PREVALENCE - A TRANSLATION-INVARIANT ALMOST EVERY ON INFINITE-DIMENSIONAL SPACES

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 C1 map on R(n) has the property that all of its periodic orbits are hyperbolic.