# Examples of non-shy sets

Fundamenta Mathematicae (1994)

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

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topDougherty, 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

top- [1] J. P. R. Christensen, On sets of Haar measure zero in abelian Polish groups, Israel J. Math. 13 (1972), 255-260.
- [2] R. Dougherty and J. Mycielski, The prevalence of permutations with infinite cycles, this volume, 89-94. Zbl0837.43008
- [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] A. Kechris, Lectures on definable group actions and equivalence relations, in preparation.
- [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] 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.

## Citations in EuDML Documents

top- Randall Dougherty, Jan Mycielski, The prevalence of permutations with infinite cycles
- Sławomir Solecki, On Haar null sets
- Eva Matoušková, Luděk Zajíček, Second order differentiability and Lipschitz smooth points of convex functionals
- Taras O. Banakh, Cardinal characteristics of the ideal of Haar null sets

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