john33 0 Posted April 8 Share Posted April 8 (edited) 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 April 8 by john33 Link to post Share on other sites

mathematic 104 Posted April 8 Share Posted April 8 Non-empty open sts must have positive measure, so your approach to the first question is correct. The second question is almost from the definition. 1 Link to post Share on other sites

joigus 443 Posted April 8 Share Posted April 8 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), intC≠Ø⇒μ(C)≥0 Think open balls. I see you already have good help, so I'll leave it at that. Cheers. Link to post Share on other sites

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