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 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 a complete space. A Borel measure μ is said to be transverse to a Borel-measurable subset S of V if
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 to be the one-dimensional Lebesgue measure of the intersection of E with the first coordinate axis:
An example of a compact set K with positive and finite μ-measure is K = B1, the closed unit ball about the origin, which has μ = 2. Now take the set S to be the second coordinate axis. Any translate + S of S will meet the first coordinate axis in precisely one point,. Since a single point has Lebesgue measure zero, μ = 0, and so μ is transverse to S.