Factorizable non-atomic copulas

Generalizing the notion of invariant sets by Darsow and Olsen, Sumetkijakan studied a subclass of singular copulas, the so-called non-atomic copulas, defined via its associated σ-algebras. It was shown that the Markov operator of every non-atomic copula is partially factorizable, i.e. it is the composition of left and right invertible Markov operators on a subspace of L1([0,1]) depending on the copula. Here, we further investigate the associated σ-algebras of the product of certain copulas and obtain (1) a sharper result on the partial factorizability of non-atomic copulas and (2) the existence and uniqueness of a completely factorizable copula that shares the same set of associated σ-algebras as that of a given non-atomic copula.