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Math Help - theorem statement

  1. #1
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    theorem statement

    Given the following theorem:

    Let E by any set of real numbers. The following assertion is equivalent to the measurability of E:
    For each epsilon>0, there is an open set O containing E for which m*(O~E)<epsilon
    (where m*() denotes Lebesgue outer measure)
    Is there any way I can manipulate this statement to get the condition that occurs when E is not measurable?

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  2. #2
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    Re: theorem statement

    Negate the statement.

    Let E\subseteq \mathbb{R} be any non-measurable set. Then there exists \varepsilon>0 such that for all open sets O with E\subseteq O, m^*(O\setminus E)\ge \varepsilon.

    A theorem that is equivalent to a definition implies a biconditional relationship. The set is measurable if and only if the theorem applies, so if you negate the statement of the theorem, you get the opposite result.
    Last edited by SlipEternal; November 12th 2013 at 09:00 PM.
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  3. #3
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    Re: theorem statement

    That's sort of what I was thinking, thank you!
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