%0 Journal Article %1 Hunt-1992-PTI %A Hunt, Brian R. %A Sauer, Tim %A Yorke, James A. %D 1992 %J Bull. Amer. Math. Soc. %K Lebesgue %N 2 %P 217-238 %R 10.1090/S0273-0979-1992-00328-2 %T Prevalence: a translation-invariant ``almost every'' on infinite-dimensional spaces %U http://www.ams.org/bull/1992-27-02/S0273-0979-1992-00328-2/home.html %V 27 %X 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 $R^n$ has the property that all of its periodic orbits are hyperbolic.

@article{Hunt-1992-PTI, 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 $R^n$ has the property that all of its periodic orbits are hyperbolic.}, added-at = {2016-03-03T19:34:43.000+0100}, author = {Hunt, Brian R. and Sauer, Tim and Yorke, James A.}, biburl = {https://www.bibsonomy.org/bibtex/2066ac6657fcff88f0b2755a369f6db0d/kasanicky.ivan}, description = {Introduces the notion of prevalence.}, doi = {10.1090/S0273-0979-1992-00328-2}, interhash = {e07649f9f53a6c7a3071817dd29f0922}, intrahash = {066ac6657fcff88f0b2755a369f6db0d}, journal = {Bull. Amer. Math. Soc.}, keywords = {Lebesgue}, number = 2, pages = {217-238}, timestamp = {2016-03-03T19:34:43.000+0100}, title = {Prevalence: a translation-invariant ``almost every'' on infinite-dimensional spaces}, url = {http://www.ams.org/bull/1992-27-02/S0273-0979-1992-00328-2/home.html}, volume = 27, year = 1992 }