fix: require boundedness for Lebesgue=Jordan-inner equalities - #679
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Chessing234 wants to merge 3 commits into
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fix: require boundedness for Lebesgue=Jordan-inner equalities#679Chessing234 wants to merge 3 commits into
Chessing234 wants to merge 3 commits into
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Jordan_inner_measure is ℝ-valued via sSup, so unbounded sets get a junk value; Lebesgue_outer_measure is EReal and can be ⊤.
…r difference." This reverts commit 0481710.
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Summary
(hbound : Bornology.IsBounded E)toLebesgue_outer_measure.eq_Jordan_inner_of_boxesandLebesgue_outer_measure.of_open.hboundinto the local call ofeq_Jordan_inner_of_boxesinsideof_open. Leavesorrybodies unchanged.Why
Lebesgue_outer_measureisEReal(can be⊤).Jordan_inner_measureisℝ, defined as ansSupof elementary measures; for sets unbounded above that collection is unbounded inℝ, and the junksSupis0.So for an unbounded almost-disjoint union of unit cubes (or
Set.univind ≥ 1), the claimed equality is false: LHS⊤, RHS0via coercion. The book allows+∞Jordan inner measure; theℝencoding here does not, so the statements need a boundedness hypothesis.