Rudin Principles of mathematical analysis p307 "For if $ A=\bigcup A^{'}_{n}$ with $A^{'}_{n} \in M_F(\mu)$, write $A_1=A^{'}_{1} $, and
$$ A_n=(A^{'}_1\cup ...\cup A^{'}_n)-(A^{'}_n \cup ... \cup A^{'}_{n-1})$$ $(n=2,3,4,...)$.
Then 
$$ A=\bigcup_{n=1}^{\infty}A_n$$
I can't understand why $A_n$ is expressed like the above? Should the correct one be 
$$ A_n=A^{'}_n-(A^{'}_1 \cup ... \cup A^{'}_{n-1})$$

At least in some editions of the book (say, the 3rd edition) there seems to be a typo. The defining identity $$ A_n=(A^{'}_1\cup ...\cup A^{'}_n)-(A^{'}_n \cup ... \cup A^{'}_{n-1})$$  should be replaced by $$ A_n=(A^{'}_1\cup ...\cup A^{'}_n)-(A^{'}_1 \cup ... \cup A^{'}_{n-1}).$$ Since $A_1=A^{'}_1$, this is equivalent to $$ A_n=A^{'}_n-(A^{'}_1 \cup ... \cup A^{'}_{n-1}).$$ Each definition leads to $$A_1\cup ...\cup A_n=A^{'}_1\cup ...\cup A^{'}_n,$$ for every $n$, and the rest follows.
