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

A non-Borel subset of ${\displaystyle V}$ is said to be prevalent if it contains a prevalent Borel subset.

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

Local prevalence and shyness

A subset ${\displaystyle S}$ of ${\displaystyle V}$ is said to be locally shy if every point ${\displaystyle v\in V}$ has a neighbourhood ${\displaystyle N_{v}}$ whose intersection with ${\displaystyle S}$ is a shy set. ${\displaystyle S}$ is said to be locally prevalent if its complement is locally shy.

Theorems involving prevalence and shyness

• If ${\displaystyle S}$ is shy, then so is every subset of ${\displaystyle S}$ and every translate of ${\displaystyle S.}$
• Every shy Borel set ${\displaystyle S}$ 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 ${\displaystyle V}$ is a separable space, then every locally shy subset of ${\displaystyle V}$ is also shy.
• A subset ${\displaystyle S}$ of ${\displaystyle n}$-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ is shy if and only if it has Lebesgue measure zero.
• Any prevalent subset ${\displaystyle S}$ of ${\displaystyle V}$ is dense in ${\displaystyle V.}$
• If ${\displaystyle V}$ is infinite-dimensional, then every compact subset of ${\displaystyle V}$ 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 ${\displaystyle [0,1]}$ into the real line ${\displaystyle \mathbb {R} }$ is nowhere differentiable; here the space ${\displaystyle V}$ is ${\displaystyle C([0,1];\mathbb {R} )}$ with the topology induced by the supremum norm.
• Almost every function ${\displaystyle f}$ in the ${\displaystyle L^{p}}$ space ${\displaystyle L^{1}([0,1];\mathbb {R} )}$ has the property that $${\displaystyle \int _{0}^{1}f(x)\,\mathrm {d} x\neq 0.}$$ Clearly, the same property holds for the spaces of ${\displaystyle k}$-times differentiable functions ${\displaystyle C^{k}([0,1];\mathbb {R} ).}$
• For ${\displaystyle 1<p\leq +\infty ,}$ almost every sequence ${\displaystyle a=\left(a_{n}\right)_{n\in \mathbb {N} }\in \ell ^{p}}$ has the property that the series $${\displaystyle \sum _{n\in \mathbb {N} }a_{n}}$$ diverges.
• Prevalence version of the Whitney embedding theorem: Let ${\displaystyle M}$ be a compact manifold of class ${\displaystyle C^{1}}$ and dimension ${\displaystyle d}$ contained in ${\displaystyle \mathbb {R} ^{n}.}$ For ${\displaystyle 1\leq k\leq +\infty ,}$ almost every ${\displaystyle C^{k}}$ function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{2d+1}}$ is an embedding of ${\displaystyle M.}$
• If ${\displaystyle A}$ is a compact subset of ${\displaystyle \mathbb {R} ^{n}}$ with Hausdorff dimension ${\displaystyle d,}$${\displaystyle m\geq ,}$ and ${\displaystyle 1\leq k\leq +\infty ,}$ then, for almost every ${\displaystyle C^{k}}$ function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m},}$${\displaystyle f(A)}$ also has Hausdorff dimension ${\displaystyle d.}$
• For ${\displaystyle 1\leq k\leq +\infty ,}$ almost every ${\displaystyle C^{k}}$ function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}$ has the property that all of its periodic points are hyperbolic. In particular, the same is true for all the period ${\displaystyle p}$ points, for any integer ${\displaystyle p.}$

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.