Continuous differential-form integrals over smooth singular chains in Euclidean domains

In coordinates, each pullback coefficient is the finite sum of products described in [F2]. Every form coefficient composed with $\sigma$ is continuous by [F1] and [F3], and each derivative of $\sigma$ is continuous by [F1]. Finite sums of continuous real functions are continuous by the triangle inequality. For a product $fg$ at a point $u_0$, first restrict to a neighbourhood where $\mathrm{\mid }g(u)-g(u_0)\mathrm{\mid }<1$, so $\mathrm{\mid }g(u)\mathrm{\mid }\le \mathrm{\mid }g(u_0)\mathrm{\mid }+1$, then use $\mathrm{\mid }f(u)g(u)-f(u_0)g(u_0)\mathrm{\mid }\le \mathrm{\mid }f(u)-f(u_0)\mathrm{\mid }(\mathrm{\mid }g(u_0)\mathrm{\mid }+1)+\mathrm{\mid }f(u_0)\mathrm{\mid }\mathrm{\mid }g(u)-g(u_0)\mathrm{\mid }$. Continuity of $f$ and $g$ makes the right side arbitrarily small. Induction proves continuity for every finite product, hence $g_{\sigma }$ is continuous on ${\mathrm{\Delta }}^k$. If two smooth extensions of $\sigma$ are chosen, they agree as functions on the relative interior of ${\mathrm{\Delta }}^k$, which is open in its affine span; hence their tangential derivatives and pullback coefficients agree there. For any boundary point $u$, the points $u_s=(1-s)u+sb$, where $b=(1\mathrm{/}(k+1),\dots ,1\mathrm{/}(k+1))$ and $0<s\le 1$, have all coordinates positive and coordinate sum below $1$, so they lie in the relative interior and converge to $u$. Continuity of both coefficient functions makes them equal at $u$ too. Thus $g_{\sigma }$ is independent of the extension. No immersion is needed; when $D{\sigma }_u$ has rank below $k$, alternation makes the pullback coefficient zero at $u$.

The set ${\mathrm{\Delta }}^k$ in [F4] is closed and bounded, hence compact by [F5]. Its boundary consists of its coordinate faces $u_i=0$ and its upper face ${\sum }_i{u}_i=1$. If $k=1$, these are the two endpoints $0,1$, covered by two intervals of arbitrarily small total length, so the boundary has content zero by [F7]. If $k\ge 2$, each coordinate face is the graph of the constant zero function over its closed bounded coordinate projection, and the upper face is the graph of $u_k=1-{\sum }_{i<k}{u}_i$ over the closed bounded simplex $\{u_i\ge 0,{\sum }_{i<k}{u}_i\le 1\}$. These bases are compact by [F5]. The upper-face function is continuous because $\mathrm{\mid }{\sum }_{i<k}(u_i-v_i)\mathrm{\mid }\le (k-1)\mathrm{\parallel }u-v{\mathrm{\parallel }}_2$, and the lower-face functions are constant; [F6] therefore makes each face content zero. Given $\epsilon >0$, cover each of the finitely many faces by a finite cube family with total volume less than $\epsilon \mathrm{/}(k+1)$; their union is a finite cover of the boundary with total volume less than $\epsilon$ by [F7]. Thus $\mathrm{\partial }{\mathrm{\Delta }}^k$ has content zero, and [F7] makes the bounded set ${\mathrm{\Delta }}^k$ Jordan measurable.

By [F9], the continuous coefficient $g_{\sigma }$ is Riemann integrable on the compact Jordan simplex from step 1.2, so each simplex integral in the Definition is finite.

A finite chain has finitely many nonzero coefficients by [F10]. Its integral is therefore a finite sum of finite real numbers; the zero chain has the empty sum, equal to $0$ by [F10]. If a chain is presented with repeated copies of the same simplex, [F10] and linearity combine their coefficients without changing the result. The positive affine parameter orientation is fixed by [F8]. The Definition covers $C^1$ coefficient forms and the continuous coefficients of their first exterior derivatives by [F12]. For smooth $\omega$, the coefficient and affine parameter integral are exactly those in [F11], so the two integral conventions agree.