Difference between revisions of "Non-Borel set"

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A '''non-Borel set''' is a [[set]] that cannot be obtained from ''simple'' sets by taking [[complement_(set theory)|complements]] and [[countable set|at most countable]] [[union_(set theory)|unions]] and [[intersection_(set theory)|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 [[interval (mathematics)|intervals]]. All Borel sets are [[measurable set|measurable]], moreover, [[universally measurable]]; however, some universally measurable sets are not Borel.
An example of a non-Borel set, due to [[Nikolai_Luzin|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==
==The example==
Every [[irrational number]] has a unique representation by a [[continued fraction]]
Every [[irrational number]] has a unique representation by a [[continued fraction]]
:<math>x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\,}}}} </math>
:<math>x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\,}}}} </math>
where <math>a_0\,</math> is some [[integer]] and all the other numbers <math>a_k\,</math> are ''positive'' integers. Let <math>A\,</math> be the set of all irrational numbers that correspond to sequences <math>(a_0,a_1,\dots)\,</math> with the following property: there exists an infinite [[subsequence]] <math>(a_{k_0},a_{k_1},\dots)\,</math> such that each element is a [[divisor]] of the next element. This set <math>A\,</math> is not Borel. For more details see [[descriptive set theory]] and the book by [[Alexander_S._Kechris|Kechris]], especially Exercise (27.2) on page 209, Definition (22.9) on page 169, and Exercise (3.4)(ii) on page 14.
where <math>a_0\,</math> is some [[integer]] and all the other numbers <math>a_k\,</math> are ''positive'' integers. Let <math>A\,</math> be the set of all irrational numbers that correspond to sequences <math>(a_0,a_1,\dots)\,</math> with the following property: there exists an infinite [[subsequence]] <math>(a_{k_0},a_{k_1},\dots)\,</math> such that each element is a [[divisor]] of the next element. This set <math>A\,</math> is not Borel. For more details see [[descriptive set theory]] and the book by [[Alexander_S._Kechris|Kechris]], especially Exercise (27.2) on page 209, Definition (22.9) on page 169, and Exercise (3.4)(ii) on page 14.

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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

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. 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.