Rademacher differentiability in Euclidean space

Fix a scalar component and a ball compactly contained in a neighborhood where it is $L$-Lipschitz. Each coordinate section is Lipschitz, hence absolutely continuous by [F2], and has a derivative almost everywhere by [F3], bounded by $L$. For measurability, write the limsup and liminf of the coordinate difference quotients as countable suprema and infima over rational increments; continuity of the section makes these equal the corresponding real-increment limits. Define $g_i$ by their common finite limit where it exists and by zero elsewhere. Fubini makes the measurable exceptional set have Lebesgue measure zero; [F11] converts it to closed-cube-cover nullity. The continuous function is an ACL representative, so [F1] gives $D_{i}u=g_i$ weakly and $u\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,\mathrm{\infty }}$.

Outside a null set, $x$ is a strong Lebesgue point of $g=(g_1,\dots ,g_n)$. For each rational vector $q\in {\mathbb{Q}}^n$, apply [F4] locally to the scalar functions $\mathrm{\mid }g-q\mathrm{\mid }$ and to the components of $g$. On a fixed bounded Lipschitz ball these functions are integrable; extend them by zero to ${\mathbb{R}}^n$ to apply [F4], and restrict to interior points, where sufficiently small ball averages are unchanged. For each application, the almost-everywhere conclusion provides a measurable Lebesgue-measure-zero set containing the exceptions. Convert that set to closed-cube-cover nullity by [F11], then remove the countable union of these cube-cover-null sets using [F6]. At each remaining point, for every rational $q$, ${\mathrm{lim\hspace{0.17em}sup}}_{r\downarrow 0}\frac{1}{\lambda (B(x,r))}{\int }_{B(x,r)}\mathrm{\mid }g(y)-g(x)\mathrm{\mid }dy\le 2\mathrm{\mid }q-g(x)\mathrm{\mid }\mathrm{.}$ Rational vectors approximate $g(x)$ by [F10], so this limit is zero. The rational balls whose closures lie in neighborhoods on which the chosen component is Lipschitz form a countable cover of $U$: local Lipschitz continuity gives a suitable ball around each point, density of rational centers/radii gives a rational subball, and [A1] selects one Lipschitz neighborhood and constant for each such ball. Apply the local argument on these balls.

Fix such a point $x$, put $p=g(x)$, and on $B(x,r)⋐U$ set $w(y)=u(y)-u(x)-p\cdot (y-x)$. The function $w$ is bounded on this ball. By [F1] and linearity of the test-function identity, its weak gradient is $g(y)-p$, which is bounded there; thus $w\in W^{1,1}(B(x,r))$ by [F9]. Also $w(x)=0$ and $w$ is $(L+\mathrm{\mid }p\mathrm{\mid })$-Lipschitz. The strong Lebesgue-point property and [F8] give ${\int }_{B(x,r)}\mathrm{\mid }Dw\mathrm{\mid }=o(r^n)$. Applying [F5] yields ${\int }_{B(x,r)}\mathrm{\mid }w-w_{B(x,r)}\mathrm{\mid }=o(r^{n+1})\mathrm{.}$

Write $M=L+\mathrm{\mid }p\mathrm{\mid }$. If $M=0$, $w$ vanishes on every sufficiently small ball. Otherwise fix $ϵ>0$. If $\mathrm{\mid }{w}_{B(x,r)}\mathrm{\mid }\ge ϵr$, then on $B(x,cr)$, with $c=\min \{1\mathrm{/}2,ϵ\mathrm{/}(2M)\}$, the Lipschitz bound and $w(x)=0$ give $\mathrm{\mid }w-w_{B(x,r)}\mathrm{\mid }\ge ϵr\mathrm{/}2$. Since $\lambda (B(x,cr))=v_n{c}^n{r}^n$ by [F8], the integral in step 3.1 would be at least $v_n{c}^nϵr^{n+1}\mathrm{/}2$, a contradiction for all sufficiently small $r$. Hence $w_{B(x,r)}=o(r)$. If some $y\in B(x,r\mathrm{/}2)$ had $\mathrm{\mid }w(y)\mathrm{\mid }\ge ϵr$, then for small $r$ we would have $\mathrm{\mid }w(y)-w_{B(x,r)}\mathrm{\mid }\ge ϵr\mathrm{/}2$. On $B(y,c^{\mathrm{\prime }}r)\subset B(x,r)$, where $c^{\mathrm{\prime }}=\min \{1\mathrm{/}4,ϵ\mathrm{/}(4M)\}$, Lipschitz continuity gives $\mathrm{\mid }w-w_{B(x,r)}\mathrm{\mid }\ge ϵr\mathrm{/}4$. Its integral is at least $v_n(c^{\mathrm{\prime }}{)}^nϵr^{n+1}\mathrm{/}4$, again contradicting step 3.1. Thus ${\sup }_{B(x,r\mathrm{/}2)}\mathrm{\mid }w\mathrm{\mid }=o(r)$. For $h\mathrm{\ne }0$, take $r=3\mathrm{\mid }h\mathrm{\mid }$, so $x+h\in B(x,r\mathrm{/}2)$; this yields the Frechet expansion $u(x+h)-u(x)-p\cdot h=o(\mathrm{\mid }h\mathrm{\mid })$.

The countable rational-ball basis [F10] and the finitely many real components of $u$ produce one exceptional cube-cover-null set by [F6] and the definition [F7]. The component derivatives assemble into the Frechet derivative. The full Axiom of Choice is used through the stated ACL, Fubini, differentiation, null-union and Poincare suppliers; no reduction to Countable Choice is asserted.