Continuity of Measure ( I Think)

In summary, the argument assumes that there is an integer N such that for all j>N :c=c-e+e< μ(Bj)< c+e+e .
  • #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.
 
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  • #2
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.
 
  • #3
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.
 

What is Continuity of Measure?

Continuity of measure is a mathematical concept that deals with the consistency of measurement within a set. It is based on the idea that the measure of a set should not change dramatically if the set is slightly altered.

Why is Continuity of Measure important?

Continuity of measure is important because it allows us to make precise and consistent measurements in mathematics. It also helps us understand the behavior of functions and sets in different scenarios.

How is Continuity of Measure related to the concept of limits?

Continuity of measure is closely related to the concept of limits. In fact, it is one of the key properties of limits. Continuity of measure guarantees that the limit of a function exists and remains the same even if the input value is changed slightly.

What are the consequences of violating Continuity of Measure?

Violating continuity of measure can lead to inconsistencies and inaccuracies in mathematical calculations. It can also affect the behavior of functions and sets, making it difficult to make accurate predictions or draw conclusions.

How is Continuity of Measure used in real-world applications?

Continuity of measure is used in many real-world applications, such as in physics, engineering, and economics. It helps to ensure accurate measurements and predictions in these fields, allowing for better understanding and control of natural phenomena and economic trends.

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