Pointwise Borel nonnegative densities

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $M$ be a Hausdorff second-countable smooth $n$-manifold, with boundary allowed. A nonnegative Borel density $r$ specifies, in every chart $x:U\to x(U)$, a Borel function $r_x:x(U)\to [0,\mathrm{\infty }]$ such that, on every overlap with $y:V\to y(V)$, $r_x(u)=r_y(y(x^{-1}(u)))\mid \det D(y\circ x^{-1})(u)\mid \mathrm{.}$ Borel means measurability for the Borel sigma-algebras, as in The Borel sigma-algebra of a topological space and Extended-real-valued measurable functions; no measure on $M$ is needed to impose this condition. Smooth boundary charts use the structure of Smooth charts, atlases, and structures with boundary.

This extends the positive cone convention of Density bundle and smooth density fields. It is an extended positive density cone, not an extended-valued section of the real line bundle. Transition factors are finite, strictly positive and smooth. For three charts $x,y,z$, the chain rule gives $J_{zx}=(J_{zy}\circ y\circ x^{-1})J_{yx}$, so repeated transitions give the same coefficient, including when it is infinite. On a countable atlas Borel coefficients satisfying the overlap law determine Borel coefficients in every chart: each transported coefficient is Borel, they agree on overlaps, and preimages are countable unions of Borel pieces. Conversely, Borel coefficients in every chart are Borel on that atlas. Nonnegative Borel products use the same convention: for $a,b\ge 0$ and $t\ge 0$, $\{ab>t\}={\bigcup }_{q\in \mathbb{Q},q>0}(\{a>q\}\cap \{b>t\mathrm{/}q\})$, also for infinite values; for $t<0$ the superlevel is the whole domain. This proves product measurability by countable unions.

Set $0\cdot \mathrm{\infty }=0$. A nonnegative finite Borel scalar $a$ multiplies coefficients by $a\circ x^{-1}$; its transition law follows by multiplying the displayed identity. In dimension zero the empty determinant is one, charts are singletons, and ${\lambda }_0$ gives the singleton mass one. The empty manifold has the unique empty coefficient family; the zero density has all coefficients zero. Coefficients are not identified almost everywhere.