In a ring R (nonempty class of sets closed under difference and finite union), any sequence (here means a function on natural numbers ) in R can be disjointlized to a disjoint sequence such that by traditional induction using the equation . But for arbitrary uncountable sequence in R, I either do not know if it is still possible to turn 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!