Bivariate measure-inducing quasi-copulas

It is well known that every bivariate copula induces a positive measure on the Borel σ-algebra on [0,1]^2, but there exist bivariate quasi-copulas that do not induce a signed measure on the same σ-algebra. In this paper we show that a signed measure induced by a bivariate quasi-copula can always be expressed as an infinite combination of measures induced by copulas. With this we are able to give the first characterization of measure-inducing quasi-copulas in the bivariate setting.