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.

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• If ${\displaystyle S}$ is shy, then so is every subset of ${\displaystyle S}$ and every translate of ${\displaystyle S.}$
• Every shy Borel set ${\displaystyle S}$ 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 ${\displaystyle V}$ is a separable space, then every locally shy subset of ${\displaystyle V}$ is also shy.
• A subset ${\displaystyle S}$ of ${\displaystyle n}$-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ is shy if and only if it has Lebesgue measure zero.
• Any prevalent subset ${\displaystyle S}$ of ${\displaystyle V}$ is dense in ${\displaystyle V.}$
• If ${\displaystyle V}$ is infinite-dimensional, then every compact subset of ${\displaystyle V}$ 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 ${\displaystyle [0,1]}$ into the real line ${\displaystyle \mathbb {R} }$ is nowhere differentiable; here the space ${\displaystyle V}$ is ${\displaystyle C([0,1];\mathbb {R} )}$ with the topology induced by the supremum norm.
• Almost every function ${\displaystyle f}$ in the ${\displaystyle L^{p}}$ space ${\displaystyle L^{1}([0,1];\mathbb {R} )}$ has the property that
  $${\displaystyle \int _{0}^{1}f(x)\,\mathrm {d} x\neq 0.}$$

  
  Clearly, the same property holds for the spaces of ${\displaystyle k}$-times differentiable functions ${\displaystyle C^{k}([0,1];\mathbb {R} ).}$
• For ${\displaystyle 1<p\leq +\infty ,}$ almost every sequence ${\displaystyle a=\left(a_{n}\right)_{n\in \mathbb {N} }\in \ell ^{p}}$ has the property that the series
  $${\displaystyle \sum _{n\in \mathbb {N} }a_{n}}$$

  
  diverges.
• Prevalence version of the Whitney embedding theorem: Let ${\displaystyle M}$ be a compact manifold of class ${\displaystyle C^{1}}$ and dimension ${\displaystyle d}$ contained in ${\displaystyle \mathbb {R} ^{n}.}$ For ${\displaystyle 1\leq k\leq +\infty ,}$ almost every ${\displaystyle C^{k}}$ function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{2d+1}}$ is an embedding of ${\displaystyle M.}$
• If ${\displaystyle A}$ is a compact subset of ${\displaystyle \mathbb {R} ^{n}}$ with Hausdorff dimension ${\displaystyle d,}$ ${\displaystyle m\geq ,}$ and ${\displaystyle 1\leq k\leq +\infty ,}$ then, for almost every ${\displaystyle C^{k}}$ function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m},}$ ${\displaystyle f(A)}$ also has Hausdorff dimension ${\displaystyle d.}$
• For ${\displaystyle 1\leq k\leq +\infty ,}$ almost every ${\displaystyle C^{k}}$ function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}$ has the property that all of its periodic points are hyperbolic. In particular, the same is true for all the period ${\displaystyle p}$ points, for any integer ${\displaystyle p.}$