# 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 for all *v* ∈ *V* we have *v* + *p* ∈ *S* for *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 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

- 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

- 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**^{2d+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**^{m},*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

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