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 United States mathematician John Milnor.

Definitions

Prevalence and shyness

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

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

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

Local prevalence and shyness

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

Theorems involving prevalence and shyness

• If [math]\displaystyle{ S }[/math] is shy, then so is every subset of [math]\displaystyle{ S }[/math] and every translate of [math]\displaystyle{ S. }[/math]
• Every shy Borel set [math]\displaystyle{ S }[/math] 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 [math]\displaystyle{ V }[/math] is a separable space, then every locally shy subset of [math]\displaystyle{ V }[/math] is also shy.
• A subset [math]\displaystyle{ S }[/math] of [math]\displaystyle{ n }[/math]-dimensional Euclidean space [math]\displaystyle{ \R^n }[/math] is shy if and only if it has Lebesgue measure zero.
• Any prevalent subset [math]\displaystyle{ S }[/math] of [math]\displaystyle{ V }[/math] is dense in [math]\displaystyle{ V. }[/math]
• If [math]\displaystyle{ V }[/math] is infinite-dimensional, then every compact subset of [math]\displaystyle{ V }[/math] 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 [math]\displaystyle{ [0, 1] }[/math] into the real line [math]\displaystyle{ \R }[/math] is nowhere differentiable; here the space [math]\displaystyle{ V }[/math] is [math]\displaystyle{ C([0, 1]; \R) }[/math] with the topology induced by the supremum norm.
• Almost every function [math]\displaystyle{ f }[/math] in the [math]\displaystyle{ L^p }[/math] space [math]\displaystyle{ L^1([0, 1]; \R) }[/math] has the property that [math]\displaystyle{ \int_0^1 f(x) \, \mathrm{d} x \neq 0. }[/math] Clearly, the same property holds for the spaces of [math]\displaystyle{ k }[/math]-times differentiable functions [math]\displaystyle{ C^k([0, 1]; \R). }[/math]
• For [math]\displaystyle{ 1 \lt p \leq +\infty, }[/math] almost every sequence [math]\displaystyle{ a = \left(a_n\right)_{n \in \N} \in \ell^p }[/math] has the property that the series [math]\displaystyle{ \sum_{n \in \N} a_n }[/math] diverges.
• Prevalence version of the Whitney embedding theorem: Let [math]\displaystyle{ M }[/math] be a compact manifold of class [math]\displaystyle{ C^1 }[/math] and dimension [math]\displaystyle{ d }[/math] contained in [math]\displaystyle{ \R^n. }[/math] For [math]\displaystyle{ 1 \leq k \leq +\infty, }[/math] almost every [math]\displaystyle{ C^k }[/math] function [math]\displaystyle{ f : \R^n \to \R^{2d+1} }[/math] is an embedding of [math]\displaystyle{ M. }[/math]
• If [math]\displaystyle{ A }[/math] is a compact subset of [math]\displaystyle{ \R^n }[/math] with Hausdorff dimension [math]\displaystyle{ d, }[/math] [math]\displaystyle{ m \geq , }[/math] and [math]\displaystyle{ 1 \leq k \leq +\infty, }[/math] then, for almost every [math]\displaystyle{ C^k }[/math] function [math]\displaystyle{ f : \R^n \to \R^m, }[/math] [math]\displaystyle{ f(A) }[/math] also has Hausdorff dimension [math]\displaystyle{ d. }[/math]
• For [math]\displaystyle{ 1 \leq k \leq +\infty, }[/math] almost every [math]\displaystyle{ C^k }[/math] function [math]\displaystyle{ f : \R^n \to \R^n }[/math] has the property that all of its periodic points are hyperbolic. In particular, the same is true for all the period [math]\displaystyle{ p }[/math] points, for any integer [math]\displaystyle{ p. }[/math]

References

• Hunt, Brian R. (1994). "The prevalence of continuous nowhere differentiable functions". Proc. Amer. Math. Soc. (American Mathematical Society) 122 (3): 711–717. doi:10.2307/2160745. 
• 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. doi:10.1090/S0273-0979-1992-00328-2.