Countably Generated Idempotent Copulas

It is known that there exists a one-to-one correspondence between the σ-fields of subsets of the unit interval and idempotent copulas. However, the literature reports few examples of idempotents. Exploiting the correspondence that also exists between idempotent copulas and conditional expectations with respect to the σ-fields just mentioned, we provide a general means of constructing idempotents, we explicitly write the copula corresponding to a countably generated σ-field and show that every idempotent copula is the limit of a sequence of such copulas.