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.

Definitions

Prevalence and shyness

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

A non-Borel subset of V is said to be prevalent if it contains a prevalent Borel subset.

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

Local prevalence and shyness

A subset S of V is said to be locally shy if every point v ∈ V has a neighbourhood N_v whose intersection with S is a shy set. S is said to be locally prevalent if its complement is locally shy.

Theorems involving prevalence and shyness

• If S is shy, then so is every subset of S and every translate of S.
• Every shy Borel set 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.
• Any shy set is also locally shy. If V is a separable space, then every locally shy subset of V is also shy.
• A subset S of n-dimensional Euclidean space Rⁿ is shy if and only if it has Lebesgue measure zero.
• Any prevalent subset S of V is dense in V.
• If V is infinite-dimensional, then every compact subset of 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 [0, 1] into the real line R is nowhere differentiable; here the space V is C([0, 1]; R) with the topology induced by the supremum norm.
• Almost every function f in the L^p space L¹([0, 1]; R) has the property that

$\int_{0}^{1} f(x) \, \mathrm{d} x \neq 0.$

Clearly, the same property holds for the spaces of k-times differentiable functions C^k([0, 1]; R).

• For 1 < p ≤ +∞, almost every sequence a = (a_n)_n∈N in ℓ^p has the property that the series

$\sum_{n \in \mathbb{N}} a_{n}$

diverges.

• Prevalence version of the Whitney embedding theorem: Let M be a compact manifold of class C¹ and dimension d contained in Rⁿ. For 1 ≤ k ≤ +∞, almost every C^k function f : Rⁿ → R^2d+1 is an embedding of M.
• If A is a compact subset of Rⁿ with Hausdorff dimension d, m ≥ d, and 1 ≤ k ≤ +∞, then, for almost every C^k function f : Rⁿ → R^m, f(A) also has Hausdorff dimension d.
• For 1 ≤ k ≤ +∞, almost every C^k function f : Rⁿ → Rⁿ has the property that all of its periodic points are hyperbolic. In particular, the same is true for all the period p points, for any integer p.

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. 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. doi:10.1090/S0273-0979-1992-00328-2.