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Math Help - show that set is measurable

  1. #1
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    show that set is measurable

    Hi all,

    Given a set E; for each open, bounded interval (a,b):
    b-a = m*((a,b)∩E) + m*((a,b)~E)
    implies E is (Lebesgue) measurable.

    (we're working with the real line here, so E is a subset of R)
    I'm having trouble with this problem. Hint given is to show that A = { E: b-a = m*((a,b)∩E) + m*((a,b)~E) } is a sigma algebra.
    I understand the hint logic - if A is a sigma algebra then the sets in A are measurable sets so E must be measurable.

    I know that all open, bounded intervals are measurable.
    I know the three requirements for a collection of sets to be a sigma algebra.

    I don't know how to connect the dots though.

    Where do I start? Thanks
    Last edited by director; November 10th 2013 at 05:17 PM.
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  2. #2
    Junior Member
    Joined
    May 2010
    Posts
    63

    Re: show that set is measurable

    Here is the definition of meausrability of a set E given in my text:

    A set E is said to be measurable provided for any set A, m*(A) = m*(A∩E) + m*(A∩(complement of E))
    It looks like (A∩(complement of E)) = (A~E) or (A\E) (just different notation)

    So, the equation in the original problem looks very close to what I have in the definition of measurable set. Except I can't say it holds for any set A, only if A=(a,b).
    If I were to restrict the measure space (correct term?) to one that contains the just open sets, then E would be measurable?

    The complement of an measurable open set is a measurable closed set... What if I add these in?
    Last edited by director; November 11th 2013 at 04:38 PM.
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