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!
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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! 

