A note on pseudo-metrics on the plane

Every c-finite measure Μ on the set G of the lines on the plane such that $$(0){text{ }}mu {text{({ g}} in G:{text{ }}P in {text{g} ) = 0}}$$ 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.