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Added: It was pointed out that this proposition is false.

Let $A$ be a bounded set in $\mathbb{R}^n$. How to prove that $A$ is Lebesgue measurable iff its Lebesgue outer measure $m^*(A)$ equals its Jordan inner measure $m_{J*}(A)$ ?

$A$ is defined to be Lebesgue measurable if for every $\epsilon >0$ there is an open set $A\subseteq O$ such that $m^*(O-A)<\epsilon$.

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closed as off-topic by user21820, Cesareo, Mostafa Ayaz, Namaste, José Carlos Santos Sep 21 '18 at 12:55

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Did you state your question correctly?

What happens for the set: the irrationals in $[0,1]$? It is Lebesgue measurable, its Lebesgue outer measure is $1$, its Jordan inner measure is $0$.

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