Continuous compact-support integration for the constitutive action

If $K=\mathrm{\varnothing }$, the form is zero and the stated zero value needs no charts. Otherwise consider the family of all pairs of concentric coordinate balls $B_i\subset V_i$ in positive charts, with $\overline{B_i}$ compact and contained in $V_i$. Every point of $K$ lies in one such inner ball: openness of a chart image permits two sufficiently small concentric Euclidean balls, whose closed inner ball is compact by [F1]. This forms an open cover without selecting a chart separately at every point. Compactness of $K$ gives finitely many such pairs whose inner balls cover $K$. Apply [F1]'s bump construction for each of these finitely many compact inner balls inside its chart image, pull back and extend by zero, to get smooth ${\rho }_i:M\to [0,1]$ with compact support inside $V_i$ and ${\rho }_i=1$ on $B_i$. Only finitely many choices are made. Set $B={\sum }_i{\rho }_i$, so $B\ge 1$ on $K$. Let $u(t)=e^{-1\mathrm{/}t}$ for $t>0$ and $u(t)=0$ for $t\le 0$, and define $q(t)=u(t-1\mathrm{/}4)\mathrm{/}(u(t-1\mathrm{/}4)+u(1\mathrm{/}2-t))$. Its denominator is positive for every real $t$; by [F6], $u$ is smooth and all its derivatives vanish at zero, so $q$ is smooth, lies in $[0,1]$, is zero for $t\le 1\mathrm{/}4$ and is one for $t\ge 1\mathrm{/}2$. Put $\eta =q(B)$ and define ${\chi }_i=\eta {\rho }_i\mathrm{/}B$ on $\{B>0\}$, extending by zero elsewhere. Since $\eta =0$ for $B\le 1\mathrm{/}4$, the extension is smooth; its support is a closed subset of the compact support of ${\rho }_i$ and remains inside $V_i$. Their sum is $\eta$, hence is one on the open neighborhood $\{B>1\mathrm{/}2\}$ of $K$. These are the required finite cutoffs.

The support of ${\chi }_i\omega$ is a closed subset of $K\cap \mathrm{supp}{\chi }_i$, hence compact and contained in its chart. Its coordinate coefficient is continuous there. The coordinate support is compact inside the open chart image, so the zero extension is continuous at artificial chart edges: outside that compact set a neighborhood has identically zero coefficient. On a closed rectangle containing this compact coordinate support the extension is therefore integrable by [F2]. The same fact and [F2]'s bounding-rectangle independence make each summand in the Definition a finite well-defined real number. The rectangle estimate in [F4] and the finite sum give the displayed bound for $\mathrm{\mid }{\mathcal{I}}_M(\omega )\mathrm{\mid }$. Applied to coefficient differences with fixed cutoffs and rectangles, this bound proves the stated uniform-limit property.

Suppose a continuous top form $\zeta$ has compact support contained in two positive charts $(U,\varphi )$ and $(V,\psi )$. On their overlap put $g=\psi \circ {\varphi }^{-1}$; this is a smooth injective diffeomorphism between open Euclidean sets, and $\det Dg>0$ because both charts are positive. If $f_{\psi }$ is its target coefficient, the defining tensor transformation law gives $f_{\varphi }=(f_{\psi }\circ g)\det Dg$. The zero-extended target coefficient is continuous with compact support contained in the target overlap, by the argument of step 2.1. All hypotheses of [F3] hold, and the transformed coefficient extended by zero is exactly the source coefficient extended by zero. Consequently their Euclidean integrals agree. This proves the required chart comparison for continuous coefficients directly, without applying the smooth-only claim in [F5] to $\zeta$.

Let $({\chi }_i,U_i)$ and $({\tau }_j,V_j)$ be two finite cutoff systems for $K$. Every ${\chi }_i{\tau }_j\omega$ has compact support contained in $U_i\cap V_j$. Near $K$ each cutoff sum is one, and outside $K$ the form vanishes, so ${\chi }_i\omega ={\sum }_j{\chi }_i{\tau }_j\omega$ and ${\tau }_j\omega ={\sum }_i{\chi }_i{\tau }_j\omega$. All sums are finite. Apply linearity [F4] and the chart comparison of step 3.1 to each product term. Both proposed integrals equal the same finite double sum, proving independence of cutoff systems and charts. Step 2.1 already proves bounding-rectangle independence. For two compactly supported forms take cutoffs near the union of their supports, which is compact; [F4] then proves linearity in the forms and scalar coefficients.

If an open $V\subseteq M$ contains $K$, the construction of step 1.1 may use only coordinate balls with compact closure inside $V$. The same finite coefficient integrals compute ${\mathcal{I}}_V(\omega {\mathrm{\mid }}_V)$ and ${\mathcal{I}}_M(\omega )$, so step 4.1 proves locality. Now suppose $\omega$ is smooth. Choose a chart-partition system $({\tau }_j,U_j)$ permitted by the smooth integral in [F5]. Only finitely many ${\tau }_j\omega$ are nonzero near $K$, as part of that integral contract. Comparing this system with the finite cutoffs ${\chi }_i$ uses the same product identities as in step 4.1: each ${\chi }_i{\tau }_j\omega$ is supported compactly in the two chart domains, and step 3.1 identifies its two coordinate integrals. The finite double sum consequently equals both ${\mathcal{I}}_M(\omega )$ and the existing smooth chart-partition integral on $M$. Countable Choice enters this comparison only through the explicit convention of [F5]; the finite construction and comparison in steps 1.1–4.1 did not assume full Choice. Thus the continuous integral extends the exact previously defined smooth integral, with the stated locality and linearity.