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Continuity of Measure ( I Think)

  1. Feb 27, 2013 #1

    WWGD

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    Hi, All:

    I think the following deals with continuity of measure, but I'm not 100%:

    Let I:=[0,1] , and let An be a sequence of pairwise-disjoint measurable sets
    whose union is I ( is me? :) ) . Let {Bj} be a sequence of measurable subsets
    of I , so that, for μ the standard Lebesgue measure:

    Limj→∞ μ( An\cap Bj )=0 , for all n .

    I want to show that above implies that : Limj→∞ μ(Bj)=0 .(**)

    This is what I have:

    We know that Ʃμ(An)=1 . So we must have some Ano in the
    collection with μ(Ano)=a>0.

    ( I am assuming that the A_n's must all be of the form [a,b) , with A1=[0,a)

    A2=[a,b) , etc. , plus a {1} thrown-in )

    Now, I am trying to argue by contradiction , assuming that the limit above in (**) equals
    some c+e ; e->0 , though I am not sure of how to show that the limit actually exists,
    tho I am assuming for now that it does:

    So, assuming limit in (**) exists and equals c+e ( e->0) , we have that there is an
    integer N such that for all j>N :

    c=c-e+e< μ(Bj)< c+e+e

    In particular, μ(Bj)>c>0 .

    Now, I can find an open set Oj, for each j , with

    μ(Cj)=μ(Oj) .

    I know the quantification here is tricky; I am then using that:


    Oj= \/(cji ,dji)

    And, since m(Bj)>c for all j>N , there is an index for the j's --

    use j=1 without loss of generality -- such that m(c1,d1)>0

    Now, this interval (c1,d1) must intersect some interval

    An , and the intersection must be of one of the forms:

    [x,y) , (x,y] , or (x,y) . In either case, the measure of the intersection is

    y-x>0 , contradicting the assumption condition (**) that

    Limj→∞ μ(Bj)=0.

    I think I'm on the right track, but not 100%. Please critique.

    Thanks.
     
  2. jcsd
  3. Feb 27, 2013 #2

    MathematicalPhysicist

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    Can you translate your question to Latex or PDF, cause it's hard to read through ascii. I am not 20 anymore that I have the patience to read that way.
     
  4. Mar 9, 2013 #3

    Bacle2

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    Use the fact that the tail of the sum m(Ai) goes to zero, since the total sum is 1

    Then use the fact that, by the limit condition, there is a K>0 with

    Lim_n->0 (Bk /\An)=0 , for all k>K . Then , from the fact that m(Ai)->0 ,

    Use Bk=(Bk /\ U Ai) , to conclude that m(Bk)->0.
     
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