Non-Borel Sets
An example of a subset of the reals which is non-Borel, due to Lusin (see Sect. 62, pages 76–78), is described below. In contrast, an example of a non-measurable set cannot be exhibited, though its existence can be proved.
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 of an infinite parity function . However, this is a proof of existence (via the choice axiom), not an explicit example.
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