Prevalent and shy sets

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.

Let ${\displaystyle V}${\displaystyle V} be a real topological vector space and let ${\displaystyle S}${\displaystyle S} be a Borel-measurable subset of ${\displaystyle V.}${\displaystyle V.} ${\displaystyle S}${\displaystyle S} is said to be prevalent if there exists a finite-dimensional subspace ${\displaystyle P}${\displaystyle P} of ${\displaystyle V,}${\displaystyle V,} called the probe set, such that for all ${\displaystyle v\in V}${\displaystyle v\in V} we have ${\displaystyle v+p\in S}${\displaystyle v+p\in S} for ${\displaystyle \lambda _{P}}${\displaystyle \lambda _{P}}-almost all ${\displaystyle p\in P,}${\displaystyle p\in P,} where ${\displaystyle \lambda _{P}}${\displaystyle \lambda _{P}} denotes the ${\displaystyle \dim(P)}${\displaystyle \dim(P)}-dimensional Lebesgue measure on ${\displaystyle P.}${\displaystyle P.} Put another way, for every ${\displaystyle v\in V,}${\displaystyle v\in V,} Lebesgue-almost every point of the hyperplane ${\displaystyle v+P}${\displaystyle v+P} lies in ${\displaystyle S.}${\displaystyle S.}

A non-Borel subset of ${\displaystyle V}${\displaystyle V} is said to be prevalent if it contains a prevalent Borel subset.

A Borel subset of ${\displaystyle V}${\displaystyle V} is said to be shy if its complement is prevalent; a non-Borel subset of ${\displaystyle V}${\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 ${\displaystyle S}${\displaystyle S} to be shy if there exists a transverse measure for ${\displaystyle S}${\displaystyle S} (other than the trivial measure).

A subset ${\displaystyle S}${\displaystyle S} of ${\displaystyle V}${\displaystyle V} is said to be locally shy if every point ${\displaystyle v\in V}${\displaystyle v\in V} has a neighbourhood ${\displaystyle N_{v}}${\displaystyle N_{v}} whose intersection with ${\displaystyle S}${\displaystyle S} is a shy set. ${\displaystyle S}${\displaystyle S} is said to be locally prevalent if its complement is locally shy.

• If ${\displaystyle S}${\displaystyle S} is shy, then so is every subset of ${\displaystyle S}${\displaystyle S} and every translate of ${\displaystyle S.}${\displaystyle S.}
• Every shy Borel set ${\displaystyle S}${\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 ${\displaystyle V}${\displaystyle V} is a separable space, then every locally shy subset of ${\displaystyle V}${\displaystyle V} is also shy.
• A subset ${\displaystyle S}${\displaystyle S} of ${\displaystyle n}${\displaystyle n}-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{n}}${\displaystyle \mathbb {R} ^{n}} is shy if and only if it has Lebesgue measure zero.
• Any prevalent subset ${\displaystyle S}${\displaystyle S} of ${\displaystyle V}${\displaystyle V} is dense in ${\displaystyle V.}${\displaystyle V.}
• If ${\displaystyle V}${\displaystyle V} is infinite-dimensional, then every compact subset of ${\displaystyle V}${\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.

• Almost every continuous function from the interval ${\displaystyle [0,1]}${\displaystyle [0,1]} into the real line ${\displaystyle \mathbb {R} }${\displaystyle \mathbb {R} } is nowhere differentiable; here the space ${\displaystyle V}${\displaystyle V} is ${\displaystyle C([0,1];\mathbb {R} )}${\displaystyle C([0,1];\mathbb {R} )} with the topology induced by the supremum norm.
• Almost every function ${\displaystyle f}${\displaystyle f} in the ${\displaystyle L^{p}}${\displaystyle L^{p}} space ${\displaystyle L^{1}([0,1];\mathbb {R} )}${\displaystyle L^{1}([0,1];\mathbb {R} )} has the property that $${\displaystyle \int _{0}^{1}f(x),円\mathrm {d} x\neq 0.}$${\displaystyle \int _{0}^{1}f(x),円\mathrm {d} x\neq 0.} Clearly, the same property holds for the spaces of ${\displaystyle k}${\displaystyle k}-times differentiable functions ${\displaystyle C^{k}([0,1];\mathbb {R} ).}${\displaystyle C^{k}([0,1];\mathbb {R} ).}
• For ${\displaystyle 1<p\leq +\infty ,}${\displaystyle 1<p\leq +\infty ,} almost every sequence ${\displaystyle a=\left(a_{n}\right)_{n\in \mathbb {N} }\in \ell ^{p}}${\displaystyle a=\left(a_{n}\right)_{n\in \mathbb {N} }\in \ell ^{p}} has the property that the series $${\displaystyle \sum _{n\in \mathbb {N} }a_{n}}$${\displaystyle \sum _{n\in \mathbb {N} }a_{n}} diverges.
• Prevalence version of the Whitney embedding theorem: Let ${\displaystyle M}${\displaystyle M} be a compact manifold of class ${\displaystyle C^{1}}${\displaystyle C^{1}} and dimension ${\displaystyle d}${\displaystyle d} contained in ${\displaystyle \mathbb {R} ^{n}.}${\displaystyle \mathbb {R} ^{n}.} For ${\displaystyle 1\leq k\leq +\infty ,}${\displaystyle 1\leq k\leq +\infty ,} almost every ${\displaystyle C^{k}}${\displaystyle C^{k}} function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{2d+1}}${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{2d+1}} is an embedding of ${\displaystyle M.}${\displaystyle M.}
• If ${\displaystyle A}${\displaystyle A} is a compact subset of ${\displaystyle \mathbb {R} ^{n}}${\displaystyle \mathbb {R} ^{n}} with Hausdorff dimension ${\displaystyle d,}${\displaystyle d,} ${\displaystyle m\geq ,}${\displaystyle m\geq ,} and ${\displaystyle 1\leq k\leq +\infty ,}${\displaystyle 1\leq k\leq +\infty ,} then, for almost every ${\displaystyle C^{k}}${\displaystyle C^{k}} function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m},}${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m},} ${\displaystyle f(A)}${\displaystyle f(A)} also has Hausdorff dimension ${\displaystyle d.}${\displaystyle d.}
• For ${\displaystyle 1\leq k\leq +\infty ,}${\displaystyle 1\leq k\leq +\infty ,} almost every ${\displaystyle C^{k}}${\displaystyle C^{k}} function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{n}} has the property that all of its periodic points are hyperbolic. In particular, the same is true for all the period ${\displaystyle p}${\displaystyle p} points, for any integer ${\displaystyle p.}${\displaystyle p.}

• Hunt, Brian R. (1994). "The prevalence of continuous nowhere differentiable functions". Proc. Amer. Math. Soc. 122 (3). American Mathematical Society: 711–717. doi:10.2307/2160745 . JSTOR 2160745.
• Hunt, Brian R. and Sauer, Tim and Yorke, James A. (1992). "Prevalence: a translation-invariant "almost every" on infinite-dimensional spaces". Bull. Amer. Math. Soc. (N.S.). 27 (2): 217–238. arXiv:math/9210220 . doi:10.1090/S0273-0979-1992-00328-2. S2CID 17534021.{{cite journal}}: CS1 maint: multiple names: authors list (link)

• Measure theory
• Functional analysis

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