Hint on a measure theory question Let $(X, \mathcal{A}, \mu)$ be a measure space, and define 
$\mu^\bullet: \mathcal{A} \to [0, +\infty]$ by 
$$
\mu^\bullet (A) = 
\sup
\left\{
\mu(B) : B \subseteq A, 
B \in \mathcal{A}, \mu(B) < + \infty 
\right\}
$$
I'm trying to show that $\mu^\bullet$ is countably additive. One thing is clear: if $A_n$ are disjoint sets in $\mathcal{A}$, then if $B \subseteq \cup_n A_n$, and $B_n = B \cap A_n$, then 
$$
\mu(B) = \sum_n \mu(B_n) \leq \sum_n \mu^\bullet(A_n),
$$
thus $\mu^\bullet(\cup_n A_n) \leq \sum_n \mu^\bullet(A_n)$. This is the easy direction (I think). 
Could I get a small hint on the reverse inequality? (I don't want the solution...)

Hint: fix $\epsilon>0$ and for each $n$ take $B_n$ with $\mu(B_n)+\epsilon/2^n>\mu(A_n)$, then try to use the fact that $\sum_n\epsilon/2^n=\epsilon$.
