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 $V$ 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 $\mu$ on $V$, there is a shy set $C$ with $\mu (C) = 1$. Further, when $V$ is locally convex, any i.i.d. sequence with law $\mu$ repeatedly visits neighborhoods of only a shy set of points if the neighborhoods shrink to $0$ at any rate.