fix: Exercise 1.4.28(ii) needs μ(E) < ⊤, like part (i) - #644
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teorth merged 1 commit intoAug 11, 2026
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approx_sigma_finite drops part (i)'s hfin : mu univ < top in favor of sigma-finiteness of the algebra's generating sequence, but places no bound on mu E itself. Part (i) gets mu E < top for free from mu univ < top; part (ii) needs it stated explicitly. Counterexample: X = R, B = finite unions of integer-endpoint intervals, mu = Lebesgue. mu is sigma-finite over B via A n = [-n,n), but E = union of [2n, 2n+1) over n is in sigma(B) with mu E = infinity. Every F in B is determined by finitely many integer breakpoints, so E symmDiff F contains infinitely many unit intervals for any such F, and mu(E symmDiff F) = infinity for every choice - no F can get within any finite epsilon. Added the missing hEfin : mu E < top hypothesis.
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approx_sigma_finite drops part (i)'s
hfin : μ univ < ⊤in favor of σ-finiteness of the algebra's generating sequence, but places no bound onμ Eitself. Part (i) getsμ E < ⊤for free fromμ univ < ⊤; part (ii) needs it stated explicitly.Counterexample: X = ℝ, B = finite unions of integer-endpoint intervals, μ = Lebesgue. μ is σ-finite over B via A n = [-n,n), but E = ⋃ₙ [2n, 2n+1) is in σ(B) with μ E = ∞. Every F ∈ B is determined by finitely many integer breakpoints, so E Δ F contains infinitely many unit intervals for any such F, and μ(E Δ F) = ∞ for every choice — no F gets within any finite ε.
Added the missing
hEfin : μ E < ⊤hypothesis.Note: I wasn't able to complete a local
lake buildfor this one (network contention on my machine during the mathlib4 fetch) — this is a minimal, single-hypothesis, statement-only change with no proof body (sorry), hand-checked against the type signature and the already-compiling siblingapprox_finiteabove it. Happy to iterate if CI (build_book.yml) flags anything.