The ACL characterisation of $W^{1,p}$

Assume (2) of the Statement. Then $u\in L^p\subseteq L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$, and each $g_i:={\mathrm{\partial }}_i{u}^{\ast }$ is measurable and lies in $L^p\subseteq L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$. By [F1] the sections of $u^{\ast }$ along almost every line in direction $i$ are absolutely continuous with one-dimensional derivative the section of $g_i$. So [F3] applies and gives that $g_i$ is the weak derivative $D_{i}u$ for every $i$; since $u\in L^p$ and $g_i\in L^p$, definition [F2] yields $u\in W^{1,p}(\mathrm{\Omega };\mathbb{K})$. Moreover, whenever a class in $L^p$ has the representative $u^{\ast }$, the weak derivative class $D_{i}u$ is unique by [F4], so $g_i={\mathrm{\partial }}_i{u}^{\ast }$ represents $D_{i}u$ almost everywhere. This proves (2)$\Rightarrow$(1) and the final identity of the Statement under (2).

Assume (1) of the Statement. By [F2], $u$ and each $D_{i}u$ have representatives in $L^p(\mathrm{\Omega };\mathbb{K})$. If $\mathrm{\Omega }\mathrm{\ne }\mathrm{\varnothing }$, let $d(x)=\mathrm{dist}(x,{\mathbb{R}}^n\setminus \mathrm{\Omega })$, with $d\equiv +\mathrm{\infty }$ when $\mathrm{\Omega }={\mathbb{R}}^n$, and, for integers $j\ge 1$, set $U_j=\{x\in \mathrm{\Omega }:\mathrm{\mid }x\mathrm{\mid }<j,d(x)>1\mathrm{/}j\}$. These sets increase and exhaust $\mathrm{\Omega }$. Their closures are bounded and lie in $\{x:\mathrm{\mid }x\mathrm{\mid }\le j,d(x)\ge 1\mathrm{/}j\}$; continuity of $d$ and [F20] show that each $K_j:={\bar{U}}_j$ is compactly contained in $\mathrm{\Omega }$. By [F19] choose ${\eta }_j\in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ equal to $1$ on a neighborhood of $K_j$. Select measurable representatives $\widehat{u},{\widehat{g}}_i$ of $u,D_{i}u$ and define ${\tilde{u}}_j={\eta }_j\widehat{u}$ on $\mathrm{\Omega }$ and $0$ outside, and $H_{j,i}={\eta }_j{\widehat{g}}_i+\widehat{u}{\mathrm{\partial }}_i{\eta }_j$ on $\mathrm{\Omega }$ and $0$ outside. These functions lie in $L^p({\mathbb{R}}^n)$ because the cutoff factors are bounded with compact support. For $\psi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$, ${\eta }_j\psi {\mathrm{\mid }}_{\mathrm{\Omega }}$ is a test function and the weak identity [F5] gives ${\int }_{{\mathbb{R}}^n}{\tilde{u}}_j{\mathrm{\partial }}_i\psi ={\int }_{\mathrm{\Omega }}u{\eta }_j{\mathrm{\partial }}_i\psi =-{\int }_{\mathrm{\Omega }}(D_{i}u){\eta }_j\psi -{\int }_{\mathrm{\Omega }}u({\mathrm{\partial }}_i{\eta }_j)\psi =-{\int }_{{\mathbb{R}}^n}{H}_{j,i}\psi \mathrm{.}$ Thus $H_{j,i}$ is the weak derivative of ${\tilde{u}}_j$ on ${\mathbb{R}}^n$. On $U_j$, ${\tilde{u}}_j=u$ and $H_{j,i}=D_{i}u$ almost everywhere. AC supplies Countable Choice via [F18] for the sequence of cutoffs.

Fix a real $\rho \in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ with $\int \rho =1$ and put ${\rho }_{\epsilon }(x)={\epsilon }^{-n}\rho (x\mathrm{/}\epsilon )$ and $v_{j,\epsilon }={\rho }_{\epsilon }\ast {\tilde{u}}_j$. By [F6] this function is smooth. For fixed $x$, the function $y\mapsto {\rho }_{\epsilon }(x-y)$ is a test function, and ${\mathrm{\partial }}_{x_i}{\rho }_{\epsilon }(x-y)=-{\mathrm{\partial }}_{y_i}{\rho }_{\epsilon }(x-y)$. Applying the weak identity of step 1.2 therefore gives ${\mathrm{\partial }}_i{v}_{j,\epsilon }(x)=\int {\tilde{u}}_j(y){\mathrm{\partial }}_{x_i}{\rho }_{\epsilon }(x-y)dy=\int H_{j,i}(y){\rho }_{\epsilon }(x-y)dy=({\rho }_{\epsilon }\ast H_{j,i})(x)\mathrm{.}$

For each fixed $j$, [F6] gives $v_{j,\epsilon }\to {\tilde{u}}_j$ and ${\mathrm{\partial }}_i{v}_{j,\epsilon }\to H_{j,i}$ in $L^p({\mathbb{R}}^n)$ as $\epsilon \downarrow 0$. Since $U_j$ is bounded, [F11] gives it finite measure, and [F10] converts these local $L^p$ convergences into $L^1(U_j)$ convergence. Hence choose ${\epsilon }_j>0$ so that, with $v_j:=v_{j,{\epsilon }_j}$, $\mathrm{\parallel }{v}_j-{\tilde{u}}_j{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}+{\sum }_i\mathrm{\parallel }{\mathrm{\partial }}_i{v}_j-H_{j,i}{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}<2^{-j},\mathrm{\parallel }{v}_j-u{\mathrm{\parallel }}_{L^1(U_j)}+{\sum }_i\mathrm{\parallel }{\mathrm{\partial }}_i{v}_j-D_{i}u{\mathrm{\parallel }}_{L^1(U_j)}<2^{-j}\mathrm{.}$ Both conditions hold for all sufficiently small ${\epsilon }_j$, and Countable Choice [F18] selects one such scale for each $j$.

Let $Q$ be an open box with $\bar{Q}\subseteq \mathrm{\Omega }$. Since $\bar{Q}$ is compact and the increasing open sets $U_j$ cover $\mathrm{\Omega }$, there is $J$ with $Q\subseteq U_J$. For $j\ge J+2$, $Q\subseteq U_{j-1}\subseteq U_j$, so ${\int }_Q(\mathrm{\mid }{v}_j-v_{j-1}\mathrm{\mid }+{\sum }_{i=1}^n\mathrm{\mid }{\mathrm{\partial }}_i{v}_j-{\mathrm{\partial }}_i{v}_{j-1}\mathrm{\mid })\le 2^{-j}+2^{-(j-1)}\mathrm{.}$ The right side is summable. Thus the measurable nonnegative function $F_Q:={\sum }_{j\ge J+2}(\mathrm{\mid }{v}_j-v_{j-1}\mathrm{\mid }+{\sum }_{i=1}^n\mathrm{\mid }{\mathrm{\partial }}_i{v}_j-{\mathrm{\partial }}_i{v}_{j-1}\mathrm{\mid })$ satisfies ${\int }_Q{F}_Q<\mathrm{\infty }$, by applying Fatou [F15] to its finite partial sums.

Fix a coordinate direction $i$ and such a box $Q$. If $n=1$, $F_Q\in L^1(Q)$ directly. For $n\ge 2$, decompose ${\mathbb{R}}^n={\mathbb{R}}^{n-1}\times \mathbb{R}$ along direction $i$. By [F9], Lebesgue measure on $Q$ is the completed product of the factor measures, so [F8] applied to $F_Q$ shows that for almost every transverse parameter $y$ the section $t\mapsto F_Q(y,t)$ is integrable on the side interval $I_{i,Q}$. For each such good $y$, ${\sum }_{j\ge J+2}{\int }_{I_{i,Q}}(\mathrm{\mid }{v}_j-v_{j-1}\mathrm{\mid }+{\sum }_{r=1}^n\mathrm{\mid }{\mathrm{\partial }}_r{v}_j-{\mathrm{\partial }}_r{v}_{j-1}\mathrm{\mid })(y,t)dt<\mathrm{\infty },$ and the same bound holds on every compact subinterval $[a,b]\subset I_{i,Q}$.

Fix a good line and a compact interval $[a,b]\subset I_{i,Q}$. For $k>j>J$, the smooth function $w=v_j-v_k$ is absolutely continuous on $[a,b]$ by [F13], so apply the fundamental theorem [F12] on that line. If $s$ minimizes $\mathrm{\mid }w\mathrm{\mid }$ on $[a,b]$, then [F17] gives $\mathrm{\mid }w(s)\mathrm{\mid }\le (b-a{)}^{-1}{\int }_a^b\mathrm{\mid }w\mathrm{\mid }$; hence $\mathrm{\parallel }w{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(a,b)}\le \frac{1}{b-a}{\int }_a^b\mathrm{\mid }w\mathrm{\mid }+{\int }_a^b\mathrm{\mid }{\mathrm{\partial }}_{i}w\mathrm{\mid }\mathrm{.}$ For complex $w$ the integral identity is applied to real and imaginary parts, while the displayed modulus estimate follows from the triangle inequality. Telescoping the differences and using step 5.1 shows that $(v_j)$ is uniformly Cauchy and $({\mathrm{\partial }}_i{v}_j)$ is Cauchy in $L^1(a,b)$.

Define the measurable Cauchy set $E:={\bigcap }_{m=1}^{\mathrm{\infty }}{\bigcup }_{N=1}^{\mathrm{\infty }}{\bigcap }_{r,s\ge N}\{x:\mathrm{\mid }{v}_r(x)-v_s(x)\mathrm{\mid }<1\mathrm{/}m\}$ and define $u^{\ast }(x):={\lim }_j{\mathbf{1}}_E(x)v_j(x)$. Each set in the definition of $E$ is measurable, and on $E$ the sequence $(v_j(x))$ is Cauchy in $\mathbb{K}$; outside $E$ the displayed sequence is identically zero. Thus its finite pointwise limit $u^{\ast }$ is measurable by applying [F14] to real and imaginary parts. On every good line of step 5.1 the convergence of $v_j$ is uniform on compact subintervals, so the line limit agrees there with $u^{\ast }$; also ${\mathrm{\partial }}_i{v}_j$ converges in $L^1(a,b)$ to some $H$. Passing to the limit in the integral identity of step 6.1 gives $u^{\ast }(t)-u^{\ast }(s)={\int }_s^{t}H$ for all $s,t\in [a,b]$. The indefinite-integral assertions in [F12], applied componentwise for complex values, shows that this section of $u^{\ast }$ is absolutely continuous. Taking the countable union of the exceptional line sets over rational boxes and the finite set of directions still gives null exceptional sets, so $u^{\ast }$ is ACL.

On almost every coordinate line through a fixed box $Q$, step 7.1 gives $v_j\to u^{\ast }$ pointwise, hence this convergence holds almost everywhere in $Q$ by [F8, F9]. Fatou [F15] and the local error bound in step 3.1 give ${\int }_Q\mathrm{\mid }{u}^{\ast }-u\mathrm{\mid }\le {\mathrm{lim\hspace{0.17em}inf}}_j{\int }_Q\mathrm{\mid }{v}_j-u\mathrm{\mid }=0,$ since $Q\subseteq U_j$ for all sufficiently large $j$. Thus $u^{\ast }=u$ almost everywhere on each rational box with closure in $\mathrm{\Omega }$, and hence on $\mathrm{\Omega }$.

Fix such a box $Q$ and direction $i$. For all sufficiently large $j$, $Q\subseteq U_j$, so step 3.1 gives ${\mathrm{\partial }}_i{v}_j\to D_{i}u$ in $L^p(Q)$. By [F16] a subsequence converges almost everywhere on $Q$ to a representative of $D_{i}u$. Fubini [F8, F9] restricts this convergence to almost every coordinate line. On the same good lines, step 5.1 makes the series of derivative increments summable in $L^1$ on every compact subinterval, so the full sequence ${\mathrm{\partial }}_i{v}_j$ converges pointwise almost everywhere there; this pointwise limit agrees with its $L^1$ limit $H$. The subsequence also converges pointwise to a representative of $D_{i}u$, so that representative equals $H$ almost everywhere on those lines. By step 7.1 this $H$ is the classical derivative of the section of $u^{\ast }$, so ${\mathrm{\partial }}_i{u}^{\ast }=D_{i}u$ almost everywhere on $Q$. The rational boxes cover $\mathrm{\Omega }$ countably, giving this identity almost everywhere on $\mathrm{\Omega }$ for every $i$. Consequently the classical derivatives, assigned value $0$ where they fail to exist, are measurable (they agree almost everywhere with measurable $L^p$ representatives) and belong to $L^p$. Hence $u$ satisfies (2).

Steps 1.1 and 1.2 with the constructions of steps 2.1–8.2 prove the two implications: (2)$\Rightarrow$(1) in step 1.1, and (1)$\Rightarrow$(2) in steps 1.2 and 2.1–8.2. The final clause of the Statement is step 1.1 under (2) and step 8.2 under (1), where the two computed representatives agree almost everywhere. The empty domain is the case $\mathrm{\Omega }=\mathrm{\varnothing }$ noted in the Statement. The Axiom of Choice is used through [F18], which supplies the Countable Choice and Dependent Choice hypotheses of [F8], [F12] and [F16] and licenses the countable cutoff and scale selections in steps 1.2 and 3.1, and through the earlier ACL reconstruction lemma [F3]. $\mathrm{\square }$