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 $v\in V$ 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 $\mathbb {R} ^{n}$ 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 $\mathbb {R}$ is nowhere differentiable; here the space $V$ is $C([0,1];\mathbb {R} )$ with the topology induced by the supremum norm.
• Almost every function $f$ in the $L^{p}$ space $L^{1}([0,1];\mathbb {R} )$ 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 $C^{k}([0,1];\mathbb {R} ).$
• For $1<p\leq +\infty ,$ almost every sequence $a=\left(a_{n}\right)_{n\in \mathbb {N} }\in \ell ^{p}$ 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 $C^{1}$ and dimension $d$ contained in $\mathbb {R} ^{n}.$ For $1\leq k\leq +\infty ,$ almost every $C^{k}$ function $f:\mathbb {R} ^{n}\to \mathbb {R} ^{2d+1}$ is an embedding of $M.$
• If $A$ is a compact subset of $\mathbb {R} ^{n}$ with Hausdorff dimension $d,$ $m\geq ,$ and $1\leq k\leq +\infty ,$ then, for almost every $C^{k}$ function $f:\mathbb {R} ^{n}\to \mathbb {R} ^{m},$ $f(A)$ also has Hausdorff dimension $d.$
• For $1\leq k\leq +\infty ,$ almost every $C^{k}$ function $f:\mathbb {R} ^{n}\to \mathbb {R} ^{n}$ 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 $p.$

References

• Hunt, Brian R. (1994). "The prevalence of continuous nowhere differentiable functions". Proc. Amer. Math. Soc. 122 (3). American Mathematical Society: 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. arXiv:math/9210220. doi:10.1090/S0273-0979-1992-00328-2. S2CID 17534021.{{cite journal}}: CS1 maint: multiple names: authors list (link)