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!



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


