Monotonicity of Lebesgue outer measure ($A\subseteq B⟹m^{\ast }(A)\le m^{\ast }(B)$)

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.