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.

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 $$v \in V$$ we have $$v + p \in S$$ for $$\lambda_P$$-almost all $$p \in P,$$ where $$\lambda_P$$ denotes the $$\dim (P)$$-dimensional Lebesgue measure on $$P.$$ Put another way, for every $$v \in V,$$ 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 $$v \in V$$ 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. Analogously, countable intersection of prevalent sets is prevalent.
 * 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 $$\R^n$$ 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 $$L^1([0, 1]; \R)$$ 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 $$1 < p \leq +\infty,$$ almost every sequence $$a = \left(a_n\right)_{n \in \N} \in \ell^p$$ has the property that the series $$\sum_{n \in \N} a_n$$ diverges.
 * Prevalence version of the Whitney embedding theorem: Let $$M$$ be a compact manifold of class $$C^1$$ and dimension $$d$$ contained in $$\R^n.$$ For $$1 \leq k \leq +\infty,$$ almost every $$C^k$$ function $$f : \R^n \to \R^{2d+1}$$ is an embedding of $$M.$$
 * If $$A$$ is a compact subset of $$\R^n$$ with Hausdorff dimension $$d,$$ $$m \geq ,$$ and $$1 \leq k \leq +\infty,$$ then, for almost every $$C^k$$ function $$f : \R^n \to \R^m,$$ $$f(A)$$ also has Hausdorff dimension $$d.$$
 * For $$1 \leq k \leq +\infty,$$ almost every $$C^k$$ function $$f : \R^n \to \R^n$$ 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 $$p.$$