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PROPERTIES OF GENERIC AND ALMOST EVERY MAPPINGS IN VARIOUS NONLOCALLY COMPACT POLISH ABELIAN GROUPS Susan L. Calcote White May 12, 2007 In a nonlocally compact Polish abelian group G, we will consider two notions of smallness of subsets of G. Those subsets of G which are topologically small are said to be meager, and those which are measure-theoretically small are Haar null. We will say that a property P holds for a generic 9 E G if the property holds on the complement of a meager subset of G, and P holds for almost every 9 E G if the property holds on the complement of a Haar null set. Thus the phrase "a randomly chosen element of G is likely to have property P" may be understood to have two different meanings in this paper. The spaces 'IlL and C(JRn) , n 2: I, the continuous self-maps of Z and JRn, respectively, are both nonlocally compact Polish abelian groups. In this paper we will study properties of generic and almost every mappings in ZZ and C(JR), and properties of generic mappings in C (JRn ). In the space ZZ, we show that the behavior of a generic ¢ E ZZ is quite different than the behavior of almost every ¢ E ZZ. We will show that in the space C (JR), the behavior of a generic f E C (JR) is analogous to the behavior of a generic ¢ E ZZ in several ways, but the analogies between the spaces ZZ and C(JR) seem to cease when the properties of almost every f E C(JR) are considered. In fact, many of the properties of functions in C(JR) that we consider in this paper are shown to be H-ambivalent; that is, the properties hold on a set which is neither Haar null nor the complement of a Haar null set. We will present preliminary results concerning the behavior of a generic f E C(JRn). We will show that several of the properties which hold for a generic f E C(JR) also hold in the more general setting of a generic f E C (JRn), although the proofs techniques differ. Finally, we will close with a discussion of future directions that this work may take.

Properties of generic and almost every mappings in various nonlogically compact Polish abelian groups.