Monotonicity of Lebesgue outer measure
= Monotonicity of Lebesgue outer measure
{title2=$A\subseteq B\Longrightarrow m^*(A)\leq m^*(B)$}
If $A\subseteq B$, every countable <interval> cover of $B$ is also a cover of $A$. Taking the <infimum> of the lengths over the larger family of covers can only decrease the result. Thus <Lebesgue outer measure> is monotone on all <subsets>, with no <measurability> assumption.