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Subject and independence

Role. The reviewer worked in a fresh context from the commissioning assignment alone, took no part in writing the reviewed page or any page in its folder, had no contact with the page's author, and was charged with refutation. The reviewer opened no folder index, no evidence folder content, no other review, no workspace file and no web page.

Subject. Path wiki/research/erdos_501/lee_lemma_2_1_reconstruction.md as it stood at 2026-09-28T05:03:27Z, the page Lee Lemma 2.1, read in full as of that time.

Artifact. The second-version, folder-name PDF lee_2026_relative_independence_erdos_problem_501.pdf in the folder of the library card Lee (2026): six pages, printed date line "June 1, 2026", printed page numbers equal to the physical ones. The text layer of all six pages was extracted. Physical p. 2 (the display (1), the definition of $B_x$, the statement of Lemma 2.1) and physical pp. 4--5 (the specialization (11), the proof of Lemma 2.1 with the displays (12)--(19) and the unlabeled upper-integral identity) were read clause by clause, every display against the page image; physical p. 3 (the definition (4) and the statement of Lemma 3.1) was read at statement depth; pp. 1 and 6 were skimmed for context only. Page images of physical pp. 2, 3, 4 and 5 were rendered at 130 dpi and read. No canonical conversion sits beside the PDF; the first-version PDF was not opened.

Allowed material read. The reconstruction page Lee Lemma 3.1 as of the same time, relied on for its Definitions and Statement sections and its section "The specialization used later" (the display (11)); the provenance paragraph of the library card; docs/verification.md "Whole-claim report" and "Audit checklist" (the shared canonical-failure-mode list and the Erdos-specific ten-item list); docs/evidence.md "Source fidelity"; docs/math_authoring.md in full; the Statement paragraph of wiki/problems/set_theory/E0501/_index.md. The page's Source paragraph links no result page under the library card, and the card folder holds none.

Exposures. Four, none used: (a) the reading command displayed the Lemma 3.1 page whole, so its Standing paragraph and its Proof section passed before the reviewer; only its Definitions, Statement and specialization sections entered this review, and (11) is taken as an input, not re-verified; (b) the library card was displayed past its provenance paragraph through its "Bears on" and "Read status" paragraphs and the first lines of its Overview, and the "Bears on" paragraph carries a page-status sentence; (c) a structural listing of the problem page displayed the first line of its Status paragraph; (d) the directory wiki/research/erdos_501/evidence/verify/ exists in the working tree with other entries, none opened.

Restatement

Conventions. A measure is countably additive with values in $[0,\infty]$; $\nu$ extends Lebesgue measure when $\nu$ is defined on every subset of $\mathbb R$ and $\nu(E)=m(E)$ for every Lebesgue measurable $E$; $m^*$ is Lebesgue outer measure, the infimum of the total length of a countable open-interval cover; the source's $\subset$ allows equality and the page writes it $\subseteq$; the upper integral of a function $g\colon\mathbb R\to[0,\infty]$ is the infimum of $\int h\,dm$ over the Lebesgue measurable majorants $h\ge g$.

Proposition. Let $\nu\colon\mathcal P(\mathbb R)\to[0,\infty]$ be a measure extending Lebesgue measure. Let $(A_y)_{y\in\mathbb R}$ be any family of subsets of $\mathbb R$, with no measurability and no boundedness assumed, such that $m^*(A_y)<1$ for every $y\in\mathbb R$. For $x\in\mathbb R$ put $B_x=\{y\in\mathbb R:x\in A_y\}$. Then for every set $C\subseteq\mathbb R$ with $\nu(C)=\infty$ there exists at least one $x\in C$ with $\nu(C\setminus B_x)=\infty$. The conclusion is existential in $x$ and depends on $C$; nothing is claimed about the size or the measure of the set of such $x$. The sets $B_x$ and $C\setminus B_x$ need not be Lebesgue measurable; $\nu(C\setminus B_x)$ is defined because $\nu$ is total.

Checklist

• Quantifiers and scope. Pass. "For every $y$" on the outer-measure hypothesis, "for every $C$ with $\nu(C)=\infty$" and "there is an $x\in C$" agree with the source's Lemma 2.1 (p. 2). The contradiction hypothesis (12) is the exact negation of the conclusion over $x\in C$, and its failure yields $\nu(C\setminus B_x)=\infty$ because the values lie in $[0,\infty]$. Boundary cases: $m^*(D_k)=\infty$ is excluded by $D_k\subseteq[-N,N]$; $\nu(C_M)=\infty$ is excluded by $\nu(C_M)\le2M$; $a=\nu(C_M)-k$ is finite and positive, so the identity is applied inside its range.
• Circularity. Pass. The proof consumes (1), the identity, elementary measure axioms and the section inequality (11); none of these mentions the family $(B_x)$ or the conclusion.
• Model and convention changes. Pass. The objects are the actual sets; the only convention differences from the source are $\subseteq$ for $\subset$ and "countably additive measure" for "measure", both meaning-preserving.
• Finite and statistical overreach. Inapplicable: the argument uses no finite verification, sampling or heuristic averaging.
• Uniformity. Pass. The parameters are chosen in the order $N$, then $k$ (depending on $N$), then $M$ (depending on $N$, $k$ and $m^*(D_k)$); the closing inequality compares fixed finite numbers and exchanges no limit with an integral.
• Extremal conclusions. Pass. The one extremum is the infimum defining the upper integral. The lower bound (18) is derived in the claim's own units, through the measurable set $\{h\ge a\}$ of an arbitrary majorant $h$; see W1. The upper half of the identity is not used.
• Consequences and composition. Pass. Each "so" and "hence" was re-derived (W1--W3). The consumed interface (11) is applied at its actual strength: a total measure on the second factor, $\sigma$-finite by $\mathbb R=\bigcup_n[-n,n]$, and an arbitrary $H\subseteq\mathbb R^2$. (1) is used only in the form $\nu(D_k)\le m^*(D_k)$.
• Computation. Inapplicable: the page carries no computation.
• Reproduction. Inapplicable: the page states no rerun command and no coverage claim.
• Source and verdict fidelity. Pass with notes. Every locator was checked: Lemma 2.1 on physical p. 2, its proof on pp. 4--5, (1) on p. 2, (11) on p. 4, (12)--(17) on p. 4, the identity and (18)--(19) on p. 5. The Standing paragraph claims author-recorded standing only. The label "(the source's (15))" attaches to a chain longer than the source's display (F7), and the supplied arguments are not marked as supplied (F3).

Weakest steps

W1. The lower bound (18). Claim: with $a=\nu(C_M)-k\in(0,\infty)$ and $S=D_k$, every Lebesgue measurable $h\ge a1_{D_k}$ has $\int h\,dm\ge a\,m^*(D_k)$. Derivation: $T=\{h\ge a\}$ is Lebesgue measurable because $h$ is; $D_k\subseteq T$ because $h\ge a$ on $D_k$; $\int h\,dm\ge\int_Th\,dm\ge a\,m(T)$ because $h\ge0$ everywhere and $h\ge a$ on $T$; and $m(T)=m^*(T)\ge m^*(D_k)$ by monotonicity of outer measure. Taking the infimum over $h$ gives $\overline{\int}a1_{D_k}\,dm\ge a\,m^*(D_k)$, and monotonicity of the upper integral turns the pointwise bound $\nu(H_x)\ge a1_{D_k}(x)$ into (18). Composition: (18) is the left end of the chain

$$(\nu(C_M)-k)\,m^*(D_k)\le\overline{\int_{\mathbb R}}\nu(H_x)\,dm(x) \le\int_{\mathbb R}m^*(H^y)\,d\nu(y)\le\nu(C_M),$$

whose middle link is (11) and whose right end is (19). Only this lower half of the identity is load-bearing.

W2. The choice (16) and the closing arithmetic. By (15), $1<m^*(D_k)<\infty$, so $r=k\,m^*(D_k)/(m^*(D_k)-1)$ is a finite positive number, and by (13) some integer $M>N$ has $\nu(C_M)>r$, with $\nu(C_M)\le2M<\infty$. Since $m^*(D_k)>m^*(D_k)-1>0$, the ratio exceeds $1$ and $r>k$, so $\nu(C_M)>k$. Multiplying $\nu(C_M)>r$ by $m^*(D_k)-1>0$ gives $\nu(C_M)m^*(D_k)-\nu(C_M)>k\,m^*(D_k)$, that is, $(\nu(C_M)-k)m^*(D_k)>\nu(C_M)$, and dividing by the finite positive $\nu(C_M)$ gives $(1-k/\nu(C_M))\,m^*(D_k)>1$; every step is reversible, so the page's "equivalent" is exact. The chain of W1, divided by the same $\nu(C_M)$, gives the same quantity $\le1$. Composition: the contradiction refutes (12), which was the only assumption beyond the hypotheses, so some $x\in C$ has $\nu(C\setminus B_x)=\infty$.

W3. The sections of $H$ and the direction of (11). With $H=\{(x,y)\in D_k\times C_M:x\in A_y\}$ and the source's (6) with $Y=\mathbb R$, $H_x=\{y:(x,y)\in H\}$ equals $\{y\in C_M:x\in A_y\}$, which is $C_M\cap B_x$, for $x\in D_k$ and $\varnothing$ otherwise; $H^y=\{x:(x,y)\in H\}$ equals $D_k\cap A_y$ for $y\in C_M$ and $\varnothing$ otherwise. In (11) the $\nu$-measured sections $H_x$ are integrated against $m$ on the first factor and the $m^*$-measured sections $H^y$ against $\nu$ on the second, so the lower bound (17)--(18) sits on the small side and the bound (19) on the large side, as required. For (17): $C_M$ is the disjoint union of $C_M\cap B_x$ and $C_M\setminus B_x$, both $\nu$-measurable since $\nu$ is total, $\nu(C_M)<\infty$ permits the subtraction, and $\nu(C_M\setminus B_x)\le\nu(C\setminus B_x)\le k$ for $x\in D_k$ by (14). For (19): $m^*(D_k\cap A_y)\le m^*(A_y)<1$, so $m^*(H^y)\le1_{C_M}(y)$ pointwise, and every function on $\mathbb R$ is $\nu$-measurable, so the integral against $\nu$ is defined and monotone.

Strongest attack

The attack aimed at (18), the only place where a set that need not be Lebesgue measurable is measured on the Lebesgue side. The upper integral is an infimum over all measurable majorants, and $D_k$ need not be Lebesgue measurable, so the reviewer tried to construct a measurable $h\ge a1_{D_k}$ with $\int h\,dm<a\,m^*(D_k)$, which would void the lower bound and with it the contradiction. Every candidate fails: $\{h\ge a\}$ is a measurable superset of $D_k$, hence of measure at least $m^*(D_k)$, and $h\ge a$ there, so the integral is at least $a\,m^*(D_k)$ (W1); the inequality that would need an envelope is the other one, and the proof never uses it. Two secondary attacks also failed: letting $\nu(C_M)$ be infinite would void the subtraction in (17), but $\nu(C_M)\le\nu([-M,M])=2M$; letting $m^*(D_k)$ be infinite would void (16), but $D_k\subseteq[-N,N]$ gives $m^*(D_k)\le2N$, and this is why the argument cuts $C$ to a window before defining $D_k$. A third attempt, swapping the roles of the two factors in (11) so that the bound (19) would have to hold for the upper integral, is blocked by the source's (6): the page's $H_x$ and $H^y$ are the source's, with $x$ on the Lebesgue factor. No defect was found.

Premises

The section inequality (11). Interface: for every measure $\nu\colon\mathcal P(\mathbb R)\to[0,\infty]$ extending Lebesgue measure and every $H\subseteq\mathbb R^2$,

$$\overline{\int_{\mathbb R}}\nu(H_x)\,dm(x)\le\int_{\mathbb R}m^*(H^y)\,d\nu(y),$$

with $H_x$ and $H^y$ as in the source's (6) with $Y=\mathbb R$. Source held: physical pp. 3--4 of the second-version PDF, read at statement depth for Lemma 3.1 and clause by clause for the specialization on p. 4. Local reconstruction: the Lemma 3.1 page as of the same time, Definitions, Statement and specialization sections, which the reviewed page names as its one input; its standing is not assessed here. Hypotheses met: $\nu$ is total, and $\sigma$-finite through $\nu([-n,n])=2n$.

The domination (1). Interface: $\nu(S)\le m^*(S)$ for every $S\subseteq\mathbb R$. Source p. 2, sketched in one sentence; proved on the page from countable subadditivity, $\nu=m$ on open intervals and the open-cover definition of $m^*$ (F1). Used once, as $\nu(D_k)\le m^*(D_k)$.

The upper-integral identity. Interface: $\overline{\int}a1_S\,dm=a\,m^*(S)$ for $a\ge0$ and $S\subseteq\mathbb R$. Source p. 5, stated without proof; the page supplies a proof (F2, F3). Only the inequality $\ge$ is consumed, at (18), with $a$ finite and positive and $m^*(S)$ finite.

Elementary measure theory, imported without a named source. Monotonicity, countable subadditivity and continuity from below of $\nu$; subtraction of a finite measure; $\nu([-M,M])=2M$; monotonicity of $m^*$ and $m^*([-N,N])=2N$; measurability of $\{h\ge a\}$ for measurable $h$; monotonicity of the upper integral (stated in the Definitions of the Lemma 3.1 page); the convention $0\cdot\infty=0$ (F6). The definition of $m^*$ by countable open-interval covers is stated on the page inside the proof of (1).

Explicit assumptions. None beyond the lemma's hypotheses; the page imports no theorem it does not name, and there is no batch acceptance order.

Findings

F1. Severity: suggested. Location: "if $S\subseteq\bigcup_iJ_i$ with open intervals $J_i$". Defect: the cover is not said to be countable, while the inequality invoked is countable subadditivity and the source (p. 2) says "countably many open intervals"; the definition of $m^*$ is over countable covers. No mathematical harm (an uncountable family of nonempty open intervals has infinite total length, so it cannot lower the infimum), but the quantifier should match the definition. Witness: source p. 2, the sentence after (1). Replacement: "if $S\subseteq\bigcup_iJ_i$ with countably many open intervals $J_i$".

F2. Severity: suggested. Location: "a measurable $T\supseteq S$ gives the majorant $a1_T$ with integral $a\,m(T)$". Defect: this proves $\overline{\int}a1_S\,dm\le a\,m(T)$ for each measurable $T\supseteq S$, and reaching $\le a\,m^*(S)$ needs the envelope fact that the infimum of $m(T)$ over measurable $T\supseteq S$ is $m^*(S)$, which the page does not state; it follows from the cover definition stated in the (1) bullet, since the union of a cover with total length at most $m^*(S)+\delta$ is an open set of measure at most $m^*(S)+\delta$. The gap is in the half of the identity that the proof never uses: (18) consumes only $\ge$. Witness: source p. 5, the identity displayed before (18), and the page's (18). Replacement: "a measurable $T\supseteq S$ gives the majorant $a1_T$ with integral $a\,m(T)$, and the open unions of covers of total length at most $m^*(S)+\delta$ make the infimum at most $a\,m^*(S)$; only the reverse inequality is used below".

F3. Severity: suggested. Location: the two Definitions bullets, "Since $m^*(D_k)/(m^*(D_k)-1)>1$", and "after multiplying by $m^*(D_k)-1>0$ and dividing by $\nu(C_M)$". Defect: these arguments are supplied by the page and not marked as supplied, while the commissioned check requires supplied steps to be marked. The source states the identity without proof (p. 5), sketches (1) in one sentence (p. 2), asserts "In particular, $\nu(C_M)>k$" without a reason (p. 4), and writes that (16) "gives" the final inequality without the algebra (p. 5). Witness: the four source passages named. Replacement: append "(the source states the identity without proof; the argument is supplied here)" to the identity bullet, "(the source sketches this in one sentence)" to the (1) bullet, and "(reason supplied)" to each of the two proof sentences.

F4. Severity: note. Location: "Two facts about $\nu$ are used". Defect: the second bullet is a fact about $m$ and $m^*$ alone; $\nu$ does not occur in it. Witness: the bullet itself. Replacement: "Two elementary facts are used".

F5. Severity: note. Location: "Only the values $\nu(C\setminus B_x)$, $\nu(C_M)$ and the outer measures $m^*(D_k)$, $m^*(A_y)$ enter". Defect: $\nu(C_N)$, $\nu(D_k)$, $\nu(C_M\cap B_x)$, $\nu(C_M\setminus B_x)$ and $m^*(H^y)$ also enter; the sentence's point, that no Lebesgue measurability of $A_y$ or $B_x$ is used, is correct. Witness: (13), (14), (17), (19) on pp. 4--5. Replacement: "Every set is measured either by $\nu$, which is total, or by the outer measure $m^*$; no measurability of the sets $A_y$ or $B_x$ for Lebesgue measure is assumed."

F6. Severity: note. Location: "for $a=0$ both sides vanish". Defect: the right side is $0\cdot m^*(S)$, which is $0$ when $m^*(S)=\infty$ only under the convention $0\cdot\infty=0$, not stated on the page. The application has $a>0$ and $m^*(D_k)<\infty$, so nothing depends on it. Witness: the identity on source p. 5, stated for all $a\ge0$ and $S\subseteq\mathbb R$. Replacement: "for $a=0$ both sides vanish, with $0\cdot\infty=0$".

F7. Severity: note. Location: "$1<\nu(D_k)\le m^*(D_k)\le2N<\infty$ (the source's (15))". Defect: the source's (15) reads $1<m^*(D_k)<\infty$; the page's chain writes the source's own justification ("By (1) and $D_k\subset[-N,N]$") into the display as explicit terms, which is faithful but makes the label cover more than the display it names. Witness: source p. 4, display (15) and the sentence before it. Replacement: "(the source's (15), with its stated justification written into the chain)".

Verdict

Source fidelity: faithful. The statement matches Lemma 2.1 on physical p. 2 in hypotheses, quantifiers and conclusion, with the source's conventions preserved; every locator and every source label (1), (11)--(19) is correct; the Standing paragraph claims no more than author-recorded standing. The seven findings are three suggested precision and labeling corrections and four notes; none is required.

The argument as reconstructed: sound. Every deduction from (12) to the contradiction was re-derived (W1--W3), the input (11) is applied inside its hypotheses, and the one incomplete argument on the page (F2) concerns the unused half of the upper-integral identity.

Limitations: the section inequality (11) is taken as an input at the standing its own page records; its proof was not verified here, since the Lemma 3.1 page's Proof section lies outside the commissioned read set. The first-version PDF was not compared. The review is noncomputational and covers this page only.

This focused review assigns no tier and changes no status.