1. ## 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

2. 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.)

3. 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 µ* nullset has the same definition as a µ nullset where µ* is an outer measure??

4. The only reasonable definition I see for a µ*-null set is this. A set A is a µ*-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 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.

5. Thank you, i think i got it from here

6. 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?

7. 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
µ*(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.