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Useful pi systems generating the borel sigma algebra

  1. May 19, 2012 #1
    1. The problem statement, all variables and given/known data

    For the purpose of my module, we define the lebesgue measure over Int(a,b) where
    Int(a,b) = (a,b] where -∞≤a≤b<∞
    (a,∞) else
    And l((Int(a,b) = b-a a≤b<∞
    ∞ a≤b =∞

    And thus we say the borel sigma algebra is generated by C={Int(a,b) -∞≤a≤b≤∞}
    We then get told that a variety of pi systems generate the borel sigma algebra, other than C={Int(a,b) -∞≤a≤b≤∞} ;
    1. C' = {(a,b] -∞<a≤b<∞}
    2. C_0 = {(a,b) -∞≤a≤b<∞ }
    3. C_open = { A [itex]\subseteq[/itex] Reals : A open}
    4 C_closed = { B [itex]\subseteq[/itex] Reals: B closed }
    5. C_half = {(-∞,b] b [itex]\in[/itex] Reals }

    We need to prove that 1-5 generate the borel sigma algebra and that lebesgue is sigma finite on 1-4 but not 5 (to do this I need to find an increasing sequence of events in each C such that the union is the sample space (the reals) and that l(Ai) < ∞ for all i .



    3. The attempt at a solution

    Basically I have proved 1-4 generate the borel sigma algebra, but 5 is proving difficult, I know that C_half [itex]\subseteq[/itex] C [itex]\subseteq[/itex] σ(C) so this proves that
    σ(C_half) [itex]\subseteq[/itex]σ(C), but now I need to show it the other way round, to show that they're equal. I can't seem to figure this out.

    Secondly, again I have shown sigma finite for 1 and 2, but i can't do 3 (and therefore 4), with all the others I did how I had been taught, i.e for 2 I said let Ai = (-i,i) then Ai is increasing, the union is the reals and then I wrote (-i,i) = U{n=1..} (-i,i-1/n] and used continuity to take lebesgue measure ... etc.
    How would you do this with open and closed sets?
    Finally for showing l isnt sigma finite on 5, is it enough to say that whatever increasing sequence you take in C_half, using continuity of the lebesgue, when you do the lebesgue measure will always get ∞ + b where b is in the reals which is clearly never going to be finite?


    Thanks a lot for any help, I'm really stuck on these and theyre really important in my module!
     
  2. jcsd
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