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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 given (rather, its existence can be proved), see non-measurable set.

The example

Every irrational number has a unique representation by a continued fraction

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\,}}}} }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_0\,} is some integer and all the other numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_k\,} are positive integers. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\,} be the set of all irrational numbers that correspond to sequences Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (a_{0},a_{1},\dots )\,} with the following property: there exists an infinite subsequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_{k_0},a_{k_1},\dots)\,} such that each element is a divisor of the next element. This set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.

References

• A. S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, 1995 (Graduate texts in Math., vol. 156).