False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-29

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FALSE: the rectangle formula determines a unique product measure without any sigma-finiteness hypothesis

Statement

For arbitrary measure spaces, any two measures on $A \otimes B$ that agree on every measurable rectangle must agree everywhere.

Facts & Assumptions

Given: Lebesgue measure $μ$ on $[0, 1]$ , counting measure $ν$ on $[0, 1]$ , the diagonal $D : = {(x, y) \in [0, 1]^{2} : x = y}$ , and the two measures $ρ, τ$ on the same product sigma-algebra supplied by the standard non-sigma-finite Lebesgue/counting construction in the listed Tao source, with $ρ (A \times B) = μ (A) ν (B) = τ (A \times B)$ on measurable rectangles and $ρ (D) = 1 \neq 0 = τ (D)$ .

[L1]

Under sigma-finiteness, the rectangle formula does determine a unique product measure. (For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique)

Refutation

technique · direct

1.1

The witness in the Given line satisfies exactly the hypothesis of the displayed claim: $ρ$ and $τ$ are measures on the same product sigma-algebra and agree on every measurable rectangle.

given

2.1

The same witness also satisfies $ρ (D) = 1 \neq 0 = τ (D)$ , so $ρ \neq τ$ . Hence the displayed universal uniqueness claim is false. This shows why the sigma-finiteness hypothesis in [L1] is essential.

L1givenstep 1.1∎

Depends on

For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Terence Tao, An Introduction to Measure Theory, Remark 1.7.12 (standard reference, not scraped)