In a ring R (nonempty class of sets closed under difference and finite union), any sequence (here means a function on natural numbers $\displaystyle \mathbb N$) $\displaystyle <E_i>$ in R can be disjointlized to a disjoint sequence $\displaystyle <F_i>$ such that $\displaystyle \bigcup E_i=\bigcup F_i$ by traditional induction using the equation $\displaystyle F_i=E_i-\bigcup \limits_{j <i} E_j$. But for arbitrary uncountable sequence $\displaystyle <E_\alpha>$ in R, I either do not know if it is still possible to turn $\displaystyle <E_\alpha>$ into a disjoint sequence with the same union or have no idea how to use transfinite induction to prove it if disjointlization holds, can you help me with this problem? Thanks!