= Countable subadditivity of Lebesgue outer measure
{title2=$m^*(\bigcup_n A_n)\leq\sum_n m^*(A_n)$}
If the sum of <Lebesgue outer measures> on the right is infinite, the inequality is automatic. Otherwise, fix $\varepsilon>0$ and cover each $A_n$ by a countable family of <intervals> whose total length is less than $m^*(A_n)+\varepsilon2^{-n}$. These covers exist by the definition of <infimum>. The combined countable family covers $\bigcup_nA_n$ and has total length at most $\sum_nm^*(A_n)+\varepsilon$. Take the infimum over covers of the union, then let $\varepsilon$ tend to zero. No <measurability> assumption on the $A_n$ is needed.
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