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| − | ==Negative results==
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| − | $\newcommand{\M}{\mathscr M}$
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| − | As was noted, the normal form of an object $M\in\M$ is a "selected representative" from the equivalence class $[M]$, usually possessing some nice properties. The set of all these "representatives" intersect each equivalence class exactly once; such set is called a ''transversal'' (for the given equivalence relation). Existence of a transversal is ensured by the [[axiom of choice]] for arbitrary equivalence relation on arbitrary set. However, a transversal in general is far from being nice. For example, consider the equivalence relation "$x-y$ is rational" for real numbers $x,y$. Its transversal (so-called [[Non-measurable set|Vitali set]]) cannot be Lebesgue measurable!
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| − | Typically, the set $\M$, endowed with its natural σ-algebra, is a [[standard Borel space]], and the set $\{(x,y)\in\M\times\M:x\sim y\}$ is a Borel subset of $\M\times\M$; this case is well-known as a "Borel equivalence relation". Still, existence of a ''Borel'' transversal is not guaranteed (for an example, use the Vitali set again).
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| − | Existence of Borel transversals and related properties of equivalence relations are investigated in [[descriptive set theory]]. According to {{Cite|K|Sect. 4}}, a lot of work in this area is philosophically motivated by problems of classification of objects up to some equivalence. A number of negative results are available. They show that in many cases, classification by a Borel transversal is impossible, and moreover, much weaker kinds of classification are also impossible. For example, a negative result is available for isomorphism of locally finite connected graphs (or trees), see {{Cite|K|Sect. 7, Item (B)(iii)}}.
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| − | |valign="top"|{{Ref|K}}|| Alexander S. Kechris, "New directions in descriptive set theory", ''Bull. Symb. Logic'' '''5''' (1999), 161–174. {{MR|}} {{ZBL|0933.03057}} | + | |valign="top"|{{Ref|B}}|| V.I. Bogachev, "Gaussian measures", AMS (????). {{MR|}} {{ZBL|}} |
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Revision as of 19:21, 14 June 2012
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V.I. Bogachev, "Gaussian measures", AMS (????).
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How to Cite This Entry:
Boris Tsirelson/sandbox2. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox2&oldid=27006