Formalize Theorem 1.3.20(i): L¹ approximation by simple functions#464
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teorth merged 3 commits intoteorth:mainfrom Mar 5, 2026
Merged
Formalize Theorem 1.3.20(i): L¹ approximation by simple functions#464teorth merged 3 commits intoteorth:mainfrom
teorth merged 3 commits intoteorth:mainfrom
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Section_1_3_5.lean: Replace sorry in both RealAbsolutelyIntegrable.approx_by_simple and ComplexAbsolutelyIntegrable.approx_by_simple. Add unsigned_approx_from_sup (simple-function approximation from LowerUnsignedLebesgueIntegral sSup) and UnsignedSimpleFunction.toRealSimple (unsigned → real conversion). Real case: pos/neg decomposition, EReal arithmetic, integral bounds. Complex case: real/imag decomposition using the real theorem, then triangle inequality for the norm bound.
Section_1_3_5.lean: Replace sorry in RealAbsolutelyIntegrable.approx_by_step (proof via approx_by_simple + RealSimpleFunction.approx_by_step_aux + PreL1.norm_sub_le_add). Add helpers: elementary_indicator_is_step, RealStepFunction.smul'/add'/simple, simple_abs_integrable_finite_measure, RealSimpleFunction.approx_by_step_aux. Two sorrys remain: PreL1.norm_smul_indicator_symmDiff_le, PreL1.norm_sub_le_add; one in approx_by_step_aux (k≠0 case). Complex case still sorry.
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Section_1_3_5.lean: Replace sorry in both RealAbsolutelyIntegrable.approx_by_simple and ComplexAbsolutelyIntegrable.approx_by_simple. Add unsigned_approx_from_sup (simple-function approximation from LowerUnsignedLebesgueIntegral sSup) and UnsignedSimpleFunction.toRealSimple (unsigned → real conversion). Real case: pos/neg decomposition, EReal arithmetic, integral bounds. Complex case: real/imag decomposition using the real theorem, then triangle inequality for the norm bound.