Intrinsic density measure and its chart restriction

For a finite nonnegative Borel scalar $a$ on $M$, $ar$ is again a density: in coordinates its coefficient is $(a\circ x^{-1})r_x$, and multiplying the transition law by $a\circ x^{-1}$ proves its law, also at $r_x=\mathrm{\infty }$ when $a=0$. Thus the overlap lemma applies to ${\phi }_{i}r$ and ${\phi }_i{\psi }_{j}r$.

Fix a Borel $E\subset U$. Transfer the $i$th gluing term, for the density ${\phi }_{i}r$, from $x_i$ to $x$ on $E\cap U_i$. In the $x$ chart the transferred integrands sum to $r_x{\mathbf{1}}_{x(E)}$: if $r_x$ is finite this is distributivity and ${\sum }_i{\phi }_i=1$; if $r_x=\mathrm{\infty }$, at least one of the finitely many nonzero weights is positive, so the sum is infinite. The nonnegative summation theorem therefore gives the chart-restriction identity.

For two partitions $({\phi }_i)$ and $({\psi }_j)$, expand the first construction on a Borel $E$ as ${\sum }_i{\sum }_j{\int }_{x_i(E\cap U_i\cap V_j)}({\phi }_i{\psi }_{j}r{)}_{x_i}$. This expansion follows from the same pointwise argument, now summing ${\psi }_j$, and the nonnegative summation theorem. The overlap lemma transfers each summand to $y_j$. Both iterated sums are the supremum of the finite rectangular subsums, so reordering and summing ${\phi }_i$ gives the second construction.

Thus the two Borel measures are identical on every Borel set. All formulas remain true for the empty set and zero density, giving zero; in dimension zero the chart formula is the singleton weight. Neither infinity nor boundary points require cancellation: the overlap lemma already treats both.