TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-30

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Chain rule for globally Lipschitz scalar maps of Sobolev functions

Sources

Juha Kinnunen, Sobolev Spaces, Chapter 2 §2.1 (chain rule, Lemma 2.1 and Remark 2.2(4)) and §2.6 (Nikodym ACL characterisation, Theorem 2.36, printed pp. 55–59). The source proves the composition formula for $C^{1}$ maps with bounded derivative by smooth approximation and records as an exercise that globally Lipschitz maps suffice; the proof below instead combines the ACL representative with the one-dimensional chain rule for absolutely continuous functions, as the design requires, and does not use the later density theorem.
John K. Hunter, Notes on Partial Differential Equations, Chapter 3, §§3.1–3.5, for the weak derivative, the ACL characterisation and the model truncated-map calculation (Proposition 3.22 and the remark after it that the $C^{1}$ hypothesis can be relaxed to $f \in W^{1, \infty}$ ).
For the level-set clause the accepted answer to the linked question proves, by a covering argument with the sets ${x \in f^{- 1} (N) : ∣ f (x) - f (y) ∣ \geq ∣ x - y ∣ / k$ near $x}$ , that a one-dimensional absolutely continuous function has $f^{'} = 0$ almost everywhere on the preimage of a null set. The proof of Claim A below is that argument, written out with the outer-measure conventions of this library.

Statement

Assume the Axiom of Choice, used for the published ACL characterisation and the Countable-Choice interfaces cited in the proof. Let $Ω \subseteq R^{n}$ be open, $n \geq 1$ , let $1 \leq p \leq \infty$ , let $u \in W^{1, p} (Ω; R)$ , and let $F : R \to R$ be Lipschitz with constant $L \geq 0$ . Put $D_{F} : = {t \in R : F is differentiable at t}, N_{F} : = R ∖ D_{F} .$

Membership. $F \circ u \in W_{l o c}^{1, p} (Ω)$ , and $F \circ u \in W^{1, p} (Ω)$ if and only if $F (u) \in L^{p} (Ω)$ . This holds automatically if $F (0) = 0$ , and also if $p < \infty$ and $λ_{n} (Ω) < \infty$ .
Level-set clause. $N_{F}$ has Lebesgue measure zero, and for every Lebesgue-null set $N \subseteq R$ (no measurability of $N$ is assumed), every measurable representative $\hat{u}$ of $u$ and every measurable representative ${\hat{v}}_{i}$ of $D_{i} u$ satisfy: the set $Z_{i, N} : = {x \in Ω : {\hat{v}}_{i} (x) \neq 0$ and $\hat{u} (x) \in N}$ has Lebesgue outer measure zero. In particular $D_{i} u = 0$ almost everywhere on the part of $Ω$ where $u$ takes values in $N_{F}$ .
Chain rule with the level-set convention. Let $\hat{u}$ and ${\hat{v}}_{i}$ be measurable representatives of $u$ and $D_{i} u$ . Define $Φ_{i} (x) : = {\begin{cases} F^{'} (\hat{u} (x)) {\hat{v}}_{i} (x), & \hat{u} (x) \in D_{F}, \\ 0, & \hat{u} (x) \in N_{F} . \end{cases}$ Then $Φ_{i}$ agrees almost everywhere with a measurable function, determines an element of $L^{p} (Ω)$ , and is a representative of the weak derivative: $D_{i} (F \circ u) = Φ_{i} almost everywhere on Ω, i = 1, \dots, n .$ That is, the classical product $F^{'} (u) D_{i} u$ is used where $F$ is differentiable at $u$ , and on the remaining preimage the product is taken as zero. Moreover, for every Borel function $g : R \to R$ with $g = F^{'}$ almost everywhere, the class of $g (u) D_{i} u$ is the same, and $D_{i} (F \circ u) = g (u) D_{i} u$ almost everywhere.

If $Ω = \emptyset$ every assertion is vacuous. The constant case $L = 0$ , the endpoint exponents $p = 1$ and $p = \infty$ and the case $n = 1$ are included. No assertion is made about the pointwise derivative of an arbitrary representative of $u$ , and none about a pointwise derivative of $F$ at a point where $F$ fails to be differentiable.

Facts & Assumptions

Given: The Axiom of Choice; an open $Ω \subseteq R^{n}$ with $n \geq 1$ ; an exponent $1 \leq p \leq \infty$ ; a class $u \in W^{1, p} (Ω; R)$ ; and an $L$ -Lipschitz function $F : R \to R$ with $L \geq 0$ .

[F1]

$W^{1, p} (Ω; K)$ is the set of classes $u \in L^{p} (Ω; K)$ such that for every first-order multi-index there is an $L^{p}$ class with a locally integrable representative satisfying the signed test identity for every test function; each such derivative determines one class $D_{i} u$ . For open $U$ with $\overline{U}$ compact in $Ω$ , the notation $W_{l o c}^{1, p} (Ω; K)$ means that the restriction of the class belongs to $W^{1, p} (U; K)$ (Integer-order Sobolev spaces and their norms).

[F2]

$L^{\infty} (μ) = {f : X \to R : f$ measurable and $∥ f ∥_{\infty} < \infty}$ for a measure space $(X, A, μ)$ (The space $L^{\infty} (μ)$ of essentially bounded measurable functions).

[F3]

For $1 \leq p < \infty$ , $L^{p} (μ)$ consists of the measurable $f$ with $\int ∣ f ∣^{p} d μ < \infty$ , and $L^{p} (μ)$ denotes the quotient of $L^{p} (μ)$ by the almost-everywhere-zero functions (The function space $L^{p} (μ)$ for $0 < p < \infty$ ).

[F4]

On a measure space, $L^{p} (μ)$ for $0 < p < \infty$ and for $p = \infty$ is the set of almost-everywhere classes of $L^{p} (μ)$ respectively $L^{\infty} (μ)$ , and for $1 \leq p \leq \infty$ the displayed quotient agrees with the usual quotient-vector-space construction (The space $L^{p} (μ)$ as the quotient by null functions).

[F5]

Assume the Axiom of Choice. Let $Ω \subseteq R^{n}$ be open, $n \geq 1$ , $1 \leq p < \infty$ and $K \in {R, C}$ . A class $u$ on $Ω$ lies in $W^{1, p} (Ω; K)$ if and only if $u \in L^{p} (Ω; K)$ and $u$ has one measurable ACL representative $u^{*}$ whose classical coordinate derivatives $\partial_{i} u^{*}$ exist almost everywhere, are measurable, and belong to $L^{p} (Ω; K)$ . In that case $\partial_{i} u^{*}$ is a representative of $D_{i} u$ for every $i$ (The ACL characterisation of $W^{1, p}$ ).

[F6]

Assume Countable Choice for the completed-product convention. A measurable representative $u^{*}$ of a class on open $Ω$ is absolutely continuous on almost every coordinate line (ACL) when its sections along almost every line in each coordinate direction are absolutely continuous on compact subintervals, with exceptional parameter sets allowed to depend on the direction and the box; the countable family of open rational boxes $Q$ with $\overline{Q} \subseteq Ω$ covers $Ω$ , and the exceptional sets may be united into one null set per direction. For $n = 1$ the condition is absolute continuity on every compact subinterval of $Ω$ ; if $Ω = \emptyset$ the condition is vacuous (Absolute continuity on almost every coordinate line).

[F7]

For $a \leq b$ , a function $f : [a, b] \to R$ is absolutely continuous when each short finite family of disjoint subintervals with total length below $δ$ has total endpoint oscillation below $ε$ (Absolute continuity on a compact interval).

[F8]

Assume Countable Choice. If $v = D^{α} u$ weakly on $Ω$ and $V \subseteq Ω$ is open, then $v ∣_{V} = D^{α} (u ∣_{V})$ weakly on $V$ (Linearity, locality, and commutation of weak derivatives).

[F9]

If $F \in A C [a, b]$ and $h$ is Lipschitz on $F ([a, b])$ , then $h \circ F \in A C [a, b]$ (A Lipschitz function after an absolutely continuous function is absolutely continuous).

[F10]

Let $(X, d_{X})$ and $(Y, d_{Y})$ be metric spaces. A function $f : X \to Y$ is Lipschitz with constant $L \geq 0$ when $d_{Y} (f (x), f (x^{'})) \leq L d_{X} (x, x^{'})$ for all $x, x^{'} \in X$ (Lipschitz map, $α$ -Hölder map for rational $0 < α \leq 1$ , and contraction).

[F11]

Every Lipschitz $f : [a, b] \to R$ is absolutely continuous ( $C^{1}$ implies Lipschitz, Lipschitz implies absolutely continuous, and absolutely continuous implies continuous and bounded variation).

[F12]

Assume Countable Choice and Dependent Choice. A real function $F$ on a compact interval $[a, b]$ is absolutely continuous if and only if $F^{'}$ exists almost everywhere, $F^{'} \in L^{1} [a, b]$ , and $F (x) - F (a) = \int_{a}^{x} F^{'} (t) d t$ for every $x \in [a, b]$ (Fundamental theorem of calculus for absolutely continuous functions).

[F13]

Assume Countable Choice and Dependent Choice. Let $a \leq b$ and $c \leq d$ , let $f : [c, d] \to R$ be a representative of an element of $L^{1} [c, d]$ , let $F = I_{f}$ be its indefinite integral on $[c, d]$ , and let $g : [a, b] \to [c, d]$ be absolutely continuous. If $F \circ g$ is absolutely continuous, then $(F \circ g)^{'} = f (g) g^{'}$ almost everywhere and $f (g) g^{'} \in L^{1} [a, b]$ ; the conclusion holds for every such representative $f$ (Chain rule for an indefinite integral after an absolutely continuous composition).

[F14]

Assume Countable Choice. Every box with $a_{i} \leq b_{i}$ is Lebesgue measurable with measure the product of the side lengths, so a box with $a_{i} < b_{i}$ has finite positive measure (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included).

[F15]

Assume Countable Choice. Every bounded Lebesgue measurable subset of $R^{n}$ has finite measure (Lebesgue measure is sigma-finite, and every metrically bounded subset of $R^{n}$ has finite outer measure).

[F16]

If $1 \leq p, q, r \leq \infty$ satisfy $1 / r = 1 / p + 1 / q$ and $f, g$ lie in the corresponding spaces, then $f g$ lies in the space for $r$ and $∥ f g ∥_{r} \leq ∥ f ∥_{p} ∥ g ∥_{q}$ (Generalized Holder inequality puts products into $L^{r}$ ).

[F17]

If $μ (X) < \infty$ , $1 \leq p < r < \infty$ and $f \in L^{r} (μ)$ , then $f \in L^{p} (μ)$ ; and if $1 \leq p < \infty$ and $f \in L^{\infty} (μ)$ , then $f \in L^{p} (μ)$ and $∥ f ∥_{p} \leq μ (X)^{1 / p} ∥ f ∥_{\infty}$ (Finite-measure $L^{r}$ includes into $L^{p}$ for $p < r$ ).

[F18]

If $f$ is measurable with $∥ f ∥_{\infty} < \infty$ , then $∣ f ∣ \leq ∥ f ∥_{\infty}$ almost everywhere; and if $∣ f ∣ \leq M$ almost everywhere, then $∥ f ∥_{\infty} \leq M$ (The essential supremum is attained as the least essential bound).

[F19]

For $1 \leq p < \infty$ the class $L^{p} (μ)$ is a real vector space under pointwise addition and scalar multiplication, and so is $L^{\infty} (μ)$ ( $L^{p}$ and $L^{\infty}$ are vector spaces for $p \geq 1$ ).

[F20]

Assume Countable Choice. Every continuous map $R^{n} \to R^{m}$ , $n, m \geq 1$ , is Borel measurable (Continuous functions on Euclidean spaces are Borel measurable).

[F21]

If $f$ is measurable and $g$ is Borel measurable on its codomain, then $g \circ f$ is measurable (Composition with a Borel measurable outer map preserves measurability).

[F22]

In ZF, every open cover of an open $Ω \subseteq R^{n}$ admits an at most countable locally finite smooth partition of unity with compact supports, each support contained in some member of the cover (Test function cutoffs and euclidean localization).

[F23]

The class $L^{1} (μ)$ is a complex vector space and the Lebesgue integral is complex-linear on it (The Lebesgue integral is linear on $L^{1} (μ)$ ).

[F24]

For nonnegative measurable $f \leq g$ one has $\int f \leq \int g$ , and $\int c f = c \int f$ for $c \geq 0$ (Monotonicity and nonnegative homogeneity of the nonnegative integral).

[F25]

A measure $μ$ on a space $X$ is finite if $μ (X) < + \infty$ (Finite, sigma-finite, and semifinite measures).

[F26]

In ZF the Axiom of Choice implies Countable Choice and the prescribed-start form of Dependent Choice (AC supplies the countable and dependent choices used in Banach integration).

[F27]

The Axiom of Choice asserts a choice function for every family of nonempty sets (The Axiom of Choice).

[F28]

For $A \subseteq R$ , $f : A \to R$ and a limit point $c$ of $A$ , the difference quotient is $q_{f, c} (x) = (f (x) - f (c)) / (x - c)$ , and $f$ is differentiable at $c$ exactly when $\lim_{x \to c} q_{f, c} (x)$ exists, in which case $f^{'} (c)$ is that limit (The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, The $ε$ - $δ$ limit $\lim_{x \to c} f (x) = L$ of $f : A \to R$ at a limit point $c$ of $A$ ).

[F29]

For a sequence $(x_{k})$ of reals, ${lim sup}_{k} x_{k} = \inf_{n} \sup_{k \geq n} x_{k}$ and ${lim inf}_{k} x_{k} = \sup_{n} \inf_{k \geq n} x_{k}$ in $\overline{R}$ , and $\lim_{k} x_{k}$ exists exactly when the two agree (Limit superior and limit inferior of a real sequence as $\inf_{n} \sup_{k \geq n} x_{k}$ and $\sup_{n} \inf_{k \geq n} x_{k}$ in $\overline{R}$ , A real sequence converges to $L \in R$ iff $lim inf x_{k} = lim sup x_{k} = L$ , and diverges to $\pm \infty$ iff both equal $\pm \infty$ ).

[F30]

Let $(X, A)$ be a measurable space and let $f_{n} : X \to \overline{R}$ be measurable for every $n$ . Then $\sup_{n} f_{n}$ , $\inf_{n} f_{n}$ , ${lim sup}_{n} f_{n}$ and ${lim inf}_{n} f_{n}$ are measurable, and the set where $(f_{n})$ converges in $\overline{R}$ is measurable (Sequential suprema, infima, limsup, liminf, and pointwise limits of measurable functions are measurable).

[F31]

Assume Countable Choice and $n \geq 1$ . The Lebesgue outer measure of $E \subseteq R^{n}$ is the infimum of $\sum_{k} μ_{0} (A_{k})$ over countable covers of $E$ by elementary sets $A_{k}$ ; in particular $λ_{n}^{*} (N) = 0$ means that for every $ε > 0$ there is a countable family of boxes covering $N$ with total elementary volume below $ε$ (Lebesgue outer measure on $R^{n}$ ).

[F32]

Assume Countable Choice. Lebesgue outer measure is monotone and countably subadditive, and agrees with elementary volume on elementary sets, so $λ_{1}^{*} (E) \leq b - a$ for $E \subseteq [a, b]$ and $λ_{1}^{*}$ of a countable union of null sets is zero (Assuming countable choice, Lebesgue outer measure is an outer measure that restricts to elementary volume).

[F33]

Assume Countable Choice. For every subset $E \subseteq R^{n}$ and every $ε > 0$ there is an open $U \supseteq E$ with $λ_{n} (U) \leq λ_{n}^{*} (E) + ε$ (Assuming countable choice, the Lebesgue outer measure of an arbitrary subset of $R^{n}$ is the infimum of the measures of the open sets containing it).

[F34]

Assume Countable Choice. Under the completed-product convention of the ACL definition, iterated integrals over $R^{n - 1} \times R$ compute the integral against $λ_{n}$ for nonnegative measurable functions (Tonelli and Fubini for the completed product, with only almost-everywhere section measurability, Absolute continuity on almost every coordinate line).

[F35]

The canonical naturals are cofinal in $R$ (Every complete ordered field is Archimedean).

Proof

technique · direct

1.1F5F6F8F12F13F14F15F20F26F27F33F34given

By [F26] the Axiom of Choice [F27] yields Countable Choice and Dependent Choice in ZF. Only these are used below: Countable Choice in the ACL definition [F6], the locality lemma [F8], the box-measure formula [F14], finiteness of Lebesgue measure on bounded sets [F15], the Borel measurability of continuous maps [F20], outer regularity [F33], the completed-product convention [F34], and the countable selection of box representatives below; Dependent Choice in the one-dimensional chain rule [F13] and the fundamental theorem [F12]; and the ACL characterisation [F5] under the Axiom of Choice itself.

1.2F7F10F11F12F32given

Since $R$ is a metric space and $F$ is $L$ -Lipschitz, [F10] gives $∣ F (s) - F (t) ∣ \leq L ∣ s - t ∣$ for all real $s, t$ , hence $∣ F (t) ∣ \leq ∣ F (0) ∣ + L ∣ t ∣$ for every $t$ . For every compact interval $[c, d]$ the restriction $F ∣_{[c, d]}$ is Lipschitz, hence absolutely continuous in the sense of [F7] by [F11]. Applying [F12] on each $[c, d]$ gives: $F^{'}$ exists almost everywhere on $[c, d]$ , $F^{'} \in L^{1} [c, d]$ , and $F (x) - F (c) = \int_{c}^{x} f_{0}$ for every $x \in [c, d]$ and every measurable $f_{0}$ with $f_{0} = F^{'}$ almost everywhere on $[c, d]$ . Since $R = ⋃_{m \geq 1} [- m, m]$ , countable subadditivity of outer measure [F32] makes $N_{F}$ a Lebesgue-null set, and for every measurable $g$ with $g = F^{'}$ almost everywhere on $R$ the restriction $g ∣_{[c, d]}$ is a representative of the $L^{1}$ class of $(F ∣_{[c, d]})^{'}$ .

1.3F28F31F32F35given

Claim A (one-dimensional level sets). Let $[a, b]$ be a compact interval, let $N \subseteq R$ satisfy $λ_{1}^{*} (N) = 0$ , and let $g : [a, b] \to R$ be any function. Then the set $Z : = {x \in [a, b] : g (x) \in N and g is differentiable at x with g^{'} (x) \neq 0}$ satisfies $λ_{1}^{*} (Z) = 0$ . Proof of Claim A. Fix $x \in Z$ and write $c : = g^{'} (x) \neq 0$ . By the definition of the derivative as a limit of difference quotients [F28] there is $δ > 0$ with $∣ g (y) - g (x) - c (y - x) ∣ \leq \frac{∣ c ∣}{2} ∣ y - x ∣$ for all $y \in [a, b]$ with $0 < ∣ y - x ∣ < δ$ . Choose $k$ so large that $1 / k < δ$ and $1 / k \leq ∣ c ∣ / 2$ . Then for every such $y$ the reverse triangle inequality gives $∣ g (y) - g (x) ∣ \geq \frac{∣ c ∣}{2} ∣ y - x ∣ \geq \frac{1}{k} ∣ y - x ∣$ . Consequently $Z \subseteq ⋃_{k \geq 1} Z_{k}$ , where $Z_{k}$ is the set of $x \in Z$ such that $∣ g (y) - g (x) ∣ \geq ∣ x - y ∣ / k$ for every $y \in (x - 1 / k, x + 1 / k) \cap [a, b]$ . Fix $k$ . By [F35] choose $M \geq 1$ with $(b - a) / M < 1 / k$ and split $[a, b]$ into $M$ consecutive intervals $J$ of length $(b - a) / M < 1 / k$ . It suffices to show $λ_{1}^{*} (Z_{k} \cap J) = 0$ for one such $J$ , since $Z_{k}$ is the union of the finitely many sets $Z_{k} \cap J$ and outer measure is subadditive [F32]. Let $ε > 0$ . Since $λ_{1}^{*} (N) = 0$ , the definition of outer measure [F31] provides countably many boxes, that is intervals $I_{n} \subseteq R$ , with $N \subseteq ⋃_{n} I_{n}$ and $\sum_{n} λ_{1} (I_{n}) < ε$ . Then $Z_{k} \cap J \subseteq ⋃_{n} g^{- 1} (I_{n})$ , so monotonicity and countable subadditivity [F32] give $λ_{1}^{*} (Z_{k} \cap J) \leq \sum_{n} λ_{1}^{*} (g^{- 1} (I_{n}) \cap Z_{k} \cap J) .$ For each $n$ , the set $E_{n} : = g^{- 1} (I_{n}) \cap Z_{k} \cap J$ is contained in the interval $J$ of length below $1 / k$ , so any two points $u, v$ of $E_{n}$ satisfy $∣ u - v ∣ < 1 / k$ and hence, by the defining property of $Z_{k}$ , $∣ u - v ∣ \leq k ∣ g (u) - g (v) ∣$ ; moreover $g (u), g (v) \in I_{n}$ , so $∣ g (u) - g (v) ∣ \leq λ_{1} (I_{n})$ . If $E_{n} = \emptyset$ , its outer measure is zero. Otherwise the pairwise estimate gives $diam (E_{n}) \leq k λ_{1} (I_{n})$ . For every $δ > 0$ , the bounded set $E_{n}$ lies in an interval of length at most $diam (E_{n}) + δ$ (use its infimum and supremum), so $λ_{1}^{*} (E_{n}) \leq diam (E_{n}) \leq k λ_{1} (I_{n})$ . Summing, $λ_{1}^{*} (Z_{k} \cap J) \leq k \sum_{n} λ_{1} (I_{n}) < k ε$ ; since $ε > 0$ was arbitrary, $λ_{1}^{*} (Z_{k} \cap J) = 0$ . Summing the finitely many $J$ gives $λ_{1}^{*} (Z_{k}) = 0$ , and countable subadditivity over $k$ gives $λ_{1}^{*} (Z) \leq \sum_{k} λ_{1}^{*} (Z_{k}) = 0$ . This proves Claim A.

2.1F20F28F29F30step 1.2given

We construct one measurable almost-everywhere version of $F^{'}$ that vanishes off a controlled set. For $m \geq 1$ and $t \in R$ put $q_{m} (t) : = m (F (t + 1 / m) - F (t))$ ; each $q_{m}$ is continuous, hence Borel measurable by [F20], and $∣ q_{m} (t) ∣ \leq L$ for all $t$ by the Lipschitz bound of step 1.2. Let $\overline{F} : = {lim sup}_{m} q_{m}$ and $\underline{F} : = {lim inf}_{m} q_{m}$ , extended-real-valued and measurable by [F30], and let $C : = {t \in R : \overline{F} (t) = \underline{F} (t) \in R}, f_{0} : = \overline{F} \cdot 1_{C},$ where $f_{0}$ is understood as $0$ outside $C$ . The set $C$ is measurable by [F30], so $f_{0}$ is measurable, and $∣ f_{0} ∣ \leq L$ everywhere because on $C$ the value is $\lim_{m} q_{m}$ , a limit of numbers bounded by $L$ , and outside $C$ the value is $0$ . If $F$ is differentiable at $t$ , then $F^{'} (t)$ is the limit of the difference quotients as the increment tends to $0$ [F28], so the sequence $q_{m} (t) = q_{F, t} (t + 1 / m)$ converges to $F^{'} (t)$ ; the criterion [F29] then gives $\overline{F} (t) = \underline{F} (t) = F^{'} (t)$ and hence $f_{0} (t) = F^{'} (t)$ . Thus $f_{0} = F^{'}$ on $D_{F}$ pointwise, and $f_{0} = F^{'}$ almost everywhere on $R$ because $N_{F}$ is null by step 1.2.

3.1F2F3F4F16F18F21step 2.1given

Fix one measurable representative $\hat{u}$ of the class $u$ and, for each $i$ , one measurable representative ${\hat{v}}_{i}$ of the class $D_{i} u$ ; this is one representative plus finitely many others, so no infinite selection is made. By [F21] the function $f_{0} (\hat{u})$ is measurable, and $∣ f_{0} (\hat{u}) ∣ \leq L$ everywhere by step 2.1, so [F2] and [F18] put $f_{0} (\hat{u})$ in $L^{\infty} (Ω;)$ with $∥ f_{0} (\hat{u}) ∥_{\infty} \leq L$ . Define the measurable function $h_{i} : = f_{0} (\hat{u}) {\hat{v}}_{i}$ . For $1 \leq p < \infty$ the case $(p, q, r) = (\infty, p, p)$ of [F16] gives $h_{i} \in L^{p} (Ω)$ with $∥ h_{i} ∥_{p} \leq L ∥ {\hat{v}}_{i} ∥_{p} < \infty$ , and for $p = \infty$ its $(\infty, \infty, \infty)$ case gives $h_{i} \in L^{\infty} (Ω)$ ; in both cases [F3] and [F4] make $h_{i}$ an element of the quotient space $L^{p} (Ω)$ . If ${\hat{u}}^{'}$ and ${\hat{v}}_{i}^{'}$ are further measurable representatives of the same two classes, then ${\hat{u}}^{'} = \hat{u}$ and ${\hat{v}}_{i}^{'} = {\hat{v}}_{i}$ almost everywhere, hence $f_{0} ({\hat{u}}^{'}) {\hat{v}}_{i}^{'} = f_{0} (\hat{u}) {\hat{v}}_{i}$ almost everywhere, and [F4] shows that the same class is obtained; we write $f_{0} (u) D_{i} u$ for it.

4.1F1F5F6F8F14F17choosestep 1.1step 3.1given

Fix one coordinate direction $i$ and a rational box $Q = \prod_{j = 1}^{n} (a_{j}, b_{j})$ with $\overline{Q} \subseteq Ω$ , as in [F6], and write $Q = Q_{\hat{i}} \times I_{i, Q}$ . By [F14] $Q$ has finite measure, and the restriction $u ∣_{Q}$ of the class $u$ lies in $L^{p} (Q)$ . Put $q : = p$ when $p < \infty$ and $q : = 1$ when $p = \infty$ . Then $u ∣_{Q}$ lies in $L^{q} (Q)$ : for finite $p$ this is $q = p$ , and for $p = \infty$ the class $u ∣_{Q} \in L^{\infty} (Q)$ is converted into $L^{1} (Q)$ by the second clause of [F17]; and by [F8] each weak derivative restricts, $D_{j} (u ∣_{Q}) = (D_{j} u) ∣_{Q}$ on $Q$ , with $(D_{j} u) ∣_{Q} \in L^{q} (Q)$ by the same two clauses. By [F1] this says $u ∣_{Q} \in W^{1, q} (Q; R)$ , and since $1 \leq q < \infty$ the ACL characterisation [F5] applies on the open box $Q$ : it provides one measurable representative $u_{Q}^{*}$ of the class $u ∣_{Q}$ , ACL in every direction, whose classical coordinate derivatives $\partial_{j} u_{Q}^{*}$ exist almost everywhere, are measurable, lie in $L^{q} (Q)$ , and represent $D_{j} (u ∣_{Q})$ almost everywhere for every $j$ . Since $u_{Q}^{*}$ and $\hat{u}$ both represent $u ∣_{Q}$ , and $\partial_{j} u_{Q}^{*}$ and ${\hat{v}}_{j}$ both represent $(D_{j} u) ∣_{Q}$ , we have $u_{Q}^{*} = \hat{u}$ and $\partial_{j} u_{Q}^{*} = {\hat{v}}_{j}$ almost everywhere on $Q$ for every $j$ , so in particular $f_{0} (u_{Q}^{*}) \partial_{i} u_{Q}^{*} = h_{i}$ almost everywhere on $Q$ . The assignment of one representative $u_{Q}^{*}$ to each rational box $Q$ is a selection from countably many nonempty sets, licensed by the Countable Choice of step 1.1.

5.1F6F12F28F32F34step 1.3step 4.1given

Level sets on a box. Fix the rational box $Q$ and the ACL representative $u_{Q}^{*}$ of step 4.1, and recall the one-dimensional input Claim A of step 1.3. Let $N \subseteq R$ be a Borel set with $λ_{1} (N) = 0$ . We claim that $w : = u_{Q}^{*}$ vanishes, as a classical partial derivative, almost everywhere on the measurable set $E_{N} : = {x \in Q : u_{Q}^{*} (x) \in N}$ . Extend the measurable function $\partial_{i} u_{Q}^{*}$ , initially defined almost everywhere on $Q$ , by the value $0$ on the null set where it does not exist, and keep the notation $\partial_{i} u_{Q}^{*}$ for the extension; this changes the function only on a null set. Then the set $Z : = {x \in Q : u_{Q}^{*} (x) \in N$ and $\partial_{i} u_{Q}^{*} (x) \neq 0}$ is measurable, because $u_{Q}^{*}$ , the extension, and $N$ are measurable. By the completed-product convention and Tonelli [F34] applied to the indicator of $Z$ , $λ_{n} (Z) = \int_{Q_{\hat{i}}} λ_{1} (Z_{y}) d λ_{n - 1} (y),$ where $Z_{y} = {t \in I_{i, Q} : u_{Q}^{*} (ι_{i} (y, t)) \in N$ and $\partial_{i} u_{Q}^{*} (ι_{i} (y, t)) \neq 0}$ is the section. For every $y$ outside the exceptional set of the ACL definition [F6] the section $g_{y} (t) : = u_{Q}^{*} (ι_{i} (y, t))$ is absolutely continuous on every compact subinterval of $I_{i, Q}$ , hence differentiable almost everywhere there by [F12]. For such $y$ and every compact subinterval $[a, b] \subseteq I_{i, Q}$ , Claim A of step 1.3 applies to $g_{y} ∣_{[a, b]}$ and the null set $N$ : the set of $t \in [a, b]$ with $g_{y} (t) \in N$ and $g_{y}$ differentiable at $t$ with $g_{y}^{'} (t) \neq 0$ has outer measure zero; and the set where $g_{y}^{'}$ fails to exist is null by [F12], while $g_{y}^{'}$ equals the partial derivative $\partial_{i} u_{Q}^{*}$ at every point of differentiability, by the definition of the partial derivative [F28] (the extension agrees with the classical partial derivative wherever the latter exists). Hence $λ_{1} (Z_{y} \cap [a, b]) = 0$ . Exhausting $I_{i, Q}$ by countably many compact subintervals, countable subadditivity [F32] gives $λ_{1} (Z_{y}) = 0$ for every such $y$ ; since the exceptional $y$ -set is null, the integral above is $0$ and $λ_{n} (Z) = 0$ . Restoring the modification on the null set where the extension differs from the true partial derivative gives the claim: $\partial_{i} u_{Q}^{*} = 0$ almost everywhere on ${x \in Q : u_{Q}^{*} (x) \in N}$ , and consequently the weak derivative satisfies $D_{i} u = 0$ almost everywhere on that set, by the representative comparison of step 4.1.

5.2F3F4F6F7F9F10F14F17F19F20F21F24step 1.2step 4.1given

We verify the membership and representative clauses for the class $F \circ u ∣_{Q}$ on $Q$ , whose exponent $q$ satisfies $1 \leq q < \infty$ . First $F \circ u ∣_{Q} \in L^{q} (Q)$ : by step 1.2 $∣ F (s) ∣ \leq ∣ F (0) ∣ + L ∣ s ∣$ for every real $s$ , so $∣ F (u) ∣ \leq ∣ F (0) ∣ + L ∣ u ∣$ pointwise; the bound function lies in $L^{q} (Q)$ because the constant belongs to $L^{q} (Q)$ by the second clause of [F17] applied to the finite-measure box of [F14], the multiple $L ∣ u ∣$ belongs to $L^{q} (Q)$ by [F19], and sums of $L^{q}$ functions belong to $L^{q} (Q)$ by [F19]; monotonicity [F24] of the nonnegative integral then gives $\int_{Q} ∣ F (u) ∣^{q} < \infty$ , so [F3] and [F4] make $F \circ u ∣_{Q}$ an element of $L^{q} (Q)$ . Second, $Φ : = F \circ u_{Q}^{*}$ is a measurable representative of that class: it is measurable by [F21] applied to the Borel function $F$ , which is continuous and hence Borel by [F20], and to the measurable $u_{Q}^{*}$ of step 4.1; and it agrees with $F \circ u$ almost everywhere on $Q$ because $u_{Q}^{*} = u$ almost everywhere there. Third, $Φ$ is ACL: by [F6] the sections of $u_{Q}^{*}$ in direction $i$ are absolutely continuous in the sense of [F7] on every compact subinterval of $I_{i, Q}$ , for every transverse parameter outside a null set depending on the direction, and $F$ is Lipschitz on the whole real line with constant $L$ by [F10], so [F9] makes each composite section $F \circ g_{y}$ absolutely continuous on every compact subinterval; the same null exceptional sets therefore serve for $Φ$ .

6.1F2F5F7F9F13F16F18F21F28F34step 1.2step 2.1step 4.1step 5.2given

The classical derivative of $Φ$ in direction $i$ exists almost everywhere on $Q$ and equals $G : = f_{0} (u_{Q}^{*}) \partial_{i} u_{Q}^{*}$ almost everywhere. Indeed, fix $y$ outside the exceptional set of [F6] and a compact subinterval $[α, β] \subseteq I_{i, Q}$ ; the section $g : = g_{y}$ is absolutely continuous on $[α, β]$ by [F7], its image $g ([α, β]) = [c, d]$ is a compact interval, and by step 1.2 the restriction of $F$ to $[c, d]$ satisfies $F (x) = F (c) + \int_{c}^{x} f_{0}$ for all $x \in [c, d]$ , so that $F ∣_{[c, d]} - F (c) = I_{f_{0}}$ for the function $f_{0} ∣_{[c, d]}$ , which is a representative of the $L^{1}$ class of $(F ∣_{[c, d]})^{'}$ ; the composite $F \circ g$ is absolutely continuous by [F9], so $(F - F (c)) \circ g$ is absolutely continuous as well. Apply [F13] to this normalized indefinite integral to obtain $((F - F (c)) \circ g)^{'} = f_{0} (g) g^{'}$ almost everywhere on $[α, β]$ , with $f_{0} (g) g^{'} \in L^{1} [α, β]$ ; since the derivative of a constant is zero, $(F \circ g)^{'} = f_{0} (g) g^{'}$ there; this holds for every compact subinterval, so $(F \circ g_{y})^{'} = f_{0} (g_{y}) g_{y}^{'}$ almost everywhere on $I_{i, Q}$ . At every point where the partial derivative $\partial_{i} u_{Q}^{*}$ exists, $g_{y}$ is differentiable there with $g_{y}^{'} = \partial_{i} u_{Q}^{*}$ , by the definition of the partial derivative [F28]. Since $\partial_{i} u_{Q}^{*}$ exists almost everywhere on $Q$ by step 4.1, and $(F \circ g_{y})^{'}$ exists almost everywhere on $I_{i, Q}$ for the good $y$ , the partial derivative $\partial_{i} Φ$ exists almost everywhere on $Q$ and agrees there with $G$ : the set where either the partial derivative fails to exist or differs from $G$ is contained in the union of the null set of bad $y$ and, for each good $y$ , a null set of $t$ , which is null for the product measure by the completed-product convention and Tonelli [F34]. The function $G$ is measurable by [F21], since $f_{0}$ is measurable by step 2.1 and $u_{Q}^{*}$ , $\partial_{i} u_{Q}^{*}$ are measurable, and $G$ lies in $L^{q} (Q)$ : $∣ f_{0} (u_{Q}^{*}) ∣ \leq L$ everywhere by step 2.1, so $f_{0} (u_{Q}^{*}) \in L^{\infty} (Q)$ by [F2] and [F18], and with $\partial_{i} u_{Q}^{*} \in L^{q} (Q)$ from step 4.1 the case $(p, q, r) = (\infty, q, q)$ of [F16] gives $G \in L^{q} (Q)$ . All clauses of the characterisation [F5] hold for the class $F \circ u ∣_{Q}$ with representative $Φ$ : it lies in $L^{q} (Q)$ by step 5.2, $Φ = F \circ u_{Q}^{*}$ is a measurable ACL representative, and its classical coordinate derivative in direction $i$ exists almost everywhere, is measurable and lies in $L^{q} (Q)$ . Hence $F \circ u ∣_{Q} \in W^{1, q} (Q; R)$ and $D_{i} (F \circ u ∣_{Q})$ is represented almost everywhere by $G$ ; by the comparison of step 4.1, $G = f_{0} (u_{Q}^{*}) \partial_{i} u_{Q}^{*} = h_{i}$ almost everywhere on $Q$ , so $D_{i} (F \circ u ∣_{Q})$ is represented almost everywhere by $h_{i}$ . The same argument applies to every coordinate direction simultaneously, because the single representative $u_{Q}^{*}$ of step 4.1 is ACL in every direction and its classical coordinate derivatives represent all $D_{j} (u ∣_{Q})$ .

6.2F32F33step 1.2step 4.1step 5.1given

The level-set clause. Let first $N \subseteq R$ be Borel with $λ_{1} (N) = 0$ . For every rational box $Q$ , step 5.1 shows that $\partial_{i} u_{Q}^{*} = 0$ almost everywhere on ${x \in Q : u_{Q}^{*} (x) \in N}$ , while $u_{Q}^{*} = \hat{u}$ and $\partial_{i} u_{Q}^{*} = {\hat{v}}_{i}$ almost everywhere on $Q$ by step 4.1; hence the set ${x \in Q : {\hat{v}}_{i} (x) \neq 0$ and $\hat{u} (x) \in N}$ is null. The rational boxes cover $Ω$ and there are countably many of them, so countable subadditivity of Lebesgue measure (via [F32]) makes $Z_{i, N} = {x \in Ω : {\hat{v}}_{i} (x) \neq 0$ and $\hat{u} (x) \in N}$ a null set. Now let $N \subseteq R$ be an arbitrary set with $λ_{1}^{*} (N) = 0$ . By outer regularity [F33], for every $m$ there is an open $G_{m} \supseteq N$ with $λ_{1} (G_{m}) \leq 1 / m$ ; the intersection $N_{0} : = ⋂_{m} G_{m}$ is a Borel set containing $N$ , and it is null because it is contained in each $G_{m}$ and outer measure is monotone [F32]. Then $Z_{i, N} \subseteq Z_{i, N_{0}}$ , which is null by the Borel case, so $λ_{n}^{*} (Z_{i, N}) = 0$ by monotonicity [F32]. This proves clause 2 in full; the case $N = N_{F}$ is available because $N_{F}$ is null by step 1.2.

7.1F1F2F18step 1.2step 2.1step 6.1given

When $p = \infty$ the exponent used so far is $q = 1$ , and we upgrade the conclusion of step 6.1 to $W^{1, \infty} (Q)$ . The class $F \circ u ∣_{Q}$ lies in $L^{\infty} (Q)$ : $∣ F (u) ∣ \leq ∣ F (0) ∣ + L ∣ u ∣$ pointwise by step 1.2 and $∣ u ∣ \leq ∥ u ∣_{Q} ∥_{\infty}$ almost everywhere on $Q$ by [F18], so $F (u)$ is bounded almost everywhere by $∣ F (0) ∣ + L ∥ u ∣_{Q} ∥_{\infty}$ and [F18] with [F2] gives $F \circ u ∣_{Q} \in L^{\infty} (Q)$ . Each weak derivative class $D_{j} (F \circ u ∣_{Q})$ lies in $L^{\infty} (Q)$ : by step 6.1 it is represented by $f_{0} (u_{Q}^{*}) \partial_{j} u_{Q}^{*}$ , whose absolute value is at most $L ∣ \partial_{j} u_{Q}^{*} ∣ \leq L ∥ \partial_{j} u_{Q}^{*} ∥_{\infty}$ almost everywhere by [F18] and the bound $∣ f_{0} ∣ \leq L$ of step 2.1, so the representative lies in $L^{\infty} (Q)$ by [F2]. Thus the class lies in $L^{\infty} (Q)$ and every first weak derivative class has an $L^{\infty}$ representative, which is membership in $W^{1, \infty} (Q; R)$ by the definition [F1].

8.1F1F6F14F18F19F22F23step 1.2step 3.1step 6.1step 7.1given

We patch the box conclusions into a global weak derivative identity. For the fixed direction $i$ and any test function $φ \in C_{c}^{\infty} (Ω)$ , cover $Ω$ by the rational boxes of [F6] and use [F22] to choose a locally finite smooth partition of unity $(χ_{k})$ subordinate to that cover, with compact supports. Only finitely many $χ_{k}$ meet the compact support of $φ$ , and $φ = \sum_{k} χ_{k} φ$ ; each summand $χ_{k} φ$ has compact support contained in some box $Q_{k}$ with $\overline{Q_{k}} \subseteq Ω$ , so it is a test function on $Q_{k}$ . Step 6.1 gives the finite-exponent weak identity and step 7.1 supplies the $p = \infty$ case; together they give, for every $ψ \in C_{c}^{\infty} (Q_{k})$ , $\int_{Q_{k}} F (u) \partial_{i} ψ = - \int_{Q_{k}} h_{i} ψ$ ; applying this to the finitely many summands and summing with the linearity of the integral [F23] gives $\int_{Ω} F (u) \partial_{i} φ = - \int_{Ω} h_{i} φ$ . Both sides are integrable: $u \in L^{p} (Ω)$ is locally integrable, $F \circ u \in L_{l o c}^{1} (Ω)$ because $∣ F (u) ∣ \leq ∣ F (0) ∣ + L ∣ u ∣$ by step 1.2 with a constant function on finite-measure pieces ([F14] and [F19]), and $h_{i} \in L_{l o c}^{p} (Ω)$ — for finite $p$ because $[h_{i}] \in L^{p} (Ω)$ by step 3.1 and for $p = \infty$ because $∣ h_{i} ∣ \leq L ∥ {\hat{v}}_{i} ∥_{\infty}$ almost everywhere by [F18]. Since $φ$ was arbitrary and the argument applies to every direction, the definition [F1] gives $D_{i} (F \circ u) = [h_{i}] = f_{0} (u) D_{i} u$ weakly on $Ω$ for every $i$ , hence almost everywhere on $Ω$ .

9.1F1F2F8F15F17F18F19F24step 1.2step 3.1step 8.1given

We record the local membership. Let $U \subseteq Ω$ be open with $\overline{U}$ compact in $Ω$ ; then $U$ is bounded, so $λ_{n} (U) < \infty$ by [F15]. The class $F \circ u ∣_{U}$ lies in $L^{p} (U)$ : for $p < \infty$ the pointwise bound $∣ F (u) ∣ \leq ∣ F (0) ∣ + L ∣ u ∣$ of step 1.2, the second clause of [F17] applied to the constant and the vector-space clauses of [F19] show first that the bound function lies in $L^{p} (U)$ , and then monotonicity [F24] gives $\int_{U} ∣ F (u) ∣^{p} < \infty$ ; for $p = \infty$ the bound $∣ F (u) ∣ \leq ∣ F (0) ∣ + L ∥ u ∣_{U} ∥_{\infty}$ holds almost everywhere on $U$ by [F18], so $F \circ u ∣_{U} \in L^{\infty} (U)$ by [F2]. Each weak derivative $D_{i} (F \circ u ∣_{U})$ is the restriction $(D_{i} (F \circ u)) ∣_{U} = [h_{i} ∣_{U}]$ by the locality lemma [F8], and $h_{i} ∣_{U} \in L^{p} (U)$ : for finite $p$ this is the restriction of the class $[h_{i}] \in L^{p} (Ω)$ from step 3.1, and for $p = \infty$ we have $∣ h_{i} ∣ \leq L ∥ {\hat{v}}_{i} ∥_{\infty}$ almost everywhere by [F18]. By the definition [F1], $F \circ u ∣_{U} \in W^{1, p} (U; R)$ for every such $U$ , that is, $F \circ u \in W_{l o c}^{1, p} (Ω)$ , and the identity of step 8.1, $D_{i} (F \circ u) = f_{0} (u) D_{i} u$ almost everywhere on $Ω$ , holds for every $i$ .

9.2F1step 3.1step 8.1given

The global membership criterion. If $F \circ u \in W^{1, p} (Ω)$ , then $F \circ u \in L^{p} (Ω)$ by the definition [F1], and $F \circ u$ is the class of $F (u)$ . Conversely, if $F (u) \in L^{p} (Ω)$ , then the class $F \circ u \in L^{p} (Ω)$ , and each derivative class $D_{i} (F \circ u) = [h_{i}]$ lies in $L^{p} (Ω)$ by step 3.1; the definition [F1] then gives $F \circ u \in W^{1, p} (Ω)$ . This proves the equivalence in clause 1 of the Statement.

10.1F4step 1.2step 2.1step 3.1step 8.1step 9.1step 6.2given

We identify the two descriptions of the derivative class. Let $\hat{u}$ and ${\hat{v}}_{i}$ be the representatives of step 3.1 and let $Φ_{i}$ be as in clause 3 of the Statement. On the measurable set ${\hat{u} \in D_{F}}$ one has $f_{0} (\hat{u}) = F^{'} (\hat{u})$ pointwise by step 2.1, so $Φ_{i} = f_{0} (\hat{u}) {\hat{v}}_{i} = h_{i}$ there. On ${\hat{u} \in N_{F}}$ the convention gives $Φ_{i} = 0$ , while $h_{i} = f_{0} (\hat{u}) {\hat{v}}_{i} = 0$ almost everywhere by clause 2 applied to the null set $N_{F}$ of step 1.2, which forces ${\hat{v}}_{i} = 0$ almost everywhere on ${\hat{u}}^{- 1} (N_{F})$ . Hence $Φ_{i} = h_{i}$ almost everywhere on $Ω$ ; in particular $Φ_{i}$ agrees almost everywhere with the measurable function $h_{i}$ , determines the same element of $L^{p} (Ω)$ as in step 3.1, and represents $D_{i} (F \circ u)$ by step 8.1 and step 9.1. Finally let $g : R \to R$ be Borel with $g = F^{'}$ almost everywhere, and put $N_{g} : = {t \in R : g (t) \neq f_{0} (t)}$ , a null set. On $Ω ∖ {\hat{u}}^{- 1} (N_{g})$ one has $g (\hat{u}) = f_{0} (\hat{u})$ pointwise, while on ${\hat{u}}^{- 1} (N_{g})$ clause 2 gives ${\hat{v}}_{i} = 0$ almost everywhere; hence $g (\hat{u}) {\hat{v}}_{i} = h_{i}$ almost everywhere, so the class of $g (u) D_{i} u$ equals the class of $h_{i}$ and $D_{i} (F \circ u) = g (u) D_{i} u$ almost everywhere.

11.1F2F3F4F6F17F18F19F24F25step 1.2step 2.1step 4.1step 9.1step 9.2given

The automatic cases and the closing discussion. If $F (0) = 0$ , then $∣ F (t) ∣ = ∣ F (t) - F (0) ∣ \leq L ∣ t ∣$ for every $t$ by step 1.2, so $∣ F (u) ∣ \leq L ∣ u ∣$ pointwise and $\int_{Ω} ∣ F (u) ∣^{p} \leq L^{p} \int_{Ω} ∣ u ∣^{p} < \infty$ by monotonicity [F24] when $p < \infty$ , while $∥ F (u) ∥_{\infty} \leq L ∥ u ∥_{\infty}$ by [F18] when $p = \infty$ ; in both cases $F (u) \in L^{p} (Ω)$ by [F2], [F3] and [F4], and clause 1 follows from step 9.2. If $p < \infty$ and $λ_{n} (Ω) < \infty$ , so that the restricted Lebesgue measure is finite in the sense of [F25], the constant $∣ F (0) ∣$ lies in $L^{\infty} (Ω)$ and hence in $L^{p} (Ω)$ by the second clause of [F17], while $L ∣ u ∣ \in L^{p} (Ω)$ by [F19]; the sum $∣ F (0) ∣ + L ∣ u ∣ \in L^{p} (Ω)$ by [F19] dominates $∣ F (u) ∣$ pointwise by step 1.2, so $\int_{Ω} ∣ F (u) ∣^{p} < \infty$ by [F24] and $F (u) \in L^{p} (Ω)$ , and clause 1 follows from step 9.2. If $p = \infty$ , then $∣ F (u) ∣ \leq ∣ F (0) ∣ + L ∥ u ∥_{\infty}$ almost everywhere on $Ω$ by [F18], so $F (u) \in L^{\infty} (Ω)$ by [F2] and clause 1 follows from step 9.2. If $L = 0$ , then $F$ is the constant $F (0)$ , so $F$ is differentiable everywhere with $F^{'} = 0$ , $N_{F} = \emptyset$ , and $f_{0} = 0$ ; clauses 1 to 3 hold with $Φ_{i} = 0$ and $D_{i} (F \circ u) = 0$ , consistently with step 9.1. The case $n = 1$ is included: the ACL definition [F6] then reads that one representative is absolutely continuous on every compact subinterval of the interval $Ω$ , and no transverse parameter occurs. If $Ω = \emptyset$ , the rational-box family of [F6] is empty, the only class is zero, the representative $\hat{u}$ may be taken to be $0$ , and every displayed assertion holds vacuously with $Z_{i, N} = \emptyset$ . The finitely many representative selections of step 3.1 need no choice principle; the only countable selection is that of the box representatives in step 4.1, licensed by the Countable Choice obtained in step 1.1 from the Axiom of Choice; the trivial representative of the zero class and the rational-box enumeration of [F6] are canonical. $□$

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

164 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Juha Kinnunen, Sobolev Spaces (2026), Chapters 1–2 (standard reference, not scraped)
John K. Hunter, Notes on Partial Differential Equations (2014), Chapter 3 (standard reference, not scraped)
Ben Johnsrude and Mateusz Kwaśnicki, answers to "Preimage of a null set by a non-monotonic absolutely continuous function" (standard reference, not scraped)