|
|
| Line 1: |
Line 1: |
| 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]].
| | I'm Yoshiko Oquendo. Alabama is exactly where he and his spouse live and he has everything that he requirements there. Bookkeeping is what he does. To perform croquet is the pastime I will never quit doing.<br><br>Here is my web page [http://sdasingles.net/index.php?m=member_profile&p=profile&id=25513 extended car warranty] |
| | |
| ==Definition==
| |
| | |
| 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
| |
| * there exists a compact subset ''K'' of ''V'' with 0 < ''μ''(''K'') < +∞; and
| |
| * ''μ''(''v'' + ''S'') = 0 for all ''v'' ∈ ''V'', where
| |
| ::<math>v + S = \{ v + s \in V | s \in S \}</math>
| |
| :is the translate of ''S'' by ''v''.
| |
| | |
| The first requirement ensures that, for example, the [[trivial measure]] is not considered to be a transverse measure.
| |
| | |
| ==Example==
| |
| | |
| 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:
| |
| | |
| :<math>\mu (E)= \lambda^{1} \big( \{ x \in \mathbf{R} | (x, 0) \in E \subseteq \mathbf{R}^{2} \} \big).</math>
| |
| | |
| 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''.
| |
| | |
| ==See also== | |
| | |
| * [[Prevalent and shy sets]]
| |
| | |
| ==References==
| |
| | |
| * {{cite journal
| |
| | author = Hunt, Brian R. and Sauer, Tim and Yorke, James A.
| |
| | title = Prevalence: a translation-invariant "almost every" on infinite-dimensional spaces
| |
| | journal = Bull. Amer. Math. Soc. (N.S.)
| |
| | volume = 27
| |
| | year = 1992
| |
| | pages = 217–238
| |
| | doi = 10.1090/S0273-0979-1992-00328-2
| |
| | issue = 2
| |
| }}
| |
| | |
| [[Category:Measures (measure theory)]]
| |
I'm Yoshiko Oquendo. Alabama is exactly where he and his spouse live and he has everything that he requirements there. Bookkeeping is what he does. To perform croquet is the pastime I will never quit doing.
Here is my web page extended car warranty