Measure-Preserving Transformations, Copulæ and Compatibility

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {D}(A,B)$$\end{document} of all copulas compatible with A and B.