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In mathematics, 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 exhibited (rather, its existence can be proved), see non-measurable set.
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. (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.
- Another non-Borel set is an inverse image f − 1[0] of an infinite parity function
. However, this is a proof of existence (via the choice axiom), not an explicit example.
References
- Alexander S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, 1995 (Graduate texts in Math., vol. 156).
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