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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- Oct 22nd 2010, 03:32 AM #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

- Oct 22nd 2010, 10:32 AM #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.)

- Oct 24th 2010, 09:19 AM #3

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- Oct 24th 2010, 11:48 AM #4

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The only reasonable definition I see for a µ*-null set is this. A set A is a µ*-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 µ*(A) = 0. And it is easy to show that Cantor set is a null set in this sense.

According to the same link, µ(A) is by definition µ*(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.

- Oct 25th 2010, 03:13 AM #5

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- Oct 28th 2010, 12:04 PM #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

µ*(E)={0 if E is countable

1 if E is uncountable

i thought that first we prove that µ* is an outermeasure, then we prove that what is required...is this true? and how to proceed later on?

- Oct 28th 2010, 12:44 PM #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 .

He gave me this outermeasure

µ*(E)={0 if E is countable

1 if E is uncountable

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.