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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMar 17th 2025
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: \tableofcontents ## Idea __Prevalence__ refers to ideas revolving around associating an [[enhanced measurable space]] to a [[completely|complete space]] [[metrizable|metrizable space]] [[topological group]]. ## Definition Suppose $G$ is a [[completely|complete space]] [[metrizable|metrizable space]] [[topological group]]. A [[Borel subset]] $S\subset G$ is __shy__ if there is a compactly supported nonzero [[Borel probability measure|Borel measure]] $\mu$ such that $\mu(xS)=0$ for all $x\in G$. ## Properties The triple $(G,B_G,S_G)$, where $B_G$ is the [[σ-algebra]] of [[Borel subsets]] and $S_G$ is the [[σ-ideal]] of shy sets is an [[enhanced measurable space]]. We may also want to [[complete|complete enhanced measurable space]] $(G,B_G,S_G)$, extending the notion of shy and prevalent sets to non-Borel sets. ## References * Brian R. Hunt, Tim Sauer, James A. Yorke, _Prevalence: a translation-invariant “almost every” on infinite-dimensional spaces_, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 2, 217–238. [doi](https://doi.org/10.1090/s0273-0979-1992-00328-2). * Brian R. Hunt, Tim Sauer, James A. Yorke, _Prevalence. An addendum to: “Prevalence: a translation-invariant ‘almost every’ on infinite-dimensional spaces”_, Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 2, 306–307. [doi](https://doi.org/10.1090/s0273-0979-1993-00396-3). Survey: * William Ott, James A. Yorke, _Prevalence_, Bulletin of the American Mathematical Society 42:03 (2005), 263–291. [doi](https://doi.org/10.1090/s0273-0979-05-01060-8). <a href="https://ncatlab.org/nlab/revision/prevalence/1">v1</a>, <a href="https://ncatlab.org/nlab/show/prevalence">current</a>

   Created:

   \tableofcontents

   Idea

   Prevalence refers to ideas revolving around associating an enhanced measurable space to a complete space metrizable space topological group.

   Definition

   Suppose is a complete space metrizable space topological group. A Borel subset is shy if there is a compactly supported nonzero Borel measure such that for all .

   Properties

   The triple , where is the σ-algebra of Borel subsets and is the σ-ideal of shy sets is an enhanced measurable space.

   We may also want to complete enhanced measurable space , extending the notion of shy and prevalent sets to non-Borel sets.

   References

   • Brian R. Hunt, Tim Sauer, James A. Yorke, Prevalence: a translation-invariant “almost every” on infinite-dimensional spaces, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 2, 217–238. doi.

   • Brian R. Hunt, Tim Sauer, James A. Yorke, Prevalence. An addendum to: “Prevalence: a translation-invariant ‘almost every’ on infinite-dimensional spaces”, Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 2, 306–307. doi.

   Survey:

   • William Ott, James A. Yorke, Prevalence, Bulletin of the American Mathematical Society 42:03 (2005), 263–291. doi.

   v1, current