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 a complete space. 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 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:

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

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. doi:10.1090/S0273-0979-1992-00328-2.