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Thread: 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 $\displaystyle \varepsilon > 0$, A can be covered by intervals whose total length (or volume) is less than $\displaystyle \varepsilon$. 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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