Prevalent and shy sets explained

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 λ _{P} -almost all p \in P, where λ _{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 \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.

[ 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 $$\backslash int\_0^1\; f(x)\; \backslash ,\; \backslash mathrm\; x\; \backslash neq\; 0.$$ Clearly, the same property holds for the spaces of k -times differentiable functions C ^{k([0,} 1 ]; \R).

• For

1 < p \leq +inf ty, almost every sequence a = \left(a _{n\right)} _{n} \in \ell ^{p} has the property that the series $$\backslash sum\_\; a\_n$$ 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 \leq k \leq +inf ty, almost every C ^{k} function f : \R ^{n} \to \R ^{2d+1} is an embedding of M.

• If

A is a compact subset of \R ^{n} with Hausdorff dimension d, m \geq , and 1 \leq k \leq +inf ty, then, for almost every C ^{k} function f : \R ^{n} \to \R ^{m,} f(A) also has Hausdorff dimension d.

• For

1 \leq k \leq +inf ty, almost every C ^{k} function f : \R ^{n} \to \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.. The prevalence of continuous nowhere differentiable functions. Proc. Amer. Math. Soc.. 122. 1994. 711 - 717. 10.2307/2160745. 3. American Mathematical Society. 2160745. free.
• Hunt, Brian R. and Sauer, Tim and Yorke, James A.. Prevalence: a translation-invariant "almost every" on infinite-dimensional spaces. Bull. Amer. Math. Soc. (N.S.). 27. 1992. 217 - 238. 10.1090/S0273-0979-1992-00328-2. 2. math/9210220. 17534021.