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## Fundamenta Mathematicae

1994 | 144 | 1 | 73-88
Tytuł artykułu

### Examples of non-shy sets

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Treść / Zawartość
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EN
Abstrakty
EN
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.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
73-88
Opis fizyczny
Daty
wydano
1994
otrzymano
1993-04-04
Twórcy
autor
• Department of Mathematics, Ohio State University, Columbus, Ohio 43210, U.S.A.
Bibliografia
• [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.
• [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.
• [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.
• [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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Bibliografia
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