Non-Borel set

A non-Borel set is a set that cannot be obtained from simple sets by taking complements and at most countable unions and intersections. (For the definition see Borel set.) Only sets of real numbers are considered in this article. Accordingly, by simple sets one may mean just intervals. All Borel sets are measurable, moreover, universally measurable; however, some universally measurable sets are not Borel.

An example of a non-Borel set, due to Lusin, is described below. In contrast, an example of a non-measurable non-Borel set can only be proved to exist, but it cannot be constructed (because the existence see non-measurable sets is not constructive).

The example

Every irrational number has a unique representation by a continued fraction

$x=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+{\cfrac {1}{\ddots \,}}}}}}}}$

where $a_{0}\,$ is some integer and all the other numbers $a_{k}\,$ are positive integers. Let $A\,$ be the set of all irrational numbers that correspond to sequences $(a_{0},a_{1},\dots )\,$ with the following property: there exists an infinite subsequence $(a_{k_{0}},a_{k_{1}},\dots )\,$ such that each element is a divisor of the next element. This set $A\,$ is not Borel.

While the construction of this set is elementary, the proof that it indeed is not a Borel set requires advanced results from descriptive set theory. (The result follows since the set is analytic, and complete in the class of analytic sets.)