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Measure theory

  1. Dec 1, 2015 #1
    1. The problem statement, all variables and given/known data
    Suppose E1 and E2 are a pair of compact sets in Rd with E1 ⊆ E2, and let a = m(E1) and b=m(E2). Prove that for any c with a<c<b, there is a compact set E withE1 ⊆E⊆E2 and m(E) = c.

    2. Relevant equations
    m(E) is ofcourese refering to the outer measure of E

    3. The attempt at a solution
    I know that for d=1 measurable subset of [0,1]. Is it worth it to look at the measure m(E∩[0,t]) as function of t?
    I really don't know how to tackle this one
     
  2. jcsd
  3. Dec 1, 2015 #2

    Ray Vickson

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    The name of the subject is "measure" theory, not "measurement" theory.
     
  4. Dec 1, 2015 #3

    Samy_A

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    Yes, I think that is the way to do it.
     
  5. Dec 1, 2015 #4

    Mark44

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    Fixed...
     
  6. Dec 1, 2015 #5

    fresh_42

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    What does it mean for ##E_1## to be compact? Does anything change if you reduce the task to ##E_2## as topological space? What do you know about the closure of finitely many open sets of finite mass? (You probably won't need to regard ##d## at all.)
     
  7. Dec 1, 2015 #6
    Could you help a bit along the way?
     
  8. Dec 1, 2015 #7

    Krylov

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    I think you may need to exploit continuity of the measure (w.r.t. inclusion).
     
  9. Dec 1, 2015 #8

    Samy_A

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    Instead of an interval you have to use a d-dimensional cube with side t. Lets call that cube C(t).
    Now define the function ##f(t)=m(E_1\cup(E_2\cap C(t))## on a well chosen interval ##[x,y]## of ##\mathbb R## so that ##f(x)=a## and ##f(y)=b##.

    Remember that ##E_1## and ##E_2## are compact, and thus bounded.

    EDIT: not to discourage you, but it does look like quite a difficult exercise. I think it should work with the function ##f## given above though.

    EDIT2: maybe the way @fresh_42 suggests will be easier.
     
    Last edited: Dec 1, 2015
  10. Dec 2, 2015 #9
    I know what you mean, but I want to learn " the hard way" in order to understand it better. I am trying to teach myself measure theory so for me it's not "just homework".

    I can't see how to apply the last information?
     
    Last edited: Dec 2, 2015
  11. Dec 2, 2015 #10

    Samy_A

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    First thing to do is to find ##x## and ##y## satisfying ##f(x)=a,f(y)=b##.
    I think that ##x## is easy, but for a arbitrary ##E_2##, it is not even sure that ##y## exists. But ##E_2## being compact (and thus bounded) makes it easy to find some ##y## large enough to satisfy ##f(y)=b##.

    Next challenge will be to prove that ##f## is continuous.
    Note that ##f## is a monotone function: ##s \leq t \Rightarrow f(s) \leq f(t)##

    Set ##E(t)=E_1\cup(E_2\cap C(t))##
    Verify that if ##s \leq t##, ##E(t)\setminus E(s) \subset C(t) \setminus C(s)##

    Also remember that if ##A \subset B##, then ##m(B \setminus A)=m(B)-m(A)## (assuming that A and B are both measurable of course).
     
  12. Dec 2, 2015 #11
    Can I use this information to conclude then?
     
  13. Dec 2, 2015 #12

    Samy_A

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    I'm not sure I understand your question. I used that in the final step proving that the function ##f## is continuous.
     
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