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}+\cfrac{1}{a_{1}+\cfrac{1}{a_{2}+\cfrac{1}{\ddots}}}

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 (http://planetmath.org/AnalyticSet2), 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).

Title	a Lebesgue measurable but non-Borel set
Canonical name	ALebesgueMeasurableButNonBorelSet
Date of creation	2013-03-22 18:37:01
Last modified on	2013-03-22 18:37:01
Owner	gel (22282)
Last modified by	gel (22282)
Numerical id	6
Author	gel (22282)
Entry type	Theorem
Classification	msc 28A05
Classification	msc 28A20