \mu(E)<\infty, then for every \epsilon>0 there is a set A that is a finite union of open intervals such that \mu(E\triangle A)<\epsilon.

Proof.  Suppose E\in M_\mu and let \epsilon>0.  By lemma 2, there is an open set U\subset \mathbb{R} so that E\subset U and \mu(U)< \mu(E)+\frac{\epsilon}{2}. This shows in particular that \mu(U)<\infty. By a lemma 1, there exists a countable family of disjoint open intervals \{(a_n,b_n): n\in \mathbb{N}\} so that U=\bigcup_{n=1}^{\infty}(a_n,b_n). Furthermore, by the countable additivity of \mu, we see that

\mu\left({U}\right)=\mu\left({\bigcup_{n=1}^{\infty}(a_n,b_n)}\right)=\sum_{n=1}^{\infty}\mu\left({(a_n,b_n)}\right)

This shows that \mu(U)=\lim\limits_{k\to \infty}\sum_{n=1}^{k}\mu\left((a_n,b_n)\right) and so we may find N sufficiently large so that \mu(U)-\sum_{n=1}^{N}\mu\left({(a_n,b_n)}\right)<\frac \epsilon 2.

Let A=\bigcup_{n=1}^{N}(a_j,b_j). Then A\subset U and A is clearly open and certainly an element of M_\mu. Since \mu(E) and \mu(U) are finite and A\subset U it follows that

\mu(A\setminus E)\le \mu(U\setminus E)=\mu(U)-\mu(E)<\frac \epsilon 2

Moreover, since E\subseteq U, we know that E\setminus A\subseteq U\setminus A and so it follows similarly that

\mu(E\setminus A)\le \mu(U\setminus A)\le\mu(U)-\mu(A)=\mu(U)-\sum_{n=1}^{N}\mu\left({(a_j,b_j)}\right)<\frac \epsilon 2

Since E\setminus A and A\setminus E are disjoint, it follows from the disjoint additivity of \mu that

\mu(E\triangle A)=\mu((E\setminus A)\cup (A\setminus E))=\mu(E\setminus A)+\mu(A\setminus E)<\frac \epsilon 2+\frac \epsilon 2=\epsilon

Although the latter principles are more striking, Littlewood’s first principle is of immense help when thinking about measurable sets. Thanks for reading.

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About Adam Azzam

This Fall I will be a first year mathematics PhD student at UCLA. I enjoy doing analysis - particularly of the functional variety.
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4 Responses to Littlewood’s Three Principles (1/3)

  1. Pingback: Littlewood’s Three Principles (2/3) | whateversuitsyourboat

  2. Ryan says:

    This proposition is really neat. Considering objects like the generalized Cantor sets that are nowhere dense (and hence contain no intervals) but can have positive Lebesgue measure, one might think that measurable sets (of finite measure) can be very strange, but in fact they really are fairly close to nice, finite collections of open intervals. Very cool!

  3. Pingback: Littlewood’s Three Principles (3/3) | whateversuitsyourboat

  4. Pingback: Littlewood’s Three Principles (3/3) | Alexander Adam Azzam

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