Bivariate measure-inducing quasi-copulas

It is well known that every bivariate copula induces a positive measure on the Borel $\sigma$-algebra on $[0,1{]}^2$, but there exist bivariate quasi-copulas that do not induce a signed measure on the same $\sigma$-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.