Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    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
    micromass,

    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

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    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:

    B_1:=A_1
    B_2:=A_2-A_1
    ......
    ......
    B_n:=A_n-[A_1\/A_2\/......\/A_(n-1)]
     
  9. Jul 3, 2011 #8

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    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

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    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...
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: How to construct a disjoint sequence from an infinite sigma-algebra?
  1. Infinite sequence (Replies: 4)

  2. Sigma algebra? (Replies: 8)

Loading...