Let $\mathcal{U}$ denote the set of all open balls with rational centre and rational radius that are contained in $A_j$ for some $j$.  This is a countable family with the same union as the original family $\{A_j : j\in J\}$.  Now for $U\in\mathcal{U}$ we have $U\cap B\subseteq A_j\cap B$ for some $j$ so $U\cap B$ has measure zero.  As $\mathcal{U}$ is countable and covers $B$ we conclude that $B$ has measure zero.