  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.
