Lebesgue measure of the union of measurable sets

Question

Let us call a subset $E$ of $[0,1]$ Lebesgue measurable if $\lambda^*(E) + \lambda^*([0,1]\setminus E) = 1$, where $\lambda^*$ is the outer measure. How can we derive from this definition the fact that countable unions of measurable sets are measurable? (I have already proved this fact for disjoint unions)