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 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:
- Every set is nearly a finite sum of intervals.
- Every function is nearly continuous.
- 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 is a Lebesgue-Stieltjes measure on . The domain of is denoted . We’ll begin with two short lemma, whose proofs are omitted:
Lemma 1. If is open and non-empty, then is a countable union of disjoint open intervals.
The proof of this actually makes for a good exercise. Hint: Pick a point in and find an interval around it (why can you do this?). Try to extend that interval as much as you can inside of . Deduce that is a disjoint union of intervals. How can you index the collection of intervals to show that there are countably many?
Lemma 2. If , then
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 , and , then for every there is a set that is a finite union of open intervals such that .
Proof. Suppose and let . By lemma 2, there is an open set so that and . This shows in particular that . By a lemma 1, there exists a countable family of disjoint open intervals so that . Furthermore, by the countable additivity of , we see that
This shows that and so we may find sufficiently large so that .
Let . Then and is clearly open and certainly an element of . Since and are finite and it follows that
Moreover, since , we know that and so it follows similarly that
Since and are disjoint, it follows from the disjoint additivity of that
Although the latter principles are more striking, Littlewood’s first principle is of immense help when thinking about measurable sets. Thanks for reading.