The gap between probability and prevalence: Loneliness in vector spaces

Abstract

The best available definition of a subset of an infinite dimensional, complete, metric vector space V V being “small” is Christensen’s Haar zero sets, equivalently, Hunt, Sauer, and Yorke’s shy sets. The complement of a shy set is a prevalent set. There is a gap between prevalence and likelihood. For any probability μ \mu on V V , there is a shy set C C with μ ( C ) = 1 \mu (C) = 1 . Further, when V V is locally convex, any i.i.d. sequence with law μ \mu repeatedly visits neighborhoods of only a shy set of points if the neighborhoods shrink to 0 0 at any rate.