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How to construct a disjoint sequence from an infinite sigma-algebra?

  1. Sep 18, 2010 #1
    Suppose [tex]\mathcal A[/tex] is an infinite [tex]\sigma[/tex]-algebra, how to construct a disjoint sequence in [tex]\mathcal A[/tex] such that each term is nonempty? Thanks!
  2. jcsd
  3. Sep 19, 2010 #2
    Last edited by a moderator: Apr 25, 2017
  4. Jul 2, 2011 #3
    Let X1, X2, X3, ..., Xn, ... are the elements of the said [itex]\sigma[/itex]-algebra.

    Then, for any positive integer n>1, let En = Xn - [itex][[/itex]Xn[itex]\bigcap[/itex][itex]([/itex]Xn-1[itex]\bigcup[/itex] ... [itex]\bigcup[/itex]X1[itex])[/itex][itex]][/itex]

    The result is a disjoint sequence {En}, where E1= X1
  5. Jul 2, 2011 #4
    And who says that the terms are nonempty?? For example, take X1=[0,2] and X2=[0,1]. Then

    [tex]X_2\setminus (X_2\cap X_1)=[0,1]\setminus [0,1]=\emptyset[/tex]
  6. Jul 2, 2011 #5

    in your example X1={0,2} and X2={0,1}, the intersection is {0} not {0,1}. Thus X2-(X2[itex]\bigcap[/itex]X1)={0,1}-{0}={1} not empty!
  7. Jul 2, 2011 #6
    I'm talking about the interval [0,1], not {0,1}.

    But, if you don't like intervals, consider X1={0,1}, X2={0}. Then

    [tex]X_2\setminus(X_2\cap X_1)=\emptyset[/tex]
  8. Jul 3, 2011 #7
    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:

  9. Jul 3, 2011 #8
    Yes, these are certainly disjoint, but perhaps empty!
  10. Jul 3, 2011 #9
    Then I imagine one can ignore the empty sets, but that depends on what zzzhhh wants.
  11. Jul 3, 2011 #10
    Yes, but you still need to end up with an infinite number of sets. If you throw away too many sets, then you might end up with a finite number of sets...
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