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.

翻译：众所周知，二元连接函数在$[0,1]^2$上的Borel $\sigma$-代数上诱导一个正测度，但存在某些二元拟连接函数无法在同一$\sigma$-代数上诱导符号测度。本文证明，由二元拟连接函数诱导的符号测度始终可表示为由连接函数诱导的测度的无穷组合。借此，我们首次给出了二元情形下测度诱导拟连接函数的完整刻画。