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.
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Transcription
Let's say that we have two parallel lines. So that's one line right over there, and then this is the other line that is parallel to the first one. I'll draw it as parallel as I can. So these two lines are parallel. This is the symbol right over here to show that these two lines are parallel. And then let me draw a transversal here. So let me draw a transversal. This is also a line. Now, let's say that we know that this angle right over here is 110 degrees. What other angles can we figure out here? Well, the first thing that we might realize is that, look, corresponding angles are equivalent. This angle, the angle between this parallel line and the transversal, is going to be the same as the angle between this parallel line and the transversal. So this right over here is also going to be 110 degrees. Now, we also know that vertical angles are equivalent. So if this is 110 degrees, then this angle right over here on the opposite side of the intersection is also going to be 110 degrees. And we could use that same logic right over here to say that if this is 110 degrees, then this is also 110 degrees. We could've also said that, look, this angle right over here corresponds to this angle right over here so that they also will have to be the same. Now, what about these other angles? So this angle right over here, its outside ray, I guess you could say, forms a line with this angle right over here. This pink angle is supplementary to this 110 degree angle. So this pink angle plus 110 is going to be equal to 180. Or we know that this pink angle is going to be 70 degrees. And then we know that it's a vertical angle with this angle right over here, so this is also 70 degrees. This angle that's kind of right below this parallel line with the transversal, the bottom left, I guess you could say, corresponds to this bottom left angle right over here. So this is also 70 degrees. And we could've also figured that out by saying, hey, this angle is supplementary to this angle right over here. And then we could use multiple arguments. The vertical angle argument, the supplementary argument two ways, or the corresponding angle argument to say that, hey, this must be 70 degrees as well.
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
- 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:
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
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. arXiv:math/9210220. doi:10.1090/S0273-0979-1992-00328-2. S2CID 17534021.
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This page was last edited on 5 July 2023, at 06:18