Let $Z$ be an arbitrary set. Let $X=\prod_{i=1}^n Z_i$ and $Z_i=Z$ for each $i=1,\ldots,n$, $n$ fixed. Consider the $n$-tuple $(\mu_1,\ldots,\mu_n)$ with $\mu_i\in\Delta_s(Z)$ for each $i=1,\ldots,n$, i.e., each $\mu_i$ is a simple probability measure on $Z$ (it has finite support). Let $J=\{\mu\in \Delta_s(X): \operatorname*{marg}_{Z_i}\mu=\mu_i \;\;\forall $i$ \}$, i.e. the set of all simple probability measures on $X$ that have $(\mu_1,\ldots,\mu_n)$ as marginals. Is there any way to characterize the set $J$?
I know that for real valued random variables this problem can be tackled with copulas, but I don't know if copulas can be used in this case. We can assume that there is some complete and transitive order $\succeq$ on $Z$.