Measure-Preserving Transformations, Copulæ and Compatibility

Abstract.

We study the relationship between copulas and measure-preserving transformations on the Borel sets of the unit interval. This also allows to investigate the connection with a restricted compatibility problem for copulas. To this end, in order to construct a 3-copula from two given 2-copulas A and B, we modify the *-operation introduced in Darsow et al., show that A * B is always compatible with A and B, and study the set $\mathcal {D}(A,B)$ of all copulas compatible with A and B.