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 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.
@article{bwmeta1.element.bwnjournal-article-fmv144i1p73bwm,
author = {Randall Dougherty},
title = {Examples of non-shy sets},
journal = {Fundamenta Mathematicae},
volume = {144},
year = {1994},
pages = {73-88},
zbl = {0842.43006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv144i1p73bwm}
}
Dougherty, Randall. Examples of non-shy sets. Fundamenta Mathematicae, Tome 144 (1994) pp. 73-88. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv144i1p73bwm/
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