a Lebesgue measurable but non-Borel set

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

$$x=a_0+\frac{1}{a_1+{\displaystyle \frac{1}{a_2+{\displaystyle \frac{1}{\mathrm{\ddots }}}}}}$$

such that there exists an infinite sequence 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 (http://planetmath.org/AnalyticSet2), meaning that it is the image of a continuous function $f:X\to ℝ$ for some Polish space $X$ and, consequently, $S$ is a universally measurable set.

This example is due to Lusin (1927).