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

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).