Orientation identifies top forms with signed densities

Orientation-free density integration and its properties: Compactly supported smooth density integration is independent of charts and partition, linear, local, nonnegative on nonnegative densities and strictly positive for a nonzero nonnegative density. It is invariant under every diffeomorphism, without choosing an orientation. The finite-parametrization formula holds under the hypotheses of prop-integration-of-top-forms-by-finite-parametrizations, with orientation preservation omitted and absolute Jacobians used.

Integral of a compactly supported top form: Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. For an oriented smooth manifold $M^n$, possibly with boundary, and $\omega \in {\mathrm{\Omega }}_c^n(M)$, choose a smooth partition $({\rho }_i)$ subordinate to connected interior or boundary charts $(U_i,{\varphi }_i)$. For $n\ge 1$ set ${\int }_M\omega ={\sum }_i{I}_{{\varphi }_i}({\rho }_i\omega )\mathrm{.}$ Each product has compact support in its chart and only finitely many are nonzero, by lem-a-locally-finite-sum-is-finite-near-the-compact-support-of-a-form. For $n=0$ set ${\int }_M\omega ={\sum }_{p\in \mathrm{supp}\omega }\epsilon (p)\omega (p)\mathrm{.}$ ${\mathrm{\Lambda }}^0{T}_p^{\ast }M\cong \mathbb{R}$ has two orientations; define $\epsilon (p)=+1$ when $1$ is positive in the chosen orientation and $\epsilon (p)=-1$ when $-1$ is positive. A zero-manifold is discrete; the singleton open cover of a compact subset has a finite subcover. Thus this sum too is finite. Empty support or empty $M$ gives zero. Independence of the choices is discharged by thm-global-form-integration-is-independent-of-the-atlas-partition-and-refinement.