User:Boris Tsirelson/Sandbox1: Difference between revisions

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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.
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The [[Heisenberg Uncertainty Principle|Heisenberg uncertainty principle]] for a particle does not allow a state in which the particle is simultaneously at a definite location and has also a definite momentum. Instead the particle has a range of momentum and spread in location attributable to quantum fluctuations.


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]].
An uncertainty principle applies to most of quantum mechanical operators that do not commute (specifically, to every pair of operators whose commutator is a non-zero scalar operator).
 
==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. (In fact, it is analytic, and complete in the class of analytic sets.) 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.
 
==References==
*A. S. Kechris, ''Classical Descriptive Set Theory'', Springer-Verlag, 1995 (Graduate texts in Math., vol. 156).

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The Heisenberg uncertainty principle for a particle does not allow a state in which the particle is simultaneously at a definite location and has also a definite momentum. Instead the particle has a range of momentum and spread in location attributable to quantum fluctuations.

An uncertainty principle applies to most of quantum mechanical operators that do not commute (specifically, to every pair of operators whose commutator is a non-zero scalar operator).

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