Complete Theorems 1.3.20(ii)-(iii): L¹ approximation by step and continuous functions#469
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teorth merged 3 commits intoteorth:mainfrom Mar 11, 2026
Merged
Complete Theorems 1.3.20(ii)-(iii): L¹ approximation by step and continuous functions#469teorth merged 3 commits intoteorth:mainfrom
teorth merged 3 commits intoteorth:mainfrom
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… and complex) Section_1_3_5.lean: Replace all sorries in step-function approximation. Real case: fill RealSimpleFunction.approx_by_step_aux (elementary box approximation, symmDiff bounds), PreL1.norm_smul_indicator_symmDiff_le, PreL1.norm_sub_le_add. Complex case: fill ComplexAbsolutelyIntegrable.approx_by_step via real/imag decomposition. Add RealStepFunction.toComplexStep, ComplexStepFunction.add/smul, RealAbsolutelyIntegrable.zero_fun, PreL1.norm_zero_le, RealAbsolutelyIntegrable.smul_indicator. Replace simple_abs_integrable_finite_measure with smul_indicator.
…y supported (real) Section_1_3_5.lean: Replace sorry in RealAbsolutelyIntegrable.approx_by_continuous_compact. Add Box.scaled_indicator_approx_continuous (Urysohn-based: compact K ⊆ box, open U ⊇ box, φ=1 on K, φ=0 outside U). Add RealStepFunction.approx_by_continuous_compact_aux (approximate step function by sum of continuous compactly supported per box). Proof chain: approx_by_step → step, then per-box Urysohn approximation. Import Mathlib.Topology.UrysohnsLemma. Complex case still sorry.
…y supported (complex) Section_1_3_5.lean: Replace sorry in ComplexAbsolutelyIntegrable.approx_by_continuous_compact. Proof via real/imag decomposition: approx_by_step for Re/Im, then RealStepFunction.approx_by_continuous_compact_aux for each. Construct g = ↑g_re + I • ↑g_im. Norm bound via PreL1.norm_sub_le_add and triangle inequality.
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Summary
Theorem 1.3.20(ii) — L¹ approximation by step functions (real and complex): Fill RealSimpleFunction.approx_by_step_aux (elementary box, symmDiff bounds), PreL1.norm_smul_indicator_symmDiff_le, PreL1.norm_sub_le_add. Complex case via real/imag decomposition. Add RealStepFunction.toComplexStep, ComplexStepFunction.add/smul, RealAbsolutelyIntegrable.zero_fun, PreL1.norm_zero_le, smul_indicator.
Theorem 1.3.20(iii) — L¹ approximation by continuous compactly supported (real): Add Box.scaled_indicator_approx_continuous (Urysohn), RealStepFunction.approx_by_continuous_compact_aux. Proof: approx_by_step → step → per-box Urysohn. Import Mathlib.Topology.UrysohnsLemma.
Theorem 1.3.20(iii) — L¹ approximation by continuous compactly supported (complex): Replace sorry in ComplexAbsolutelyIntegrable.approx_by_continuous_compact. Proof via real/imag decomposition: approx_by_step for Re/Im, then RealStepFunction.approx_by_continuous_compact_aux for each. Construct g = ↑g_re + I • ↑g_im. Norm bound via PreL1.norm_sub_le_add and triangle inequality.