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 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} ig( { x in mathbf{R} | (x, 0) in E subseteq mathbf{R}^{2} } ig).

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".

ee also

* Prevalent and shy sets

References

* cite journal
author = Hunt, Brian R. and Sauer, Tim and Yorke, James A.
title = Prevalence: a translation-invariant "almost every" on infinite-dimensional spaces
journal = Bull. Amer. Math. Soc. (N.S.)
volume = 27
year = 1992
number = 2
pages = 217–238
doi = 10.1090/S0273-0979-1992-00328-2


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