We give an example of a subset of the real numbers which is Lebesgue measurable, but not Borel measurable.

Let $S$ consist of the set of all irrational real numbers with continued fraction of the form \begin{equation*} x=a_0+\sfrac{1}{a_1+\sfrac{1}{a_2+\sfrac{1}{\ddots}}} \end{equation*}such that there exists an infinite sequence $0<i_1<i_2<\cdots$ where each $a_{i_k}$ divides $a_{i_{k+1}}$ . It can be shown that this set is Lebesgue measurable, but not Borel measurable.

In fact, it can be shown that $S$ is an analytic set, meaning that it is the image of a continuous function $f\colon X\rightarrow\mathbb{R}$ for some Polish space $X$ and, consequently, $S$ is a universally measurable set.

This example is due to Lusin (1927).