Non-Borel set: Difference between revisions

Revision as of 00:30, 19 June 2009

The example

Every irrational number has a unique representation by a continued fraction

x=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+{\cfrac {1}{\ddots \,}}}}}}}}

where $a_{0}\,$ is some integer and all the other numbers $a_{k}\,$ are positive integers. Let $A\,$ be the set of all irrational numbers that correspond to sequences $(a_{0},a_{1},\dots )\,$ with the following property: there exists an infinite subsequence $(a_{k_{0}},a_{k_{1}},\dots )\,$ such that each element is a divisor of the next element. This set $A\,$ 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.

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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==
 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>
 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.

Non-Borel set: Difference between revisions

Revision as of 00:30, 19 June 2009

The example

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