Past exam of the options I course of the University of Oxford / 2011 / D / 1 / c / Solution

If $A\subseteq E$ and $E$ is a null set, every interval cover of $E$ also covers $A$. Thus monotonicity of Lebesgue outer measure gives $0\le m^{\ast }(A)\le m^{\ast }(E)=0$. Hence every subset of a null set is null, and in particular is Lebesgue measurable.