Prevalent and shy sets

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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 λP-almost all p ∈ P, where λP 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 Nv 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 shy set is also locally shy. If V is a separable space, then every locally shy subset of V is also shy.
  • 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 function f in the Lp space L1([0, 1]; R) has the property that
Clearly, the same property holds for the spaces of k-times differentiable functions Ck([0, 1]; R).
  • For 1 < p ≤ +∞, almost every sequence a = (an)nN in ℓp has the property that the series
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 Ck function f : Rn → R2d+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 Ck function f : Rn → Rm, f(A) also has Hausdorff dimension d.
  • For 1 ≤ k ≤ +∞, almost every Ck 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

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