Measure theory 1.2.23: add translation invariance to Lebesgue_measure.unique#537
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teorth merged 1 commit intoJun 22, 2026
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Exercise 1.2.23 assumes a translation-invariant measure; without it, weighted measures on the unit cube satisfy the other hypotheses but are not Lebesgue measure. Co-authored-by: Cursor <cursoragent@cursor.com>
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Fixes narinluangrath's report on #517.
Bug
Lebesgue_measure.uniqueonly assumed empty-set zero, nonnegativity, countable additivity on Lebesgue measurable sets, and unit-cube normalization. That does not characterize Lebesgue measure.Root cause
The standard uniqueness proof subdivides the unit cube into congruent boxes; that step needs translation invariance, which was missing from the hypotheses.
Why this fix is correct
Any density-weighted measure
m(E) = ∫_E fwithf ≥ 0,∫_{[0,1]^d} f = 1, and non-constantfsatisfies the old hypotheses but is not Lebesgue measure. Addingm (E + {x}) = m Ematches Tao's Exercise 1.2.23 and the Jordan-measure uniqueness statements earlier in the project.Test plan
lake build Analysis.MeasureTheory.Section_1_2_2Made with Cursor