Transverse measure
# 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

       $v+S=\{v+s\in V|s\in S\}$
    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 $R^2$ with its usual Euclidean norm/metric structure. Define a measure μ on $R^2$ by setting μ(E) to be the one-dimensional Lebesgue measure of the intersection of E with the first coordinate axis:

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

An example of a compact set K with positive and finite μ-measure is K = $B_1(0),$ 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 $(v_1,$ $v_2)$ + S of S will meet the first coordinate axis in precisely one point, $(v_1,$ 0). Since a single point has Lebesgue measure zero, $μ((v_1,$ $v_2)$ + S) = 0, and so μ is transverse to S.

