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Brian R. Hunt, Tim Sauer, and James A. Yorke

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Bull. Amer. Math. Soc. 27 (1992), 217-238; 28 (1993), 306-307.
*

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.

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*Last updated: June 23, 1998*