Orientation-free density integration and its properties

Integral of a compactly supported smooth density: Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $\delta$ be a compactly supported smooth density on $M^n$, with boundary allowed. Choose a chart partition $({\rho }_i)$ and write ${\rho }_i\delta =f_i\mathrm{\mid }d{x}_i\mathrm{\mid }$. For $n\ge 1$ define ${\int }_M\delta ={\sum }_i{\int }_{{\mathbb{R}}^n}{\tilde{f}}_i(x_i)d{x}_i\mathrm{.}$ The zero extensions are Riemann integrable, including at genuine faces, by lem-chart-supported-coefficients-have-well-defined-riemann-integrable-half-space-extensions. The compact-support/local-finiteness argument of lem-a-locally-finite-sum-is-finite-near-the-compact-support-of-a-form applies to density supports as closed sets, so the sum is finite. For $n=0$ sum the scalar density values over the finite support, without orientation signs. Empty support gives zero. Choice independence is discharged by thm-density-integration-is-defined-without-an-orientation.

Pullback of densities by local diffeomorphisms: For a local diffeomorphism $F:M^n\to N^n$, pullback of smooth densities is smooth and in coordinates satisfies $F^{\ast }(f\mathrm{\mid }dy\mathrm{\mid })=(f\circ F)\mathrm{\mid }\det DF\mathrm{\mid }\mathrm{\mid }dx\mathrm{\mid }\mathrm{.}$ It is real-linear, obeys $F^{\ast }(a\delta )=(a\circ F)F^{\ast }\delta$ for smooth functions $a$ on $N$, and $(F\circ G{)}^{\ast }=G^{\ast }{F}^{\ast }$ for composable local diffeomorphisms.

Coordinate independence of chart integrals: On an oriented smooth $n$-manifold, including $n=0$ and genuine boundary, a smooth top form with compact support contained in two connected charts has the same signed chart integral in both charts.

Local finiteness near compact support: If $(C_i{)}_{i\in I}$ is a locally finite family of closed subsets of a manifold and $K$ is compact, only finitely many $C_i$ meet $K$. There is an open neighborhood of $K$ disjoint from all the other $C_i$. In particular, for a smooth partition of unity $({\rho }_i)$ and $\omega \in {\mathrm{\Omega }}_c^k(M)$, only finitely many ${\rho }_i\omega$ are nonzero.

Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in ${\mathbb{R}}^m$: Let $Q={\prod }_{j<m}[a_j,b_j]$ be nondegenerate. For integrable $f,g:Q\to \mathbb{R}$ and scalars $\alpha ,\beta$, the function $\alpha f+\beta g$ is integrable and its integral is $\alpha {\int }_{Q}f+\beta {\int }_{Q}g$. If $f\le g$, then ${\int }_{Q}f\le {\int }_{Q}g$. Also $\mathrm{\mid }f\mathrm{\mid }$ is integrable and $\mathrm{\mid }{\int }_{Q}f\mathrm{\mid }\le {\int }_Q\mathrm{\mid }f\mathrm{\mid }$. If $a_r<c<b_r$, cutting $Q$ at the coordinate hyperplane $x_r=c$ gives two nondegenerate subrectangles; integrability on $Q$ is equivalent to integrability on both restrictions, and their integral values add to the integral over $Q$.

Computing form integrals by finite parametrizations: Let $n\ge 1$, let $M^n$ be oriented, and let $\omega \in {\mathrm{\Omega }}_c^n(M)$. For $1\le i\le m$ let $D_i\subset {\mathbb{R}}^n$ be bounded open Jordan domains and $F_i:{\bar{D}}_i\to M$ continuous and smooth up to the boundary in target coordinates: near each parameter point, a target coordinate representative extends smoothly to a Euclidean neighborhood. Suppose $F_i{\mathrm{\mid }}_{D_i}$ is an orientation-preserving diffeomorphism onto an open $W_i\subset M$, the $W_i$ are pairwise disjoint, and $\mathrm{supp}\omega \subset {\bigcup }_i{\bar{W}}_i$. Then ${\int }_M\omega ={\sum }_{i=1}^m{\int }_{D_i}{F}_i^{\ast }\omega \mathrm{.}$ An empty family is allowed when the support is empty. No nonsingularity of $D{F}_i$ on $\mathrm{\partial }{D}_i$, and no $M$-valued extension across a genuine target boundary, is assumed.