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Following is the Carathéodory's criterion for Lebesgue measurability

$A\subset \mathbb{R}$ is Lebesgue measurable $\textbf{iff}$ for any set B (measurable or not) $m_*(B)=m_*(B\cap A)+m_*(B -A)$.

where $m_*(B)$ is the outer measure of $A$.

How one could show that this is equivalent to other criteria of Lebesgue measurability such as

$A\subset \mathbb{R}$ is Lebesgue measurable $\textbf{iff}$ for any $\epsilon>0$ there exists an open set $O$ that contains $A$ and $m_*(O-A)<\epsilon$.

The above criterion can be found in Stein Shakarchi "Real Analysis" book printed 2007, page 21, Theorem 3.4.

Susan_Math123
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  • Maybe I'm misunderstanding something, but isn't this definition of Lebesgue measure? – Ennar Oct 20 '15 at 16:15
  • @Ennar I don't think so. In my last paragraph I used one of the definitions to prove, but not successful yet. – Susan_Math123 Oct 20 '15 at 16:21
  • Well, how do you define Lebesgue measure? – Ennar Oct 20 '15 at 16:23
  • @Ennar I see what you are talking about. The one I mentioned above is the definition of Lebesgue measure in Wiki https://en.wikipedia.org/wiki/Lebesgue_measure. But my definition is that if A is Lebesgue measurable then for every positive ε there exist an open set G and a closed set F such that G⊇A⊇F and λ(G\A)<ε and λ(A\F)<ε. – Susan_Math123 Oct 20 '15 at 16:25
  • What you state is regularity of Lebesgue measure. See. I'm not aware of characterization of Lebesgue measure via regularity alone. Where is this definition from? – Ennar Oct 20 '15 at 16:39
  • @Ennar Property 10 in https://en.wikipedia.org/wiki/Lebesgue_measure . You can also see that in Stein Shakarchi book "real analysis" page 21 Theorem 3.4. – Susan_Math123 Oct 20 '15 at 17:18
  • You should include this definition in your question because, as far as I know, standard approach is through Caratheodory construction described in Wikipedia, and the definition you are using is proved as a theorem. The current form of question might be confusing to others as it was to me. – Ennar Oct 20 '15 at 18:45
  • @Ennar Thanks. I added that to the problem statement. – Susan_Math123 Oct 22 '15 at 13:58
  • See here: https://math.stackexchange.com/questions/305221/equivalent-definitions-of-lebesgue-measurability – amsmath Sep 05 '19 at 13:20

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