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$F_{\sigma}$ is a countable union of closed sets, and $G_{\delta}$ is a countable intersection of open sets.

I have proven the forward implication already. For the second one I know that I can pick one of the closed sets that make up $B$, say the closed set $F$ with $F \subset B$, and one of the open sets that make up $C$, say the open set $G$ with $C \subset G$. It then follows that $F \subset A \subset G$. I am having trouble showing that the measure of $G$~$F$ is less than an arbitrary $\epsilon > 0$. I know I need to use the fact that $C$~$B$ is a null set but I am not sure how.

Any help is appreciated.