zzzhhh
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Suppose \mathcal A is an infinite \sigma-algebra, how to construct a disjoint sequence in \mathcal A such that each term is nonempty? Thanks!
stat22 said:Let X1, X2, X3, ..., Xn, ... are the elements of the said \sigma-algebra.
Then, for any positive integer n>1, let En = Xn - [Xn\bigcap(Xn-1\bigcup ... \bigcupX1)]
The result is a disjoint sequence {En}, where E1= X1
stat22 said:micromass,
in your example X1={0,2} and X2={0,1}, the intersection is {0} not {0,1}. Thus X2-(X2\bigcapX1)={0,1}-{0}={1} not empty!
Bacle said:I think the standard way, given a collection A:={A_1,...,A_n,..} (I think this works only for countable collections ) to define B:={B_1,...,B_n,...} by:
B_1:=A_1
B_2:=A_2-A_1
...
...
B_n:=A_n-[A_1\/A_2\/...\/A_(n-1)]
Bacle said:Then I imagine one can ignore the empty sets, but that depends on what zzzhhh wants.