countable additivity for an outer measure

Let $\mu^*$ be an outer measure on an non-empty set $X$. Let $(E_n)_{n=1}^{\infty}$ be a sequence of pairwise disjoint $\mu^*-$measurable subsets of $X$. Let $B_n\subset E_n$ be arbitrary(not necessarily measurable), show that

$$\mu^*\left(\bigcup_{n=1}^{\infty}B_n\right)=\sum_{n=1}^{\infty}\mu^*(B_n)$$

Intuitively I see that these $B_n$'s are positively separated, however $\mu^*$ is not assumed to be a metric outer measure. How can we solve this?

Answer

Hint: It suffices to show

$$\sum_{n=1}^N \mu^\ast(B_n) \leqslant \mu^\ast\left(\bigcup_{n=1}^\infty B_n\right)$$

for all $N$. To see that, consider

$$A_N = \bigcup_{n=1}^N B_n$$

and use the measurability of the $E_n$ to conclude

$$\mu^\ast (A_N) = \sum_{n=1}^N \mu^\ast(B_n).$$