Complete proofs for absolutely integrable functions in Section 1.3.4 #433
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- Implement RealMeasurable.measurable_pos: prove positive part is unsigned measurable using continuity of max(·,0) composition - Implement RealMeasurable.measurable_neg: prove negative part is unsigned measurable using continuity of max(-·,0) composition - Implement RealAbsolutelyIntegrable.pos: prove positive part is absolutely integrable via monotonicity of integral - Implement RealAbsolutelyIntegrable.neg: prove negative part is absolutely integrable via monotonicity of integral All four proofs replace sorry placeholders with complete implementations.
- Implement RealSimpleFunction.absolutelyIntegrable_iff': bidirectional equivalence between simple function and general absolute integrability using constant sequences and simple integral equality - Implement ComplexSimpleFunction.absolutelyIntegrable_iff': same for complex functions - Implement RealSimpleFunction.AbsolutelyIntegrable.integ_eq: prove simple function integral equals general Lebesgue integral via pos/neg decomposition - Implement ComplexSimpleFunction.AbsolutelyIntegrable.integ_eq: prove complex simple function integral equals general integral via real/imaginary decomposition All four proofs replace sorry placeholders with complete implementations.
…ions - Implement RealAbsolutelyIntegrable.add: sum is absolutely integrable via triangle inequality and integral additivity - Implement ComplexAbsolutelyIntegrable.add: same for complex functions - Implement RealAbsolutelyIntegrable.sub: difference is absolutely integrable via norm_sub_le - Implement ComplexAbsolutelyIntegrable.sub: same for complex functions - Implement RealAbsolutelyIntegrable.smul: scalar multiple is absolutely integrable via norm_mul and integral homogeneity - Implement ComplexAbsolutelyIntegrable.smul: same for complex functions - Implement RealAbsolutelyIntegrable.of_neg: negation via smul with -1 - Implement ComplexAbsolutelyIntegrable.of_neg: same for complex functions - Implement ComplexAbsolutelyIntegrable.conj: conjugation preserves absolute integrability via norm_conj All nine proofs replace sorry placeholders with complete implementations using measurability, norm bounds, and integral properties.
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Summary
Positive/Negative Parts:
Simple Function Equivalences:
Closure Properties:
All proofs replace
sorryplaceholders with complete implementations using measurability, norm bounds, continuity, and integral properties.