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Math Help - nullsets and putermeasures

  1. #1
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    nullsets and putermeasures

    Hi,
    i have a true or false question that seems a bit tricky for me:
    suppose m* is a Lebesgue Stieltjes outermeasure, then is it possible to say that a m* nullset is countable?

    the problem is i need to figure out what's a m*-nullset first!
    Thank you
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  2. #2
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    See this definition of a null set.

    Cantor set is an example of an uncountable set that has Lebesgue measure 0, which implies that it is a null set. (I don't remember how trivial is the proof of implication, but it may not be needed for the Cantor set.)
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  3. #3
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    i guess i wasn't clear with my question because i know that the cantor set is an example for uncountable nullset. but is a * nullset has the same definition as a nullset where * is an outer measure??
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  4. #4
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    The only reasonable definition I see for a *-null set is this. A set A is a *-null set if for every \varepsilon > 0, A can be covered by intervals whose total length (or volume) is less than \varepsilon. First, this is the definition in the Wikipedia link above. Second, it is the same as saying *(A) = 0. And it is easy to show that Cantor set is a null set in this sense.

    According to the same link, (A) is by definition *(A) for a Lebesgue-measurable set A, which Cantor set certainly is. So, proving that Cantor set has Lebesgue measure 0 involves showing two facts: it is measurable and its outer measure is 0. I tend to think that the second facts implies the first, but it was a long time ago since I studied this.
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  5. #5
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    Thank you, i think i got it from here
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  6. #6
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    I asked my professor about using the cantor set an an example and he said he wants me to do it in anothjer way . He gave me this outermeasure
    *(E)={0 if E is countable
    1 if E is uncountable

    i thought that first we prove that * is an outermeasure, then we prove that what is required...is this true? and how to proceed later on?
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  7. #7
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    I asked my professor about using the cantor set an an example and he said he wants me to do it in anothjer way .
    What do you think he means: that your solution is wrong or that it is right but he wants a different approach?

    He gave me this outermeasure
    *(E)={0 if E is countable
    1 if E is uncountable
    Well, he has the right to give a definition. I am not sure how this is related to Lebesgue measure.

    Maybe we are missing some definitions here. I would advise you to find all relevant definitions: outer measure, null set, Lebesgue measure, Lebesgue–Stieltjes measure, etc.
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