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 AmericanmathematicianJohn Milnor.
Definitions
Prevalence and shyness
Let V be a realtopological vector space and letS be a Borel-measurablesubset 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-dimensional Lebesgue measure on P. Put another way, for every v ∈ V, Lebesgue-almost every point of the hyperplanev + 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.
A subset S of V is said to be locally shy if every point v ∈ V has a neighbourhoodNv 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.
Almost every function f in the Lp space L1 has the property that
For 1 < p ≤ +∞, almost every sequence a = n∈N in ℓp has the property that the series
Prevalence version of the Whitneyembedding theorem: Let M be a compact manifold of class C1 and dimensiond 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 dimensiond, m ≥ d, and 1 ≤ k ≤ +∞, then, for almost every Ck function f : Rn → Rm, f 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.
