# Difference between revisions of "Non-Borel set"

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A '''non-Borel set''' is a | 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 set|measurable]], moreover, [[universally measurable]]; however, some universally measurable sets are not Borel. | ||

An example of a non-Borel set, due to [[ | An example of a non-Borel set, due to [[Nikolai Luzin|Lusin]], is described below. In contrast, an example of a non-measurable non-Borel set can only be proved to exist, but it cannot be constructed (because the existence see [[non-measurable set|non-measurable sets]] is not constructive). | ||

==The example== | ==The example== | ||

Every | 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 | 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. | ||

While the construction of this set is elementary, the proof that it indeed is not a Borel set | |||

requires advanced results from [[descriptive set theory]]. | |||

(The result follows since the set is analytic, and complete in the class of analytic sets.) |

## Latest revision as of 10:44, 2 December 2010

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 non-Borel set can only be proved to exist, but it cannot be constructed (because the existence see non-measurable sets is not constructive).

## 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.

While the construction of this set is elementary, the proof that it indeed is not a Borel set requires advanced results from descriptive set theory. (The result follows since the set is analytic, and complete in the class of analytic sets.)