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It's a well known theorem that a set with finite outer measure is measurable if and only if its outer and inner measure agree.

I need to justify the assumption that the outer measure of the set is finite, i.e., an example of a non-measurable set with outer and inner measure both infinite.

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A simple way to do this is to take a non-measurable set $A \subseteq [0,1]$ and consider $B = ( - \infty , 0 ) \cup A \cup ( 1 , + \infty )$. Clearly the inner and outer Lebesgue measure of $B$ are infinite, and $B$ is also not measurable.

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