Transverse measure
In mathematics, a measure on a real vector space is said to be transverse to a given set if it assigns measure zero to every translate of that set, while assigning finite and positive (i.e. non-zero) measure to some compact set.
Definition
Let V be a real vector space together with a metric space structure with respect to which it is complete. 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]\displaystyle{ 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 R2 with its usual Euclidean norm/metric structure. Define a measure μ on R2 by setting μ(E) to be the one-dimensional Lebesgue measure of the intersection of E with the first coordinate axis:
- [math]\displaystyle{ \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 = B1(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 (v1, v2) + S of S will meet the first coordinate axis in precisely one point, (v1, 0). Since a single point has Lebesgue measure zero, μ((v1, v2) + S) = 0, and so μ is transverse to S.
See also
References
- Hunt, Brian R. and Sauer, Tim and Yorke, James A. (1992). "Prevalence: a translation-invariant "almost every" on infinite-dimensional spaces". Bull. Amer. Math. Soc. (N.S.) 27 (2): 217–238. doi:10.1090/S0273-0979-1992-00328-2.
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