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

Let's take the usual ring of finite unions $E$ of halfopen rectangles $[a,b)\times [c,d)$ on the plane. The closed halfplane $x+y\ge 0$ is a union of continuum of such rectangles (all possible rectangles contained in that closed halfplane) but, since each $E$ contained in this halfplane can intersect the boundary line by only finitely many points, you cannot get the halfplane as a union of countably many ring elements. On the other hand, any disjoint family of ring elements is at most countable. 


"Disjointlization" ... the world would be a better place without this word! 

