Prevalence: a translation-invariant ``almost every'' on infinite-dimensional spaces

Abstract

We present a measure-theoretic condition for a property to hold ``almost everywhere'' on an infinite-dimensional vector space, with particular emphasis on function spaces such as $C^k$ and $L^p$. Like the concept of ``Lebesgue almost every'' on finite-dimensional spaces, our notion of ``prevalence'' is translation invariant. Instead of using a specific measure on the entire space, we define prevalence in terms of the class of all probability measures with compact support. Prevalence is a more appropriate condition than the topological concepts of ``open and dense'' or ``generic'' when one desires a probabilistic result on the likelihood of a given property on a function space. We give several examples of properties which hold ``almost everywhere'' in the sense of prevalence. For instance, we prove that almost every $C^1$ map on $\reals^n$ has the property that all of its periodic orbits are hyperbolic.