LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 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.

ACL representatives recover their weak gradients by Fubini

Sources

Juha Kinnunen, Sobolev Spaces, Chapter 2 §2.6, Theorem 2.36, converse proof, printed p. 59 (PDF pp. 60–61). On almost every coordinate line the source applies one-dimensional integration by parts and then Fubini to get the weak derivative identity. Here the argument is written on coordinate boxes, with the completion and endpoint hypotheses made explicit.
John K. Hunter, Notes on Partial Differential Equations, Chapter 3 §3.5, Definition 3.23, printed p. 59 (PDF p. 63), for the function and weak derivative conventions used by the Sobolev interface.

Statement

Assume the Axiom of Choice. Let $Ω \subseteq R^{n}$ be open, $n \geq 1$ , $1 \leq p \leq \infty$ , and $K \in {R, C}$ . Let $u \in L_{l o c}^{p} (Ω; K)$ have one measurable ACL representative $u^{*}$ , and suppose measurable functions $g_{i} \in L_{l o c}^{p} (Ω; K)$ , $1 \leq i \leq n$ , satisfy the following: on almost every line parallel to the $i$ th coordinate axis, the one-dimensional derivative of the ACL section of $u^{*}$ exists and equals the section of $g_{i}$ almost everywhere. Then $g_{i}$ is the weak derivative $D_{i} u$ for every $i$ , that is, $\int_{Ω} u \partial_{i} φ d x = - \int_{Ω} g_{i} φ d x (φ \in C_{c}^{\infty} (Ω)) .$ The integrals are bilinear, without conjugation. If $Ω = \emptyset$ , the assertion is vacuous.

The Axiom of Choice is used only to invoke the cited Countable Choice and Dependent Choice interfaces for completed-product Fubini and one-dimensional absolute-continuity integration by parts; no representative is selected in the proof.

Facts & Assumptions

Given: AC, an open $Ω \subseteq R^{n}$ , $n \geq 1$ , $1 \leq p \leq \infty$ , the a.e. class $u$ , one ACL representative $u^{*}$ , and its measurable local $L^{p}$ line derivatives $g_{i}$ as in the Statement.

[F1]

An ACL representative is absolutely continuous on compact subintervals of almost every coordinate line in each rational coordinate box (Absolute continuity on almost every coordinate line).

[F2]

For absolutely continuous real functions $F, G$ on $[a, b]$ , $\int_{a}^{b} F G^{'} + \int_{a}^{b} F^{'} G = F (b) G (b) - F (a) G (a) .$ The complex-valued version follows by applying this real identity to real and imaginary parts (Integration by parts for absolutely continuous functions).

[F3]

If $f$ is integrable for the completed product of sigma-finite measures, its sections are integrable almost everywhere and the iterated integrals equal its product integral (Tonelli and Fubini for the completed product, with only almost-everywhere section measurability).

[F4]

For positive integers $m, n$ , Euclidean Lebesgue measure on $R^{m + n}$ is the completion of the product of the factor Lebesgue measures (The Euclidean Lebesgue measure is the completion of the product of the factor Lebesgue measures).

[F5]

Every open cover of $Ω$ has a locally finite smooth partition of unity with compact supports subordinate to its members (Test function cutoffs and euclidean localization).

[F6]

Complex Hölder gives $\int ∣ f g ∣ \leq ∥ f ∥_{p} ∥ g ∥_{p^{'}}$ for conjugate exponents including $(1, \infty)$ and $(\infty, 1)$ (Complex Holder, Minkowski, and the quotient norm).

[F7]

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).

[F8]

AC supplies a choice function for every family of nonempty sets (The Axiom of Choice), and the cited consequence supplies Countable Choice and prescribed-start Dependent Choice (AC supplies the countable and dependent choices used in Banach integration).

[F9]

The weak derivative identity is the signed test-function identity for every $φ \in C_{c}^{\infty} (Ω)$ (Weak derivative of a locally integrable function).

Proof

technique · Fubini applied to one-dimensional integration by parts

1.1given

If $Ω = \emptyset$ or $φ = 0$ , the identity holds with both sides zero.

Otherwise cover $Ω$ by its rational coordinate boxes and use [F5] to write $φ$ as a locally finite sum of compactly supported smooth functions, each supported in one such box. Only finitely many summands meet the compact support of $φ$ , so it is enough by linearity to prove the identity for a test function $ψ$ supported in a single coordinate box $Q$ . [F5, given]

1.2F1given

Fix such a $Q$ and a coordinate $i$ .

When $u = 0$ and $g_{i} = 0$ as a.e. classes, both integrals vanish. Otherwise, the support of $ψ$ lies in a compact sub-box, so its restriction to every $i$ -coordinate section vanishes near the two endpoints of the side interval of $Q$ . Choose a compact interval $[a, b]$ strictly inside that side interval and containing the projection of $supp ψ$ onto the $i$ th coordinate. For almost every transverse point the section of $u^{*}$ is absolutely continuous on $[a, b]$ ; by hypothesis its derivative equals the section of $g_{i}$ almost everywhere. Apply [F2] on $[a, b]$ . Since the section of $ψ$ vanishes at $a, b$ , This gives $\int_{I_{i, Q}} u^{*} (y, t) \partial_{i} ψ (y, t) d t = - \int_{I_{i, Q}} g_{i} (y, t) ψ (y, t) d t$ for almost every transverse $y$ . When $n = 1$ this is directly the same one-dimensional identity, with no transverse integral. For $n \geq 2$ , both products are in $L^{1} (Q)$ : the test and its derivative are bounded, $Q$ has finite measure by [F7], and local $L^{p}$ with [F6] gives local $L^{1}$ . By [F4], Lebesgue measure on $Q$ is the completion of the corresponding product measure, so [F3] integrates the line identity over the transverse variables. Since $u^{*} = u$ almost everywhere, this gives $\int_{Q} u \partial_{i} ψ d x = - \int_{Q} g_{i} ψ d x .$ AC supplies the Countable Choice and Dependent Choice hypotheses required by [F2] and [F3], exactly as stated in [F8]. [F1, F2, F3, F4, F6, F7, F8, given]

2.1F5F6F9givenstep 1.1step 1.2

Sum the finitely many partition identities.

Their sum is the original test function, so the same identity holds on $Ω$ . By [F9] this says precisely that $D_{i} u = g_{i}$ weakly. The argument applies to every coordinate; the endpoints $p = 1$ and $p = \infty$ are included by [F6]. [F5, F6, F9, given, step 1.1, step 1.2] $□$

Depends on

Used by

The ACL characterisation of W^1,p Theorem

Dependency tree · two levels

64 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), Chapter 2 §2.6 (standard reference, not scraped)
John K. Hunter, Notes on Partial Differential Equations (2014), Chapter 3 §3.5 (standard reference, not scraped)