Examples of non-shy sets

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(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. If G is a Polish group which is locally compact, then one can define a Borel measure on G, the Haar measure, which is invariant (on one side) and gives finite non-zero measure to non-empty open sets with compact closure. This measure is unique up to a multiplicative constant, so the collection of measure-zero sets is uniquely determined, and gives an invariant (on both sides) property of “smallness” for subsets of G which is probably closer to the intuitive idea of “smallness” than the category analogue, meagerness. The definition of Haar measure does not extend to groups which are not locally compact, but there is a suitable extension of the property of being of Haar measure zero; this is given in Christensen [1] and again in Hunt, Sauer, and Yorke [3]. In the former paper, a universally measurable subset S of an abelian Polish group G is called a Haar zero set if there is a probability measure on G which gives every translate of S measure 0; the latter paper uses an equivalent definition for Borel subsets of separable complete metric linear spaces, and calls such sets shy . Topsøe and Hoffmann-Jørgensen [6] 1991 Mathematics Subject Classification: Primary 20B07. The author was supported by NSF grant number DMS-9158092 and by a fellowship from the Sloan Foundation.