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
v + S = { v + s ∈ V | s ∈ S } {\displaystyle v+S=\{v+s\in V|s\in S\}}
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:
μ ( E ) = λ 1 ( { x ∈ R | ( x , 0 ) ∈ E ⊆ R 2 } ) . {\displaystyle \mu (E)=\lambda ^{1}{\big (}\{x\in \mathbf {R} |(x,0)\in E\subseteq \mathbf {R} ^{2}\}{\big )}.}
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 Prevalent and shy sets
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. arXiv:math/9210220. doi:10.1090/S0273-0979-1992-00328-2. S2CID 17534021.{{cite journal}}: CS1 maint: multiple names: authors list (link)
