Lipschitz graph area and calculus under Countable Choice

We first need the integral identity for Lipschitz interval functions without invoking the stronger published general absolutely continuous FTC. If $b$ is $L$-Lipschitz on $[a,d]$, extend it constantly past the endpoints. Its difference quotients are bounded by $L$ and converge to $b^{\mathrm{\prime }}$ almost everywhere by [F1]. For any $a\le s<t\le d$, dominated convergence gives ${\int }_s^t{b}^{\mathrm{\prime }}(r)dr={\lim }_{ϵ\downarrow 0}\frac{1}{ϵ}({\int }_t^{t+ϵ}b(r)dr-{\int }_s^{s+ϵ}b(r)dr)=b(t)-b(s)\mathrm{.}$ Coordinate difference quotients of $h$ are measurable; [F1] and [F2] show their limits $g_i$ exist almost everywhere, with $\mathrm{\mid }{g}_i\mathrm{\mid }\le L$. Define them to be zero on the exceptional sets. Applying the preceding identity and one-dimensional integration by parts on each line, then [F2], gives $\int h{\mathrm{\partial }}_i\eta =-\int g_i\eta$ for smooth compactly supported $\eta$. The same line identities hold for every coordinate line of $h$, outside the appropriate line-null exceptional set of derivatives.

For a cube $C$ of side $2r$ and a Lipschitz $w$, join $x$ to $y$ by the $m$ coordinate segments replacing their coordinates in order. Step 1.1 bounds $\mathrm{\mid }w(x)-w(y)\mathrm{\mid }$ by the sum of the line integrals of $\mathrm{\mid }{\mathrm{\partial }}_{i}w\mathrm{\mid }$. Integrate in $x,y\in C$. For coordinate $i$, the unused coordinates contribute $(2r{)}^{m-1}$, while the measure of endpoint pairs whose interval contains a fixed coordinate value is at most $(2r{)}^2$. Thus, after division by $\mathrm{\mid }C\mathrm{\mid }$, the triangle inequality yields ${\int }_C\mathrm{\mid }w-w_C\mathrm{\mid }\le 2r{\sum }_{i=1}^m{\int }_C\mathrm{\mid }{\mathrm{\partial }}_{i}w\mathrm{\mid },$ where $w_C$ is its mean. Outside a null set, $g=(g_i)$ has strong Lebesgue points: apply [F3] to its coordinates and to $\mathrm{\mid }g-q\mathrm{\mid }$ for every rational vector $q$, then approximate $g(x)$ by such $q$. The countable union of the exceptional null sets is null under Countable Choice. Ball differentiation also gives cube differentiation by the fixed volume comparison. At such a point $x$, set $p=g(x)$ and $w(y)=h(y)-h(x)-p\cdot (y-x)$. The displayed inequality gives ${\int }_{C(x,r)}\mathrm{\mid }w-w_C\mathrm{\mid }=o(r^{m+1})$. As $w(x)=0$ and $w$ has a fixed Lipschitz bound, a mean with $\mathrm{\mid }{w}_C\mathrm{\mid }\ge ϵr$ would give $\mathrm{\mid }w-w_C\mathrm{\mid }\ge ϵr\mathrm{/}2$ on a cube of side proportional to $r$ centered at $x$, contradicting this estimate. Thus $w_C=o(r)$. If $\mathrm{\mid }w(y)\mathrm{\mid }\ge ϵr$ at some $y$ in the half cube, the Lipschitz bound gives the same contradiction on a cube of side proportional to $r$ centered at $y$. Hence ${\sup }_{C(x,r\mathrm{/}2)}\mathrm{\mid }w\mathrm{\mid }=o(r)$, proving differentiability with derivative $p$. Difference quotients then give $\mathrm{\mid }p\mathrm{\mid }\le L$. Countably many rational boxes cover $Q$, so the assertion is global.

We prove the area identity first for bounded Borel base sets within one $L$-Lipschitz box. On the differentiability set, for any $0<ϵ<1\mathrm{/}4$, form countably many Borel pieces on each of which $Dh$ lies within $ϵ$ of a fixed rational vector $p$, the differentiability remainder is at most $ϵ\mathrm{\mid }z-y\mathrm{\mid }$ whenever $\mathrm{\mid }z-y\mathrm{\mid }<1\mathrm{/}k$, and the points lie in one cube of diameter less than $1\mathrm{/}k$. Such pieces cover the differentiability set: the remainder condition follows from differentiability, can be tested on countably many rational $z$ by continuity, and is measurable; use increasing integers $k$ and a rational-vector enumeration. Subtract earlier pieces to make them disjoint. For two points $y,z$ of a single piece $E$, set $A_{p}v=(v,p\cdot v)$. Then $\mathrm{\mid }F(z)-F(y)-A_p(z-y)\mathrm{\mid }\le 2ϵ\mathrm{\mid }z-y\mathrm{\mid }\le 2ϵ\mathrm{\mid }{A}_p(z-y)\mathrm{\mid }$. Consequently $F\circ A_p^{-1}:A_p(E)\to F(E)$ is at most $(1+2ϵ)$-Lipschitz, and its inverse is at most $(1-2ϵ{)}^{-1}$-Lipschitz. The sets $F(E)$ are Borel because $F$ is a homeomorphism from $Q$ onto its graph.

Choose an orthonormal coordinate system on the range plane of $A_p$. Its square coordinate matrix has determinant of absolute value $J_p=\sqrt{\det (A_p^T{A}_p)}=\sqrt{1+\mathrm{\mid }p{\mathrm{\mid }}^2}$. Thus [F4] gives ${\sigma }_m(A_p(E))=J_p\mathrm{\mid }E\mathrm{\mid }$. Apply [F4] to the two maps in step 3.1 to obtain $(1-2ϵ{)}^m{J}_p\mathrm{\mid }E\mathrm{\mid }\le {\sigma }_m(F(E))\le (1+2ϵ{)}^m{J}_p\mathrm{\mid }E\mathrm{\mid }\mathrm{.}$ The function $p\mapsto \sqrt{1+\mathrm{\mid }p{\mathrm{\mid }}^2}$ is $1$-Lipschitz, so $\mathrm{\mid }{J}_p-\sqrt{1+\mathrm{\mid }Dh(y){\mathrm{\mid }}^2}\mathrm{\mid }\le ϵ$ on $E$. The excluded base-null set has graph image of zero ${\sigma }_m$ by [F4], since $F$ is Lipschitz. Sum the displayed inequalities over the disjoint pieces, using $\mathrm{\mid }p\mathrm{\mid }\le L+ϵ$. The difference between ${\sigma }_m(F(B))$ and ${\int }_B\sqrt{1+\mathrm{\mid }Dh{\mathrm{\mid }}^2}$ is bounded by $C_{m,L}ϵ\mathrm{\mid }B\mathrm{\mid }$. Letting $ϵ\downarrow 0$ proves equality. Exhaustion and countable localization extend it to arbitrary Borel $B\subseteq Q$; indicator functions, simple functions and monotone convergence prove the asserted integral formula.

For a compactly supported smooth scalar $b$ in the chart, vertical integration gives ${\int }_G{\mathrm{\partial }}_{z}b=-{\int }_{Q}b(y,h(y))dy$. For a horizontal coordinate put $B_i(y)={\int }_{h(y)}^{\mathrm{\infty }}b(y,z)dz$. This is Lipschitz and compactly supported in $Q$; step 2.1 and ordinary differentiation of the smooth integral give ${\mathrm{\partial }}_i{B}_i={\int }_{h(y)}^{\mathrm{\infty }}{\mathrm{\partial }}_{i}bdz-b(y,h(y))g_i$ almost everywhere. Its weak derivative integrates to zero by step 1.1 and [F2]. Apply these identities to the components of $\varphi$ and sum. Step 4.1 converts the resulting flux into the unit-normal formula. The graph measure is a graph pushforward of the locally bounded measurable density $\sqrt{1+\mathrm{\mid }Dh{\mathrm{\mid }}^2}dy$; compact and open approximation in the base show it is locally Radon on the graph with its relative topology. The density is between $1$ and $\sqrt{1+L^2}$ on every local box, which also proves the null-set assertion.