普遍性：infinite-dimensional空间上的一个平移不变“几乎所有”
Prevalence: a translation-invariant ``almost every'' on infinite-dimensional spaces
摘要：
我们提出了一个测量理论的条件，一个属性举行“几乎处处”的无限维向量空间，特别强调函数空间，如$C^k$和$L^p$。就像有限维空间中的“勒贝格几乎所有”概念一样，我们的“普遍性”概念也是平移不变的。而不是使用一个特定的措施，在整个空间，我们定义流行的类的所有概率测度紧支持。当人们希望得到函数空间上给定性质的概率结果时，普遍性是一个比"开放和稠密“或”一般“的拓扑概念更合适的条件。我们给出几个在普遍性意义上“几乎处处”成立的性质的例子。例如，我们证明了几乎所有$\reals^n$上的$C^1$映射都具有周期轨道都是双曲的性质。
英文摘要：
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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