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.

  • $\begingroup$ Maybe I'm misunderstanding something, but isn't this definition of Lebesgue measure? $\endgroup$ – Ennar Oct 20 '15 at 16:15
  • $\begingroup$ @Ennar I don't think so. In my last paragraph I used one of the definitions to prove, but not successful yet. $\endgroup$ – Susan_Math123 Oct 20 '15 at 16:21
  • $\begingroup$ Well, how do you define Lebesgue measure? $\endgroup$ – Ennar Oct 20 '15 at 16:23
  • $\begingroup$ @Ennar I see what you are talking about. The one I mentioned above is the definition of Lebesgue measure in Wiki 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. $\endgroup$ – Susan_Math123 Oct 20 '15 at 16:25
  • $\begingroup$ 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? $\endgroup$ – Ennar Oct 20 '15 at 16:39

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