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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.
In the following, "almost every" is taken to mean that the stated property holds of a prevalent subset of the space in question.
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cite journal}}
: CS1 maint: multiple names: authors list (
link)![]() | This article includes a
list of references,
related reading, or
external links, but its sources remain unclear because it lacks
inline citations. (June 2020) |
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.
In the following, "almost every" is taken to mean that the stated property holds of a prevalent subset of the space in question.
{{
cite journal}}
: CS1 maint: multiple names: authors list (
link)