Proof of Independence of the Manifold Integral from Chart and Partition Choices

We use throughout that the iterated integral of Integral of a Compactly Supported Continuous n-Form on a Euclidean or Half-Space Domain is additive over finite sums of compactly supported continuous $n$-forms on a common admissible domain: choosing one box containing all the supports (possible by claim 2 of Zero Extension Continuity and Box Independence of the Iterated Integral applied to the sum and the coordinatewise hull construction of its claim 4), the zero extensions add, and the one-dimensional Riemann integral is additive in the integrand — a basic consequence of its definition via Riemann sums, applied at each stage of the iteration. We call this additivity below.

Claim 1. Fix $i$. The coefficient function of $(\chi_i\omega)_{\alpha_i}$ is $x\mapsto\chi_{i,\alpha_i}(x)\,f_{\alpha_i}(x)$ where $f_{\alpha_i}$ is the coefficient function of $\omega_{\alpha_i}$; both factors are continuous on $\Omega_{\alpha_i}$ (their local smooth extensions are continuous, and continuity is local), so the product is continuous by Sums and Products of Continuous Real-Valued Functions. Let $S_i\subseteq M$ be the support of $\chi_i$ as in Smooth Partitions of Unity on a Compact Smooth Manifold with Boundary; it is a closed subset of the compact space $M$ (its complement is open by its definition), hence compact by Closed Subset of a Compact Space is Compact, and $S_i\subseteq U_{\alpha_i}$. Then $K_i=\varphi_{\alpha_i}(S_i)$ is compact by Continuous Image of a Compact Space is Compact, and $K_i\subseteq\Omega_{\alpha_i}$. If $x\in\Omega_{\alpha_i}\setminus K_i$, the point $q=\varphi_{\alpha_i}^{-1}(x)$ lies outside $S_i$, so $\chi_i$ vanishes on a neighborhood of $q$, and hence $(\chi_i\omega)_{\alpha_i}$ vanishes on a neighborhood of $x$ in $\Omega_{\alpha_i}$. Since $K_i$ is closed in $\mathbb{R}^n$ by Compact Subset of $\mathbb{R}^n$ is Closed, it follows from the support definition in Continuous n-Form, Support, and Zero Extension on a Euclidean or Half-Space Domain that $\operatorname{supp}\bigl((\chi_i\omega)_{\alpha_i}\bigr)\subseteq K_i$; a closed subset of the compact set $K_i$ is compact (Closed Subset of a Compact Space is Compact), so $(\chi_i\omega)_{\alpha_i}$ is compactly supported in $\Omega_{\alpha_i}$.

Claim 2. Immediate from claim 1 and Integral of a Compactly Supported Continuous n-Form on a Euclidean or Half-Space Domain: each summand is a well-defined real number, and $S$ is a finite sum.

Claim 3. Since $\sum_{j=1}^{N'}\chi_j'=1$ on $M$, we have, chart representative by chart representative, $(\chi_i\omega)_{\alpha_i}=\sum_{j=1}^{N'}(\chi_j'\chi_i\omega)_{\alpha_i}$, where $\chi_j'\chi_i\omega$ is the smooth $n$-form with representatives $\chi'_{j,\alpha}\chi_{i,\alpha}\,\omega_\alpha$. By additivity,

It therefore suffices to prove, for each pair $(i,j)$ with $\eta=\chi_j'\chi_i\omega$ not identically zero,

The supports of $\chi_i$ and $\chi_j'$ meet, so $U=U_{\alpha_i}\cap U_{\beta_j}\ne\varnothing$, and, as in claim 1, the support of $\eta$ in $M$ is a compact set contained in $U$; write $K_\alpha=\varphi_{\alpha_i}(\operatorname{supp}$ of $\eta)$ and $K_\beta=\varphi_{\beta_j}(\text{the same set})$, compact subsets of $A_\alpha=\varphi_{\alpha_i}(U)$ and $A_\beta=\varphi_{\beta_j}(U)$ respectively.

First, $\int_{\Omega_{\alpha_i}}\eta_{\alpha_i}=\int_{A_\alpha}\eta_{\alpha_i}$, where on the right $\eta_{\alpha_i}$ is restricted to the admissible domain $A_\alpha$ (open in the same ambient set): both integrals are iterated integrals over a common box of zero extensions that agree as functions, since both vanish off $K_\alpha\subseteq A_\alpha$ (as in the preliminary observation of the proof of Zero Extension Continuity and Box Independence of the Iterated Integral); box independence is claim 4 of Zero Extension Continuity and Box Independence of the Iterated Integral. Likewise $\int_{\Omega_{\beta_j}}\eta_{\beta_j}=\int_{A_\beta}\eta_{\beta_j}$. (If one of $A_\alpha$, $A_\beta$ is open in $\mathbb{R}^n$ while the corresponding $\Omega$ is treated as open in $H^n$, the two readings of the zero-extension integral agree for the same reason: all versions vanish off the common compact support and the iterated integrals over a common box coincide.)

Next, let $\tau=\varphi_{\beta_j}\circ\varphi_{\alpha_i}^{-1}:A_\alpha\to A_\beta$. By property 3 of Smooth Differential k-Form on a Smooth Manifold with Boundary, $\tau$ is a smooth diffeomorphism of admissible domains and $\tau^{*}(\eta_{\beta_j})=\eta_{\alpha_i}$ on $A_\alpha$. Moreover $\tau$ is orientation-preserving: at points of $A_\alpha$ mapping to the interior of the half-space this is the positive compatibility of the oriented smooth atlas (using claim 3 of Boundary of a Smooth Manifold with Boundary as a Smooth Manifold of Dimension n-1 to identify interior images when $n\ge 2$; for $n=1$, or when $\partial M=\varnothing$, every relevant point is handled the same way), and at points of the boundary hyperplane positivity follows by the continuity-and-invertibility argument of claim 1 of Transition Maps of Induced Boundary Charts of an Oriented Atlas are Orientation-Preserving, which applies verbatim for every $n\in\mathbb{N}$.

Finally, $\eta_{\beta_j}$ restricted to $A_\beta$ is continuous and compactly supported in $A_\beta$ (its support is contained in the compact set $K_\beta$, by the argument of claim 1), so Pullback Invariance of the Integral under Orientation-Preserving Smooth Diffeomorphisms yields

Combining the three displayed equalities for every pair $(i,j)$ and summing gives claim 3. $\blacksquare$