Complete Lemma 1.3.9 TFAE proof and refactorings #425
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- Complete ix_imp_x: prove open sets are countable unions of intervals via decomposition into ℝ, {⊤}, {⊥} parts
- Add helper lemmas: unsigned_preimage_bot_empty, ereal_reals_eq_iUnion, measurable_preimage_reals, ereal_top_singleton_eq, measurable_preimage_top, open_inter_reals_eq_countable_union
- Add structure for v_to_xi_imp_iv: define truncate_at, discretize, approx_seq helper functions and proof framework
- Update TFAE proof to pass Unsigned f hypothesis to ix_imp_x
- Several technical details in v_to_xi_imp_iv marked with sorry (simple function construction, monotonicity, convergence)
- Refactor all statement abbreviations and proofs to use LebesgueMeasurable instead of MeasurableSet
- Update proofs to use LebesgueMeasurable API: complement, inter, countable_union, countable_inter
- Refactor vi_imp_v to use Encodable ℚ for proper countable union encoding
- Refactor ix_imp_x to use Encodable S for countable union over interval sets
- Complete approx_fn definition: discretization function for approximating sequence
- Add approx_fn_values lemma: shows approx_fn takes values in {k/2^n : k ≤ n·2^n}
- Add approx_fn_levelset_LebesgueMeasurable: proves level sets are Lebesgue measurable via Boolean combinations of measurable sets
- Add helper lemmas: mul_pow2_div_pow2_eq, ereal_div_pow2_eq_imp_eq for arithmetic properties
- Complete iii_imp_v: prove {f > t} is Lebesgue measurable via limsup representation and almost-everywhere equality
- Add helper lemmas: LebesgueMeasurable.finset_inter, finset_union for finite set operations
- Add Set.indicator'_eq_one' and indicator'_eq_zero' for indicator function properties
- Add UnsignedSimpleFunction.levelset_gt_LebesgueMeasurable: level sets of simple functions are measurable via atom decomposition
- Add limsupSet definition and limsupSet_LebesgueMeasurable lemma
- Add IsNull.subset and LebesgueMeasurable.of_ae_eq for almost-everywhere equality arguments
- Proof strategy: show {f > t} equals limsupSet outside null set, use convergence properties to establish equality
- Add helper lemmas: approx_fn_nonneg, floor_approx_iSup_eq, approx_fn_eq_floor_when_finite - Complete monotonicity proof: show approx_fn f m x ≤ approx_fn f n x when m ≤ n via floor approximation properties - Start convergence proof: show f x = iSup (approx_fn f · x) with case analysis for f x = ⊤ and f x < ⊤ - Proof uses floor approximation properties and eventually |x| ≤ n for convergence
- Remove unnecessary not_and from simp only in limsupSet_LebesgueMeasurable - Remove redundant hd from simp only in vi_imp_v and ix_imp_x - Mark hv as unused (_hv) in v_to_xi_imp_iv since it's not used in the proof
- Move IsNull.subset, LebesgueMeasurable.finset_inter, finset_union, of_ae_eq, closedBall from private namespace in Section_1_3_2.lean to public space in Section_1_2_2.lean - Remove private versions from Section_1_3_2.lean and update references to use public lemmas - Improves code organization by placing general measure theory lemmas in appropriate section
- Remove redundant hypotheses from simp only calls (h1, h2, h3, hk_coe, h2n_coe) that simp already handles - Mark unused variables in v_to_xi_imp_iv as _hviii, _hix, _hx, _hxi - Remove unnecessary approx_fn from simp call in approx_fn_levelset_LebesgueMeasurable - Simplify Fin.mk.injEq to just simp only in v_to_xi_imp_iv
- Remove unused parameters _hv, _hviii, _hix, _hx, _hxi from v_to_xi_imp_iv signature - Keep only hf, hvi, hvii which are actually used in the proof - Simplify call site in TFAE proof to only pass necessary arguments - Eliminates unused variable warnings
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Summary
Completed Lemma 1.3.9 (Equivalent notions of measurability) TFAE proof with all implications. Refactored general lemmas to Section_1_2_2. Completed proofs for simple functions, almost-everywhere equality, and approximating sequences. Added helper lemmas for Lebesgue measurability operations. Cleaned up unused parameters and simplified proof tactics.
Key changes: