Littlewood’s Three Principles

We may summarize the properties of measurable sets and measurable functions by the following three principles, which give an intuitive understanding.

Littlewood’s Three Principles

The following principles hold:

1. Every (measurable) set is nearly a finite union of intervals.
2. Every (measurable) function is nearly continuous.
3. Every convergent sequence is nearly uniformly convergent.

A precise description of the third principle is given by the Egorov’s theorem:

Egorov's Theorem

Suppose is a sequence of measurable functions defined on a measurable set with , and assume that pointwisely a.e. on . Given , we can find a closed set such that and uniformly on .

Lusin's Theorem

Suppose is measurable and finite valued on with . Given , we can find a closed set such that and is continuous on .

Proof By the theorem, let be a sequence of step functions so that a.e. Then we can find sets so that and is continuous outside

Although there exist Lebesgue measurable sets that are not Borel sets, you are unlikely to encounter one. Similarly, a Lebesgue measurable function that is not Borel measurable is unlikely to arise in anything you do.