# 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

where is some integer and all the other numbers are positive integers. Let be the set of all irrational numbers that correspond to sequences with the following property: there exists an infinite subsequence such that each element is a divisor of the next element. This set 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.)