# Continuity of Measure ( I Think)

by WWGD
Tags: continuity, measure
 P: 311 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.
 P: 3,128 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.
 Sci Advisor P: 1,167 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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