Measures of concordance determined by 𝐷₄-invariant measures on (0,1)²

Abstract

A measure, μ \mu , on ( 0 , 1 ) 2 (0,1)^2 is said to be D 4 D_4 -invariant if its value for any Borel set is invariant with respect to the symmetries of the unit square. A function, κ \kappa , generated in a certain way by a measure, μ \mu , on ( 0 , 1 ) 2 (0,1)^2 is shown to be a measure of concordance if and only if the generating measure is positive, regular, D 4 D_4 -invariant, and satisfies certain inequalities. The construction examined here includes Blomqvist’s beta as a special case.