Countable partition construction of the Borel set function

For a density $r$ as in Pointwise Borel nonnegative densities, choose a countable locally finite chart cover $(U_i,x_i)$ and a subordinate smooth partition of unity $({\phi }_i)$ with ${\phi }_i\ge 0$, $\mathrm{supp}{\phi }_i\subset U_i$ and ${\sum }_i{\phi }_i=1$. Define the proposed set function on $\mathcal{B}(M)$ by ${\mu }_{r,(x_i,{\phi }_i)}(E)={\sum }_i{\int }_{x_i(E\cap U_i)}({\phi }_i\circ x_i^{-1})r_{x_i}d{\lambda }_n\mathrm{.}$ Each term is the nonnegative integral of The nonnegative Lebesgue integral, with $0\cdot \mathrm{\infty }=0$. Its coefficient, extended by zero outside the chart image, is Borel; smoothness of that zero extension is not required. Empty sums and the empty-set value are zero. For $n=0$ use singleton charts and the mass-one coordinate convention.

Here is why the choices exist under countable choice, including at a boundary. From a countable base select a chart and a relatively compact ball or half-ball for each basis member whose closure fits inside such a chart; these members cover $M$. Their finite unions of compact closures give compact sets whose interiors cover $M$. Passing recursively to the least sufficiently large index gives an exhaustion $K_m\subset \mathrm{Int}{K}_{m+1}$. Cover each compact annulus $K_m\setminus \mathrm{Int}{K}_{m-1}$ by finitely many chart balls or half-balls whose closures lie in $\mathrm{Int}{K}_{m+1}\setminus K_{m-2}$ (take $K_0=K_{-1}=\mathrm{\varnothing }$). Such small coordinate neighborhoods exist at every point of the annulus. Countable choice selects one finite cover per annulus. The resulting countable chart cover is locally finite: $\mathrm{Int}{K}_j$ misses all families indexed $m\ge j+2$, and only finitely many sets come from each remaining annulus. Apply Smooth partitions of unity exist on manifolds with boundary to this cover; the boundaryless specialization is also Smooth partitions subordinate to a countable coordinate cover. If the subordinate partition has several terms per chart, aggregate those terms; local finiteness makes each sum smooth, and its support remains in the assigned chart.

Countable additivity is discharged by The glued set function is a Borel measure ↗. Independence of both choices and the intrinsic notation ${\mu }_r$ are discharged by Intrinsic density measure and its chart restriction ↗.