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

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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

e a l s^{n}

has the property that all of its periodic orbits are hyperbolic.

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