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Is it possible for every $0\leq a< b \leq \infty$ to find a set $A\subset R^n$ such that the inner Lebesgue measure of $A$ is equal $a$ and the outer Lebesgue measure is equal $b$. It is true [Halmos, Measure theory, Chap.3, $\S$16, Th. 5] if $a=0$ and $0 < b \leq \infty$. Is it true in general?

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Thank you very much for your help. – Richard Aug 6 '11 at 14:00
up vote 5 down vote accepted

It is true. For any $c$ with $0 \le c \le \infty$, we can find a subset $C$ of $\mathbb{R}^+$ with inner measure $0$ and outer measure $c$.

To solve your problem for $\mathbb{R}$, just let $c=b-a$, shift $C$ by $a$ to the right, and add the interval $[0,a]$.

An analogous idea works for $\mathbb{R}^n$.

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