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 set cannot be given (rather, its existence can be proved), see non-measurable set.

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. (In fact, it is analytic, and complete in the class of analytic sets.) For more details see descriptive set theory and the book by Kechris, especially Exercise (27.2) on page 209, Definition (22.9) on page 169, and Exercise (3.4)(ii) on page 14.