TITLE: constructing a disjoint set
QUESTION [0 upvotes]: Let $F_1,F_2,\dots$ be sets in some sigma algebra, let $E_i = F_i \setminus\bigcup_{j<i} F_j$. It is true that now $E_i$ and $F_i$ are disjoint, but is it true that $\bigcup_{i=1}^n E_i = \bigcup_{i=1}^n F_i$? the infinite case I can see, but not the finite case

REPLY [1 votes]: $\text{“}E_i$ and $F_i$ are disjoint$\text{''}$ could be construed to mean $E_i\cap F_i=\varnothing$, and that is not true.  It is true that $E_1,E_2,E_3,\ldots$ are pairwise disjoint.
Suppose $x\in\bigcup_i F_i$.  Then there is some smallest index $i_0$ such that $x\in F_i$.  For that smallest index $i_0$ we have $x\in E_{i_0}$; therefore $x\in\bigcup_i E_i$.
{"set_name": "stack_exchange", "score": 0, "question_id": 1766793}