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| In [[mathematics]], a [[measure (mathematics)|measure]] on a [[real number|real]] [[vector space]] is said to be '''transverse''' to a given set if it assigns [[measure zero]] to every [[Translation (geometry)|translate]] of that set, while assigning finite and [[Positive number|positive]] (i.e. non-zero) measure to some [[compact space|compact set]].
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| ==Definition==
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| Let ''V'' be a real vector space together with a [[metric space]] structure with respect to which it is a [[complete space]]. A [[Borel measure]] ''μ'' is said to be '''transverse''' to a Borel-measurable subset ''S'' of ''V'' if
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| * there exists a compact subset ''K'' of ''V'' with 0 < ''μ''(''K'') < +∞; and
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| * ''μ''(''v'' + ''S'') = 0 for all ''v'' ∈ ''V'', where
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| ::<math>v + S = \{ v + s \in V | s \in S \}</math>
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| :is the translate of ''S'' by ''v''.
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| The first requirement ensures that, for example, the [[trivial measure]] is not considered to be a transverse measure.
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| ==Example==
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| As an example, take ''V'' to be the [[Euclidean plane]] '''R'''<sup>2</sup> with its usual Euclidean norm/metric structure. Define a measure ''μ'' on '''R'''<sup>2</sup> by setting ''μ''(''E'') to be the one-dimensional [[Lebesgue measure]] of the intersection of ''E'' with the first coordinate axis:
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| :<math>\mu (E)= \lambda^{1} \big( \{ x \in \mathbf{R} | (x, 0) \in E \subseteq \mathbf{R}^{2} \} \big).</math>
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| An example of a compact set ''K'' with positive and finite ''μ''-measure is ''K'' = ''B''<sub>1</sub>(0), the [[closed unit ball]] about the origin, which has ''μ''(''K'') = 2. Now take the set ''S'' to be the second coordinate axis. Any translate (''v''<sub>1</sub>, ''v''<sub>2</sub>) + ''S'' of ''S'' will meet the first coordinate axis in precisely one point, (''v''<sub>1</sub>, 0). Since a single point has Lebesgue measure zero, ''μ''((''v''<sub>1</sub>, ''v''<sub>2</sub>) + ''S'') = 0, and so ''μ'' is transverse to ''S''.
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| ==See also== | |
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| * [[Prevalent and shy sets]]
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| ==References==
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| * {{cite journal
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| | author = Hunt, Brian R. and Sauer, Tim and Yorke, James A.
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| | title = Prevalence: a translation-invariant "almost every" on infinite-dimensional spaces
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| | journal = Bull. Amer. Math. Soc. (N.S.)
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| | volume = 27
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| | year = 1992
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| | pages = 217–238
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| | doi = 10.1090/S0273-0979-1992-00328-2
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| | issue = 2
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| }}
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| [[Category:Measures (measure theory)]]
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