In mathematics, the notions of prevalence and shyness are notions of "almost everywhere" and "measure zero" that are well-suited to the study of infinite-dimensional spaces and make use of the translation-invariant Lebesgue measure on finite-dimensional real spaces. The term "shy" was suggested by the American mathematician John Milnor.
Definitions
Prevalence and shyness Let V {\displaystyle V} be a real topological vector space and let S {\displaystyle S} be a Borel-measurable subset of V . {\displaystyle V.} S {\displaystyle S} is said to be prevalent if there exists a finite-dimensional subspace P {\displaystyle P} of V , {\displaystyle V,} called the probe set, such that for all v ∈ V {\displaystyle v\in V} we have v + p ∈ S {\displaystyle v+p\in S} for λ P {\displaystyle \lambda _{P}} -almost all p ∈ P , {\displaystyle p\in P,} where λ P {\displaystyle \lambda _{P}} denotes the dim ( P ) {\displaystyle \dim(P)} -dimensional Lebesgue measure on P . {\displaystyle P.} Put another way, for every v ∈ V , {\displaystyle v\in V,} Lebesgue-almost every point of the hyperplane v + P {\displaystyle v+P} lies in S . {\displaystyle S.}
A non-Borel subset of V {\displaystyle V} is said to be prevalent if it contains a prevalent Borel subset. A Borel subset of V {\displaystyle V} is said to be shy if its complement is prevalent; a non-Borel subset of V {\displaystyle V} is said to be shy if it is contained within a shy Borel subset. An alternative, and slightly more general, definition is to define a set S {\displaystyle S} to be shy if there exists a transverse measure for S {\displaystyle S} (other than the trivial measure).
Local prevalence and shyness A subset S {\displaystyle S} of V {\displaystyle V} is said to be locally shy if every point v ∈ V {\displaystyle v\in V} has a neighbourhood N v {\displaystyle N_{v}} whose intersection with S {\displaystyle S} is a shy set. S {\displaystyle S} is said to be locally prevalent if its complement is locally shy.
Theorems involving prevalence and shyness If S {\displaystyle S} is shy, then so is every subset of S {\displaystyle S} and every translate of S . {\displaystyle S.}
Every shy Borel set S {\displaystyle S} admits a transverse measure that is finite and has compact support. Furthermore, this measure can be chosen so that its support has arbitrarily small diameter. Any finite or countable union of shy sets is also shy. Analogously, countable intersection of prevalent sets is prevalent. Any shy set is also locally shy. If V {\displaystyle V} is a separable space, then every locally shy subset of V {\displaystyle V} is also shy. A subset S {\displaystyle S} of n {\displaystyle n} -dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} is shy if and only if it has Lebesgue measure zero. Any prevalent subset S {\displaystyle S} of V {\displaystyle V} is dense in V . {\displaystyle V.}
If V {\displaystyle V} is infinite-dimensional, then every compact subset of V {\displaystyle V} is shy. In the following, "almost every" is taken to mean that the stated property holds of a prevalent subset of the space in question.
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