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I proved this problem: Let $E =\bigcup\limits_{i=1}^{\infty} E_i$ where $E_i$ are Lebesgue measurable subsets of $[0,1]$, and $E_1 \subseteq E_2 \subseteq E_3 \subseteq ....$ Show that $m^*(E_i) \rightarrow m^*(E) $ as $i \rightarrow \infty$ where $m^*$ is Lebesgue outer measure.

But now, i need some hints how to spell out second part: Is the conclusion above still true if the sets $E_i$ are not necessarily measurable?

I think the answer is No, and i try using the fact: that for any arbitrary set $E$, there exists a Lebesgue measurable set $A$ such that $E\subseteq A$ and $m^*(A) = m^*(E)$.

How do I continue from here? Thanks

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If the sets $E_i$ are not Lebesque measurable then the question doesn't make any sense...the sequence $m^*(E_i)$ is not well-defined.

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