Examples of non-shy sets

Randall Dougherty

Fundamenta Mathematicae (1994)

  • Volume: 144, Issue: 1, page 73-88
  • ISSN: 0016-2736

Abstract

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Christensen has defined a generalization of the property of being of Haar measure zero to subsets of (abelian) Polish groups which need not be locally compact; a recent paper of Hunt, Sauer, and Yorke defines the same property for Borel subsets of linear spaces, and gives a number of examples and applications. The latter authors use the term “shyness” for this property, and “prevalence” for the complementary property. In the present paper, we construct a number of examples of non-shy Borel sets in various groups, and thereby answer several questions of Christensen and Mycielski. The main results are: in many (most?) non-locally-compact Polish groups, the ideal of shy sets does not satisfy the countable chain condition (i.e., there exist uncountably many disjoint non-shy Borel sets); in function spaces C ( ω 2 , G ) where G is an abelian Polish group, the set of functions f which are highly non-injective is non-shy, and even prevalent if G is locally compact.

How to cite

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Dougherty, Randall. "Examples of non-shy sets." Fundamenta Mathematicae 144.1 (1994): 73-88. <http://eudml.org/doc/212016>.

@article{Dougherty1994,
abstract = {Christensen has defined a generalization of the property of being of Haar measure zero to subsets of (abelian) Polish groups which need not be locally compact; a recent paper of Hunt, Sauer, and Yorke defines the same property for Borel subsets of linear spaces, and gives a number of examples and applications. The latter authors use the term “shyness” for this property, and “prevalence” for the complementary property. In the present paper, we construct a number of examples of non-shy Borel sets in various groups, and thereby answer several questions of Christensen and Mycielski. The main results are: in many (most?) non-locally-compact Polish groups, the ideal of shy sets does not satisfy the countable chain condition (i.e., there exist uncountably many disjoint non-shy Borel sets); in function spaces $C(^ω 2,G)$ where G is an abelian Polish group, the set of functions f which are highly non-injective is non-shy, and even prevalent if G is locally compact.},
author = {Dougherty, Randall},
journal = {Fundamenta Mathematicae},
keywords = {prevalence; shyness; Haar measure; Polish groups; non-locally-compact groups; countable chain condition; function spaces},
language = {eng},
number = {1},
pages = {73-88},
title = {Examples of non-shy sets},
url = {http://eudml.org/doc/212016},
volume = {144},
year = {1994},
}

TY - JOUR
AU - Dougherty, Randall
TI - Examples of non-shy sets
JO - Fundamenta Mathematicae
PY - 1994
VL - 144
IS - 1
SP - 73
EP - 88
AB - Christensen has defined a generalization of the property of being of Haar measure zero to subsets of (abelian) Polish groups which need not be locally compact; a recent paper of Hunt, Sauer, and Yorke defines the same property for Borel subsets of linear spaces, and gives a number of examples and applications. The latter authors use the term “shyness” for this property, and “prevalence” for the complementary property. In the present paper, we construct a number of examples of non-shy Borel sets in various groups, and thereby answer several questions of Christensen and Mycielski. The main results are: in many (most?) non-locally-compact Polish groups, the ideal of shy sets does not satisfy the countable chain condition (i.e., there exist uncountably many disjoint non-shy Borel sets); in function spaces $C(^ω 2,G)$ where G is an abelian Polish group, the set of functions f which are highly non-injective is non-shy, and even prevalent if G is locally compact.
LA - eng
KW - prevalence; shyness; Haar measure; Polish groups; non-locally-compact groups; countable chain condition; function spaces
UR - http://eudml.org/doc/212016
ER -

References

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  1. [1] J. P. R. Christensen, On sets of Haar measure zero in abelian Polish groups, Israel J. Math. 13 (1972), 255-260. 
  2. [2] R. Dougherty and J. Mycielski, The prevalence of permutations with infinite cycles, this volume, 89-94. Zbl0837.43008
  3. [3] B. Hunt, T. Sauer, and J. Yorke, Prevalence: a translation-invariant "almost every" on infinite-dimensional spaces, Bull. Amer. Math. Soc. 27 (1992), 217-238. Zbl0763.28009
  4. [4] A. Kechris, Lectures on definable group actions and equivalence relations, in preparation. 
  5. [5] J. Mycielski, Some unsolved problems on the prevalence of ergodicity, instability and algebraic independence, Ulam Quart. 1 (3) (1992), 30-37. Zbl0846.28006
  6. [6] F. Topsøe and J. Hoffmann-Jørgensen, Analytic spaces and their application, in: C. A. Rogers ( et al.), Analytic Sets, Academic Press, London, 1980, 317-401. 

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