This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (June 2020) (Learn how and when to remove this message)

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.

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).

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.

- 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