prevalent and shy sets

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 $v\; \backslash in\; V$ we have $v\; +\; p\; \backslash in\; S$ for $\backslash lambda\_P$-almost all $p\; \backslash in\; P,$ where $\backslash lambda\_P$ denotes the $\backslash dim\; (P)$-dimensional Lebesgue measure on $P.$ Put another way, for every $v\; \backslash in\; V,$ 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).

A subset $S$ of $V$ is said to be locally shy if every point $v\; \backslash in\; V$ 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.

• 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 $\backslash R^n$ 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 $\backslash R$ is nowhere differentiable; here the space $V$ is $C([0,\; 1];\; \backslash R)$ with the topology induced by the supremum norm.
• Almost every function $f$ in the $L^p$ space $L^1([0,\; 1];\; \backslash R)$ has the property that $$\backslash int\_0^1\; f(x)\; \backslash ,\; \backslash mathrm\{d\}\; x\; \backslash neq\; 0.$$ Clearly, the same property holds for the spaces of $k$-times differentiable functions $C^k([0,\; 1];\; \backslash R).$
• For almost every sequence $a\; =\; \backslash left(a\_n\backslash right)\_\{n\; \backslash in\; \backslash N\}\; \backslash in\; \backslash ell^p$ has the property that the series $$\backslash sum\_\{n\; \backslash in\; \backslash N\}\; a\_n$$ diverges.
• Prevalence version of the Whitney embedding theorem: Let $M$ be a compact manifold of class $C^1$ and dimension $d$ contained in $\backslash R^n.$ For $1\; \backslash leq\; k\; \backslash leq\; +\backslash infty,$ almost every $C^k$ function $f\; :\; \backslash R^n\; \backslash to\; \backslash R^\{2d+1\}$ is an embedding of $M.$
• If $A$ is a compact subset of $\backslash R^n$ with Hausdorff dimension $d,$ $m\; \backslash geq\; ,$ and $1\; \backslash leq\; k\; \backslash leq\; +\backslash infty,$ then, for almost every $C^k$ function $f\; :\; \backslash R^n\; \backslash to\; \backslash R^m,$ $f(A)$ also has Hausdorff dimension $d.$
• For $1\; \backslash leq\; k\; \backslash leq\; +\backslash infty,$ almost every $C^k$ function $f\; :\; \backslash R^n\; \backslash to\; \backslash R^n$ 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 $p.$

{{reflist}}

{{reflist|group=note}}

• {{cite journal

| last = Hunt

| first = Brian R.

| title = The prevalence of continuous nowhere differentiable functions

| journal = Proc. Amer. Math. Soc.

| volume = 122

| year = 1994

| pages = 711–717

| doi = 10.2307/2160745

| issue = 3

| publisher = American Mathematical Society

| jstor = 2160745

|doi-access = free

}}

• {{cite journal

| author = Hunt, Brian R. and Sauer, Tim and Yorke, James A.

| title = Prevalence: a translation-invariant "almost every" on infinite-dimensional spaces

| journal = Bull. Amer. Math. Soc. (N.S.)

| volume = 27

| year = 1992

| pages = 217–238

| doi = 10.1090/S0273-0979-1992-00328-2

| issue = 2

|arxiv = math/9210220

| s2cid = 17534021

}}

{{Topological vector spaces}}

{{Functional analysis}}

Category:Measure theory

Category:Functional analysis