Past exam of the options I course of the University of Oxford / 2008 / D / D1 / a / Solution

A null set has Lebesgue outer measure zero: for every $\epsilon >0$ it can be covered by countably many open intervals with total length below $\epsilon$. If $E_j$ are null, cover $E_j$ by intervals of total length below $\epsilon 2^{-j}$, for $j\ge 1$. The union of all these covers is countable and covers ${\bigcup }_j{E}_j$ with total length below $\epsilon$. As $\epsilon$ is arbitrary, the countable union is null.