In class, I have learned that Lebesgue outer measure has upward measure continuity. That is, if $A_n$ increase to $A$, $m^*(A_n)$ converges to $m^*(A)$ even though $A_n$ is not measurable. I am wondering whether the same result holds for the inner measure. Is there any counterexample where $m_*(A_n)$ doesn't converge to $m_*(A)$ with $A_n$ increasing to $A$ ?
2026-03-25 17:34:09.1774460049
Does the continuity from below hold for inner measure?
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