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 "v" + "p" ∈ "S" for all "v" ∈ "V" and "λ"_"P"-almost all "p" ∈ "P", where "λ"_"P" denotes the dim("P")-dimensional Lebesgue measure on "P". Put another way, for every "v" ∈ "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" ∈ "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.

* 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::$int\_\{0\}^\{1\}\; f(x)\; ,\; mathrm\{d\}\; x\; eq\; 0.$:Clearly, the same property holds for the spaces of "k"-times differentiable functions "C"^"k"( [0, 1] ; R).

* For 1 < "p" ≤ +∞, almost every sequence "a" = ("a"_"n")_"n"∈N in ℓ^"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 R^"n". For 1 ≤ "k" ≤ +∞, almost every "C"^"k" function "f" : R^"n" → R^2"d"+1 is an embedding of "M".

* If "A" is a compact subset of R^"n" with Hausdorff dimension "d", "m" ≥ "d", and 1 ≤ "k" ≤ +∞, then, for almost every "C"^"k" function "f" : R^"n" → R^"n", "f"("A") also has Hausdorff dimension "d".

* For 1 ≤ "k" ≤ +∞, almost every "C"^"k" function "f" : R^"n" → 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

* cite journal
last = Hunt
first = Brian R.
title = The prevalence of continuous nowhere differentiable functions
journal = Proc. Amer. Math. Soc.
volume = 122
year = 1994
number = 3
pages = 711–717
doi = 10.2307/2160745
* 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
number = 2
pages = 217–238
doi = 10.1090/S0273-0979-1992-00328-2