Inmathematics, ameasure on arealvector space is said to betransverse to a given set if it assignsmeasure zero to everytranslate of that set, while assigning finite andpositive (i.e. non-zero) measure to somecompact set.

LetV be a real vector space together with ametric space structure with respect to which it iscomplete. ABorel measureμ is said to betransverse to a Borel-measurable subsetS ofV if

• there exists a compact subsetK ofV with 0 < μ(K) < +∞; and
• μ(v + S) = 0 for allv ∈ V, where

${\displaystyle v+S=\{v+s\in V|s\in S\}}$

is the translate ofS byv.

The first requirement ensures that, for example, thetrivial measure is not considered to be a transverse measure.

As an example, takeV to be theEuclidean planeR² with its usual Euclidean norm/metric structure. Define a measureμ onR² by settingμ(E) to be the one-dimensionalLebesgue measure of the intersection ofE with the first coordinate axis:

${\displaystyle \mu (E)={\lambda }^1(\{x\in \mathbf{R}|(x,0)\in E\subseteq {\mathbf{R}}^2\}).}$

An example of a compact setK with positive and finiteμ-measure isK = B₁(0), theclosed unit ball about the origin, which hasμ(K) = 2. Now take the setS to be the second coordinate axis. Any translate (v₁, v₂) + S ofS 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 soμ is transverse toS.

• Prevalent and shy sets

• 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)

• Measures (measure theory)

• CS1 maint: multiple names: authors list