Area formula and endpoint bound with multiplicity

Use the ZFC, componentwise distance and normalized measure conventions in [F1]. Refine the countable extension cover into source coordinate balls with compact closures inside extension domains and with images in target charts, and subtract earlier balls to obtain a disjoint Borel partition $(A_i)$. Further such refinements remain countable. Bounded derivatives on smaller coordinate balls make each coordinate extension Lipschitz. The Jacobian supplier identifies its derivative with $d_{A}F$ almost everywhere on each partition piece; the multiplicity supplier gives measurability of each image and its count. With source metric matrix $H$, target metric matrix $K$ and coordinate derivative $B$, the coordinate density of $J_{2}Fd{\mu }_C$ is $\sqrt{\det (B^{T}KB)}$, since the factor $\sqrt{\det H}$ cancels its reciprocal in $J_{2}F$.

Fix $0<\epsilon <1$ and put $a=1-\epsilon$, $b=1+\epsilon$. Around each target point take linearly normalized coordinates with metric matrix equal to the identity at the center. Smoothness gives a coordinate ball $B_R$ on which $a^{2}I\le K\le b^{2}I$. Restrict to $B_r$ with $r<R\mathrm{/}3$: straight segments give the upper distance bound $d_k(v^{-1}z,v^{-1}w)\le b\mathrm{\mid }z-w\mathrm{\mid }$. Curves staying in $B_R$ have length at least $a\mathrm{\mid }z-w\mathrm{\mid }$; a curve leaving it has initial displacement at least $R-r>2r\ge \mathrm{\mid }z-w\mathrm{\mid }$ before its first exit and obeys the same lower bound. Thus the restricted ambient component distance is at least $a\mathrm{\mid }z-w\mathrm{\mid }$ as well. Applying [F3] to the coordinate map and its inverse yields $a^2{\mathcal{H}}_{\mathrm{E}}^2(E)\le {\mathcal{H}}_k^2(v^{-1}E)\le b^2{\mathcal{H}}_{\mathrm{E}}^2(E)$ on this patch. Refine the partition in step 1.1 so every image lies in such a patch. The same quadratic-form bounds give $a^2\sqrt{\det (B^{T}B)}\le \sqrt{\det (B^{T}KB)}\le b^2\sqrt{\det (B^{T}B)}$; if $B$ has rank below two all three determinants vanish.

Write $f_i=v_i\circ F\circ u_i^{-1}$ on $u_i(A_i)$ and let $N_i$ count preimages in $A_i$. Apply [F2] to the Lipschitz coordinate extension on this Borel subset of ${\mathbb{R}}^2$, with Euclidean target dimension three and normalized Euclidean measures. It gives $T_i:={\int }_{u_i(A_i)}\sqrt{\det (D{f}_i^{T}D{f}_i)}d{\lambda }_2={\int }_{f_i(u_i(A_i))}{N}_i(v_i^{-1}z)d{\mathcal{H}}_{\mathrm{E}}^2(z)$. Let $I_i={\int }_{A_i}{J}_{2}Fd{\mu }_C$ and $R_i={\int }_{F(A_i)}{N}_{i}d{\mathcal{H}}_k^2$. Steps 1.1 and 2.1 give $a^2{T}_i\le I_i\le b^2{T}_i$ and $a^2{T}_i\le R_i\le b^2{T}_i$; the measure inequalities apply to nonnegative integrals by simple-function approximation. Hence $(a\mathrm{/}b{)}^2{I}_i\le R_i\le (b\mathrm{/}a{)}^2{I}_i$, also for infinite values. Since the source pieces are disjoint, ${\sum }_i{N}_i=N(F,A;\cdot )$, with counts extended by zero off their images, so summing gives $(a\mathrm{/}b{)}^{2}I\le R\le (b\mathrm{/}a{)}^{2}I$ for the two integrals $I,R$ in the Statement. Although the partition depends on $\epsilon$, these full integrals do not. Let $\epsilon$ decrease to zero: finite values are squeezed to equality, and an infinite value forces the other to be infinite. This proves the metric identity without assuming disjoint images.

Each piece in step 1.1 is parametrized by a subset of ${\mathbb{R}}^2$; its coordinate extension is Lipschitz and the target inverse chart is Lipschitz for the component distance by step 2.1. These countably many maps cover $D$, proving countable two-rectifiability in the stated convention. Since $N\ge 1$ on $D$, the identity gives $\mathrm{\mid }D\mathrm{\mid }\le I$. If $N=1$ almost everywhere on $D$, its right integral is $\mathrm{\mid }D\mathrm{\mid }$, giving equality. If also $J_{2}F\le 1$ almost everywhere, then $I\le {\mu }_C(A)=\mathrm{\mid }A\mathrm{\mid }$. No injectivity or smoothness of the image cut was used.