nLab prevalence

Idea

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

Definition

Suppose $G$ is a completely metrizable topological group. A Borel subset $S\subset G$ is shy if there is a compactly supported nonzero Borel probability 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? the 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.

• 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.