Countable boundary null partitions of a separable metric space

S is nonempty since $\mu$(S)=1. Fix a countable dense list $a_i$. For a fixed center, spheres at distinct radii are disjoint; at most r spheres have mass at least 1/r. Thus the radii with positive sphere mass form a countable union of finite, increasing-order lists. For each k,i, AC chooses $r_{k,i}\in (2^{-k-2},2^{-k-1})$ outside this countable exceptional set. The balls $B(a_i,r_{k,i})$ cover S by density, and each has diameter at most 2^{-k} and boundary contained in its null sphere.

At level k disjointize this ordered cover: $D_{k,i}=B(a_i,r_{k,i})\setminus {\bigcup }_{j<i}B(a_j,r_{k,j})$. These sets partition S; discard empty members. Their boundaries lie in the finite union of the first i sphere boundaries, hence are null by F1. Let $P_k$ consist of all nonempty intersections $D_{1,i_1}\cap \cdots \cap D_{k,i_k}$. These form a countable Borel partition, refine the preceding one, and have diameter at most 2^{-k}; their boundaries are again contained in finitely many null boundaries.

Every partition atom is Borel, so the $\sigma$-algebra they generate is contained in Borel(S). Conversely if U is open and x belongs to U, choose a ball about x contained in U and then k with 2^{-k} below its radius. The $P_k$ atom containing x lies in that ball, hence in U. Thus U is the union of the atoms, over countably many levels and members, that are contained in U. It lies in the generated $\sigma$-algebra, proving equality.