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Help with some Lebesgue measure problems


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Hi everyone, I'm trying to solve some Lebesgue problems from my exercise book and I got stuck in some of them:

  • Prove that if a set A has zero measure, then its interior is empty. I've thinking on suppose the contrary and find an open subset of A with positive measure, but I'm not really sure if it's the right way.
  • True or false: f is integrable if and only if |f| is integrable over Rn. I know that if f is measurable, then it's correct, but here there is no previous condition so I don't know if the statement is true or false.

Could you give me some tips to solve it? Thanks in advance.

Edited by john33
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55 minutes ago, john33 said:

Hi everyone, I'm trying to solve some Lebesgue problems from my exercise book and I got stuck in some of them:

  • Prove that if a set A has zero measure, then its interior is empty. I've thinking on suppose the contrary and find an open subset of A with positive measure, but I'm not really sure if it's the right way.
  • True or false: f is integrable if and only if |f| is integrable over Rn. I know that if f is measurable, then it's correct, but here there is no previous condition so I don't know if the statement is true or false.

Could you give me some tips to solve it? Thanks in advance.

For the first one I would try the inverse direction; something like (as @mathematicsuggests),

 

intCØμ(C)0

Think open balls.

I see you already have good help, so I'll leave it at that.

Cheers.

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