Abstract
Every c-finite measure Μ on the set G of the lines on the plane such that
for every point P∃R 2 generates a pseudo-metric F on the plane when one puts F P 1, P 2=\(\tfrac{1}{2}\) μ({g∈G:g separates the points P 1 and P 2}) The pseudo-metrics which are generated in this way possess the property of linear additivity, that is F(P 1,P 3)=F(P 1,P 2)+F(P 2,P 3) for P 1,P 2,P 3 on a line, P 2 between P 1 and P 3, and are continuous with respect to the Euclidean topology in R 2 × R 2. In this paper we prove the converse: every linear additive and continuous pseudo-metric F is generated as above by some c-finite measure Μ on G for which (0) holds.
The method of proof shows that values of linearly additive and continuous pseudo-metric F inside every bounded convex polygon C are determined completely by the values of F on (δC)2.
The representation of pseudo-metrics by measures is useful in derivation of inequalities for the former.
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The author is greatly indebted to the referees and to K. Krickeberg for indicating the relation of the present work to the construction of multiparametric Brownian motion [6], [7] as well as for their many remarks towards improving the present manuscript.
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Ambartzumian, R.V. A note on pseudo-metrics on the plane. Z. Wahrscheinlichkeitstheorie verw Gebiete 37, 145–155 (1976). https://doi.org/10.1007/BF00536777
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DOI: https://doi.org/10.1007/BF00536777