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[edit]

Prevalence and shyness[edit]

Let be a real topological vector space and let be a Borel-measurable subset of is said to be prevalent if there exists a finite-dimensional subspace of called the probe set, such that for all we have for -almost all where denotes the -dimensional Lebesgue measure on Put another way, for every Lebesgue-almost every point of the hyperplane lies in

A non-Borel subset of is said to be prevalent if it contains a prevalent Borel subset.

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

Local prevalence and shyness[edit]

A subset of is said to be locally shy if every point has a neighbourhood whose intersection with is a shy set. is said to be locally prevalent if its complement is locally shy.

Theorems involving prevalence and shyness[edit]

• If is shy, then so is every subset of and every translate of
• Every shy Borel set 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 is a separable space, then every locally shy subset of is also shy.
• A subset of -dimensional Euclidean space is shy if and only if it has Lebesgue measure zero.
• Any prevalent subset of is dense in
• If is infinite-dimensional, then every compact subset of 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 into the real line is nowhere differentiable; here the space is with the topology induced by the supremum norm.
• Almost every function in the space has the property that Clearly, the same property holds for the spaces of -times differentiable functions
• For almost every sequence has the property that the series diverges.
• Prevalence version of the Whitney embedding theorem: Let be a compact manifold of class and dimension contained in For almost every function is an embedding of
• If is a compact subset of with Hausdorff dimension and then, for almost every function also has Hausdorff dimension
• For almost every function has the property that all of its periodic points are hyperbolic. In particular, the same is true for all the period points, for any integer

References[edit]

• 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)