Convolution

📁 Source: Mathlib/Analysis/Convolution.lean

Statistics

Metric	Count
Definitions «term_⋆[_,_]_», «term_⋆[_]_», «term_⋆_», ConvolutionExists, ConvolutionExistsAt, convolution, posConvolution	7
Theorems continuous_convolution_left_of_integrable, continuous_convolution_right_of_integrable, convolutionExistsAt, convolutionExistsAt', convolution_integrand_fst, continuous_convolution_left, continuous_convolution_right, convolution, convolutionExistsAt, convolutionExists_left, convolutionExists_left_of_continuous_right, convolutionExists_right, convolutionExists_right_of_continuous_left, convolution_integrand_bound_left, convolution_integrand_bound_right, convolution_integrand_bound_right_of_subset, convolution_integrand, convolution_integrand', convolution_integrand_snd, convolution_integrand_snd', convolution_integrand_swap_snd, convolution_integrand_swap_snd', add_distrib, distrib_add, add_distrib, distrib_add, integrable, integrable_swap, of_norm, of_norm', ae_convolution_exists, convolution_integrand, integrable_convolution, continuousOn_convolution_right_with_param, continuousOn_convolution_right_with_param_comp, convolutionExistsAt_flip, convolutionExistsAt_iff_integrable_swap, convolution_assoc, convolution_assoc', convolution_congr, convolution_def, convolution_eq_right', convolution_eq_swap, convolution_flip, convolution_integrand_bound_right_of_le_of_subset, convolution_lsmul, convolution_lsmul_swap, convolution_mono_right, convolution_mono_right_of_nonneg, convolution_mul, convolution_mul_swap, convolution_neg_of_neg_eq, convolution_precompR_apply, convolution_smul, convolution_tendsto_right, convolution_zero, dist_convolution_le, dist_convolution_le', integrable_posConvolution, integral_convolution, integral_posConvolution, posConvolution_eq_convolution_indicator, smul_convolution, support_convolution_subset, support_convolution_subset_swap, zero_convolution	66
Total	73

BddAbove

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
continuous_convolution_left_of_integrable 📖	mathematical	BddAbove Real Real.instLE Set.range Norm.norm NormedAddCommGroup.toNorm Continuous UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup MeasureTheory.Integrable SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup	MeasureTheory.convolution SubNegMonoid.toSub AddGroup.toSubNegMonoid AddCommGroup.toAddGroup	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul RingHomIsometric.ids MeasureTheory.convolution_flip ContinuousNeg.measurableNeg IsTopologicalAddGroup.toContinuousNeg ContinuousAdd.measurableAdd continuous_convolution_right_of_integrable
continuous_convolution_right_of_integrable 📖	mathematical	BddAbove Real Real.instLE Set.range Norm.norm NormedAddCommGroup.toNorm MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Continuous	MeasureTheory.convolution SubNegMonoid.toSub AddGroup.toSubNegMonoid	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul continuous_iff_continuousAt RingHomIsometric.ids MeasureTheory.Measure.instOuterMeasureClass Filter.univ_mem' ContinuousLinearMap.le_of_opNorm₂_le_of_le le_rfl le_ciSup MeasureTheory.continuousAt_of_dominated Filter.Eventually.of_forall MeasureTheory.AEStronglyMeasurable.convolution_integrand_snd' ContinuousAdd.measurableAdd ContinuousNeg.measurableNeg IsTopologicalAddGroup.toContinuousNeg MeasureTheory.Integrable.aestronglyMeasurable Continuous.aestronglyMeasurable BorelSpace.opensMeasurable PseudoEMetricSpace.pseudoMetrizableSpace MeasureTheory.Integrable.mul_const MeasureTheory.Integrable.const_mul MeasureTheory.Integrable.norm ContinuousAt.comp' ContinuousAt.clm_apply continuousAt_const continuousAt_id' Continuous.continuousAt ContinuousAt.sub IsTopologicalAddGroup.to_continuousSub
convolutionExistsAt 📖	mathematical	BddAbove Real Real.instLE Set.image Norm.norm NormedAddCommGroup.toNorm Set.preimage SubNegMonoid.toSub AddGroup.toSubNegMonoid AddCommGroup.toAddGroup MeasurableSet Set Set.instHasSubset Function.support NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup ContinuousLinearMap.addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup ContinuousLinearMap.module smulCommClass_self CommRing.toCommMonoid EuclideanDomain.toCommRing Field.toEuclideanDomain DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NormedSpace.toIsBoundedSMul MeasureTheory.IntegrableOn SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup MeasureTheory.AEStronglyMeasurable	MeasureTheory.ConvolutionExistsAt	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul convolutionExistsAt' MeasurableAdd₂.toMeasurableAdd MeasureTheory.AEStronglyMeasurable.mono_ac MeasureTheory.Measure.QuasiMeasurePreserving.absolutelyContinuous MeasureTheory.quasiMeasurePreserving_sub_left_of_right_invariant MeasureTheory.IsAddLeftInvariant.isAddRightInvariant MeasureTheory.AEStronglyMeasurable.mono_measure MeasureTheory.Measure.map_mono MeasureTheory.Measure.restrict_le_self Measurable.sub measurableDiv₂_of_add_neg measurable_const measurable_id'
convolutionExistsAt' 📖	mathematical	BddAbove Real Real.instLE Set.image Norm.norm NormedAddCommGroup.toNorm Set.preimage AddSemigroup.toAdd AddMonoid.toAddSemigroup SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid NegZeroClass.toNeg SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid AddGroup.toSubtractionMonoid MeasurableSet Set Set.instHasSubset Function.support NegZeroClass.toZero SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup ContinuousLinearMap.addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup ContinuousLinearMap.module smulCommClass_self CommRing.toCommMonoid EuclideanDomain.toCommRing Field.toEuclideanDomain DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NormedSpace.toIsBoundedSMul SubNegMonoid.toSub MeasureTheory.IntegrableOn SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup MeasureTheory.AEStronglyMeasurable MeasureTheory.Measure.map MeasureTheory.Measure.restrict	MeasureTheory.ConvolutionExistsAt	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul MeasureTheory.ConvolutionExistsAt.eq_1 MeasureTheory.integrableOn_iff_integrable_of_support_subset PseudoEMetricSpace.pseudoMetrizableSpace RingHomIsometric.ids MeasureTheory.Measure.instOuterMeasureClass Filter.univ_mem' Set.le_indicator ContinuousLinearMap.le_of_opNorm₂_le_of_le le_rfl le_ciSup_set Set.mem_preimage neg_sub sub_add_cancel Function.notMem_support norm_zero le_refl MeasureTheory.Integrable.mono' MeasureTheory.integrable_indicator_iff MeasureTheory.Integrable.integrableOn MeasureTheory.Integrable.mul_const MeasureTheory.Integrable.const_mul MeasureTheory.Integrable.norm MeasureTheory.AEStronglyMeasurable.convolution_integrand_snd' MeasureTheory.Integrable.aestronglyMeasurable

Continuous

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
convolution_integrand_fst 📖	mathematical	Continuous UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup	DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup ContinuousLinearMap.addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup ContinuousLinearMap.module smulCommClass_self CommRing.toCommMonoid EuclideanDomain.toCommRing Field.toEuclideanDomain DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup NormedSpace.toIsBoundedSMul SubNegMonoid.toSub AddGroup.toSubNegMonoid	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul comp₂ ContinuousLinearMap.continuous₂ continuous_const comp' sub continuous_id'

Convolution

Definitions

Name	Category	Theorems
«term_⋆[_,_]_» 📖	CompOp	—
«term_⋆[_]_» 📖	CompOp	—
«term_⋆_» 📖	CompOp	—

HasCompactSupport

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
continuous_convolution_left 📖	mathematical	HasCompactSupport NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup Continuous UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup MeasureTheory.LocallyIntegrable SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup	MeasureTheory.convolution SubNegMonoid.toSub AddGroup.toSubNegMonoid AddCommGroup.toAddGroup	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul RingHomIsometric.ids MeasureTheory.convolution_flip ContinuousNeg.measurableNeg IsTopologicalAddGroup.toContinuousNeg ContinuousAdd.measurableAdd continuous_convolution_right
continuous_convolution_right 📖	mathematical	HasCompactSupport NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup MeasureTheory.LocallyIntegrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Continuous	MeasureTheory.convolution SubNegMonoid.toSub AddGroup.toSubNegMonoid	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul continuousOn_univ Continuous.continuousOn Continuous.comp continuous_snd MeasureTheory.continuousOn_convolution_right_with_param_comp continuous_id image_eq_zero_of_notMem_tsupport
convolution 📖	mathematical	HasCompactSupport NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup	MeasureTheory.convolution SubNegMonoid.toSub AddGroup.toSubNegMonoid	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul IsCompact.of_isClosed_subset IsCompact.add isCompact isClosed_closure closure_minimal HasSubset.Subset.trans Set.instIsTransSubset MeasureTheory.support_convolution_subset_swap Set.add_subset_add subset_closure IsCompact.isClosed
convolutionExistsAt 📖	mathematical	HasCompactSupport NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup ContinuousLinearMap.addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup ContinuousLinearMap.module smulCommClass_self CommRing.toCommMonoid EuclideanDomain.toCommRing Field.toEuclideanDomain DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NormedSpace.toIsBoundedSMul SubNegMonoid.toSub AddGroup.toSubNegMonoid MeasureTheory.LocallyIntegrable SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Continuous	MeasureTheory.ConvolutionExistsAt	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul IsTopologicalAddGroup.toContinuousNeg BddAbove.convolutionExistsAt' ContinuousAdd.measurableAdd ContinuousNeg.measurableNeg IsCompact.bddAbove_image instClosedIciTopology HasSolidNorm.orderClosedTopology instHasSolidNormReal Real.instIsOrderedAddMonoid Homeomorph.isCompact_preimage Continuous.continuousOn Continuous.norm IsClosed.measurableSet BorelSpace.opensMeasurable isClosed_closure subset_closure MeasureTheory.LocallyIntegrable.integrableOn_isCompact PseudoEMetricSpace.pseudoMetrizableSpace ContinuousOn.aestronglyMeasurable_of_isCompact Continuous.comp Homeomorph.continuous isClosed_tsupport sub_eq_add_neg MeasureTheory.MeasurePreserving.aestronglyMeasurable_comp_iff Measurable.measurePreserving Continuous.measurable Homeomorph.measurableEmbedding
convolutionExists_left 📖	mathematical	HasCompactSupport NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup Continuous UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup MeasureTheory.LocallyIntegrable SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup	MeasureTheory.ConvolutionExists SubNegMonoid.toSub AddGroup.toSubNegMonoid AddCommGroup.toAddGroup	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul RingHomIsometric.ids MeasureTheory.convolutionExistsAt_flip ContinuousNeg.measurableNeg IsTopologicalAddGroup.toContinuousNeg ContinuousAdd.measurableAdd convolutionExists_right
convolutionExists_left_of_continuous_right 📖	mathematical	HasCompactSupport NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup MeasureTheory.LocallyIntegrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Continuous	MeasureTheory.ConvolutionExists SubNegMonoid.toSub AddGroup.toSubNegMonoid	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul convolutionExistsAt mono ContinuousLinearMap.map_zero₂
convolutionExists_right 📖	mathematical	HasCompactSupport NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup MeasureTheory.LocallyIntegrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Continuous	MeasureTheory.ConvolutionExists SubNegMonoid.toSub AddGroup.toSubNegMonoid	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul convolutionExistsAt mono comp_homeomorph ContinuousLinearMap.map_zero
convolutionExists_right_of_continuous_left 📖	mathematical	HasCompactSupport NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup Continuous UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup MeasureTheory.LocallyIntegrable SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup	MeasureTheory.ConvolutionExists SubNegMonoid.toSub AddGroup.toSubNegMonoid AddCommGroup.toAddGroup	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul RingHomIsometric.ids MeasureTheory.convolutionExistsAt_flip ContinuousNeg.measurableNeg IsTopologicalAddGroup.toContinuousNeg ContinuousAdd.measurableAdd convolutionExists_left_of_continuous_right
convolution_integrand_bound_left 📖	mathematical	HasCompactSupport NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup Continuous UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup Set Set.instMembership	Real Real.instLE Norm.norm NormedAddCommGroup.toNorm DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup ContinuousLinearMap.addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup ContinuousLinearMap.module smulCommClass_self CommRing.toCommMonoid EuclideanDomain.toCommRing Field.toEuclideanDomain DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NormedSpace.toIsBoundedSMul SubNegMonoid.toSub AddGroup.toSubNegMonoid Set.indicator Real.instZero Set.add AddSemigroup.toAdd AddMonoid.toAddSemigroup SubNegMonoid.toAddMonoid Set.neg NegZeroClass.toNeg AddGroup.toSubtractionMonoid tsupport Real.instMul ContinuousLinearMap.hasOpNorm ContinuousLinearMap.toSeminormedAddCommGroup RingHomIsometric.ids ContinuousLinearMap.toNormedSpace iSup Real.instSupSet	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul RingHomIsometric.ids ContinuousLinearMap.opNorm_flip mul_right_comm convolution_integrand_bound_right
convolution_integrand_bound_right 📖	mathematical	HasCompactSupport NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup Continuous UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup Set Set.instMembership	Real Real.instLE Norm.norm NormedAddCommGroup.toNorm DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup ContinuousLinearMap.addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup ContinuousLinearMap.module smulCommClass_self CommRing.toCommMonoid EuclideanDomain.toCommRing Field.toEuclideanDomain DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NormedSpace.toIsBoundedSMul SubNegMonoid.toSub AddGroup.toSubNegMonoid Set.indicator Real.instZero Set.add AddSemigroup.toAdd AddMonoid.toAddSemigroup SubNegMonoid.toAddMonoid Set.neg NegZeroClass.toNeg AddGroup.toSubtractionMonoid tsupport Real.instMul ContinuousLinearMap.hasOpNorm ContinuousLinearMap.toSeminormedAddCommGroup RingHomIsometric.ids ContinuousLinearMap.toNormedSpace iSup Real.instSupSet	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul convolution_integrand_bound_right_of_subset Set.Subset.rfl
convolution_integrand_bound_right_of_subset 📖	mathematical	HasCompactSupport NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup Continuous UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup Set Set.instMembership Set.instHasSubset Set.add AddSemigroup.toAdd AddMonoid.toAddSemigroup SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid Set.neg NegZeroClass.toNeg AddGroup.toSubtractionMonoid tsupport	Real Real.instLE Norm.norm NormedAddCommGroup.toNorm DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup ContinuousLinearMap.addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup ContinuousLinearMap.module smulCommClass_self CommRing.toCommMonoid EuclideanDomain.toCommRing Field.toEuclideanDomain DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NormedSpace.toIsBoundedSMul SubNegMonoid.toSub Set.indicator Real.instZero Real.instMul ContinuousLinearMap.hasOpNorm ContinuousLinearMap.toSeminormedAddCommGroup RingHomIsometric.ids ContinuousLinearMap.toNormedSpace iSup Real.instSupSet	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul MeasureTheory.convolution_integrand_bound_right_of_le_of_subset le_ciSup Continuous.bddAbove_range_of_hasCompactSupport instClosedIciTopology HasSolidNorm.orderClosedTopology instHasSolidNormReal Real.instIsOrderedAddMonoid Continuous.norm norm

MeasureTheory

Definitions

Name	Category	Theorems
ConvolutionExists 📖	MathDef	4 math math: HasCompactSupport.convolutionExists_left_of_continuous_right, HasCompactSupport.convolutionExists_right, HasCompactSupport.convolutionExists_right_of_continuous_left, HasCompactSupport.convolutionExists_left
ConvolutionExistsAt 📖	MathDef	6 math math: BddAbove.convolutionExistsAt', convolutionExistsAt_flip, Integrable.ae_convolution_exists, convolutionExistsAt_iff_integrable_swap, HasCompactSupport.convolutionExistsAt, BddAbove.convolutionExistsAt
convolution 📖	CompOp	61 math math: convolution_flip, ContDiffBump.ae_convolution_tendsto_right_of_locallyIntegrable, convolution_eq_right', BddAbove.continuous_convolution_right_of_integrable, contDiffOn_convolution_left_with_param, convolution_mul, BddAbove.continuous_convolution_left_of_integrable, smul_convolution, convolution_tendsto_right, convolution_smul, convolution_lsmul, convolution_precompR_apply, HasCompactSupport.hasDerivAt_convolution_right, contDiffOn_convolution_right_with_param, contDiffOn_convolution_left_with_param_comp, posConvolution_eq_convolution_indicator, convolution_def, ContDiffBump.convolution_tendsto_right, Real.fourier_mul_convolution_eq, ContDiffBump.normed_convolution_eq_right, hasFDerivAt_convolution_right_with_param, HasCompactSupport.hasFDerivAt_convolution_right, SchwartzMap.convolution_apply, continuousOn_convolution_right_with_param_comp, dist_convolution_le, HasCompactSupport.convolution, convolution_mono_right_of_nonneg, contDiffOn_convolution_right_with_param_comp, Real.fourier_bilin_convolution_eq, convolution_assoc', dist_convolution_le', HasCompactSupport.continuous_convolution_left, ConvolutionExistsAt.add_distrib, HasCompactSupport.contDiff_convolution_left, convolution_zero, convolution_lsmul_swap, convolution_neg_of_neg_eq, Real.fourier_bilin_convolution_eq_integral, ContDiffBump.convolution_eq_right, continuousOn_convolution_right_with_param, ContDiffBump.dist_normed_convolution_le, HasCompactSupport.hasDerivAt_convolution_left, Integrable.integrable_convolution, ConvolutionExistsAt.distrib_add, convolution_mul_swap, convolution_mono_right, ConvolutionExists.distrib_add, support_convolution_subset_swap, SchwartzMap.fourier_convolution_apply, convolution_eq_swap, HasCompactSupport.contDiff_convolution_right, HasCompactSupport.continuous_convolution_right, support_convolution_subset, zero_convolution, contDiffOn_convolution_right_with_param_aux, ContDiffBump.convolution_tendsto_right_of_continuous, convolution_congr, ConvolutionExists.add_distrib, Real.fourier_smul_convolution_eq, integral_convolution, HasCompactSupport.hasFDerivAt_convolution_left
posConvolution 📖	CompOp	2 math math: posConvolution_eq_convolution_indicator, integrable_posConvolution

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
continuousOn_convolution_right_with_param 📖	mathematical	IsCompact NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup LocallyIntegrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup ContinuousOn instTopologicalSpaceProd Function.HasUncurry.uncurry Function.hasUncurryInduction Function.hasUncurryBase SProd.sprod Set Set.instSProd Set.univ	convolution SubNegMonoid.toSub AddGroup.toSubNegMonoid	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul ContinuousOn.congr continuousOn_const integral_eq_zero_of_ae Filter.Eventually.of_forall Measure.instOuterMeasureClass map_zero AddMonoidHomClass.toZeroHomClass DistribMulActionSemiHomClass.toAddMonoidHomClass SemilinearMapClass.distribMulActionSemiHomClass ContinuousSemilinearMapClass.toSemilinearMapClass ContinuousLinearMap.continuousSemilinearMapClass Mathlib.Tactic.Push.not_forall_eq Function.support_subset_iff' ContinuousOn.comp_continuous Continuous.prodMk_right eq_zero_or_locallyCompactSpace_of_support_subset_isCompact_of_addGroup T5Space.toT1Space T6Space.toT5Space instT6SpaceOfMetrizableSpace EMetricSpace.metrizableSpace WeaklyLocallyCompactSpace.exists_compact_mem_nhds instWeaklyLocallyCompactSpaceOfLocallyCompactSpace IsCompact.vadd_set IsTopologicalAddTorsor.toContinuousVAdd instIsTopologicalAddTorsor IsCompact.neg IsTopologicalAddGroup.toContinuousNeg ContinuousOn.comp ContinuousOn.prodMk ContinuousOn.fst continuousOn_id' ContinuousOn.sub IsTopologicalAddGroup.to_continuousSub ContinuousOn.snd continuousOn_integral_bilinear_of_locally_integrable_of_compact_support BorelSpace.opensMeasurable Mathlib.Tactic.Contrapose.contrapose₄ neg_sub sub_add_cancel LocallyIntegrable.integrableOn_isCompact PseudoEMetricSpace.pseudoMetrizableSpace ContinuousWithinAt.mono_of_mem_nhdsWithin mem_of_mem_nhds mem_nhdsWithin_prod_iff self_mem_nhdsWithin nhdsWithin_le_nhds Set.Subset.rfl
continuousOn_convolution_right_with_param_comp 📖	mathematical	ContinuousOn IsCompact NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup LocallyIntegrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup instTopologicalSpaceProd Function.HasUncurry.uncurry Function.hasUncurryInduction Function.hasUncurryBase SProd.sprod Set Set.instSProd Set.univ	convolution SubNegMonoid.toSub AddGroup.toSubNegMonoid	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul ContinuousOn.comp continuousOn_convolution_right_with_param ContinuousOn.prodMk continuousOn_id
convolutionExistsAt_flip 📖	mathematical	—	ConvolutionExistsAt SubNegMonoid.toSub AddGroup.toSubNegMonoid AddCommGroup.toAddGroup ContinuousLinearMap.flip NormedAddCommGroup.toSeminormedAddCommGroup RingHom.id Semiring.toNonAssocSemiring DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHomIsometric.ids SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul RingHomIsometric.ids integrable_comp_sub_left sub_sub_cancel
convolutionExistsAt_iff_integrable_swap 📖	mathematical	—	ConvolutionExistsAt SubNegMonoid.toSub AddGroup.toSubNegMonoid AddCommGroup.toAddGroup Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup ContinuousLinearMap.addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup ContinuousLinearMap.module smulCommClass_self CommRing.toCommMonoid EuclideanDomain.toCommRing Field.toEuclideanDomain DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup NormedSpace.toIsBoundedSMul	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul RingHomIsometric.ids convolutionExistsAt_flip
convolution_assoc 📖	—	DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField DenselyNormedField.toNormedField RCLike.toDenselyNormedField RingHom.id Semiring.toNonAssocSemiring UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup ContinuousLinearMap.addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup ContinuousLinearMap.module smulCommClass_self CommRing.toCommMonoid EuclideanDomain.toCommRing Field.toEuclideanDomain DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup NormedSpace.toIsBoundedSMul AEStronglyMeasurable Filter.Eventually ConvolutionExistsAt DenselyNormedField.toNontriviallyNormedField SubNegMonoid.toSub AddGroup.toSubNegMonoid ae Measure Measure.instFunLike Measure.instOuterMeasureClass Real Real.normedAddCommGroup Real.denselyNormedField InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal Norm.norm NormedAddCommGroup.toNorm ContinuousLinearMap.mul NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing Real.normedCommRing NonUnitalSeminormedRing.toSeminormedAddCommGroup IsScalarTower.right Real.instCommSemiring CommSemiring.toSemiring Algebra.id Algebra.to_smulCommClass convolution	—	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul Measure.instOuterMeasureClass IsScalarTower.right Algebra.to_smulCommClass convolution_assoc' Filter.Eventually.mono ConvolutionExistsAt.of_norm ContinuousLinearMap.aestronglyMeasurable_comp₂ AEStronglyMeasurable.comp_snd AEStronglyMeasurable.comp_fst AEStronglyMeasurable.comp_measurable AEStronglyMeasurable.mono_ac Measure.QuasiMeasurePreserving.absolutelyContinuous QuasiMeasurePreserving.prod_of_left Measurable.sub measurableDiv₂_of_add_neg Measurable.const_sub measurableSub_of_add_neg MeasurableAdd₂.toMeasurableAdd Measurable.snd measurable_id' Measurable.fst Filter.Eventually.of_forall quasiMeasurePreserving_sub_left_of_right_invariant AEStronglyMeasurable.integral_prod_right' AEStronglyMeasurable.norm AEStronglyMeasurable.prod_swap MeasurePreserving.map_eq measurePreserving_sub_prod integrable_prod_iff' Measure.QuasiMeasurePreserving.ae ContinuousLinearMap.integrable_comp RingHomIsometric.ids Integrable.mono' Integrable.const_mul mul_mul_mul_comm Real.norm_of_nonneg integral_nonneg Real.instIsOrderedAddMonoid IsStrictOrderedModule.toIsOrderedModule IsStrictOrderedRing.toIsStrictOrderedModule Real.instIsStrictOrderedRing instClosedIciTopology HasSolidNorm.orderClosedTopology instHasSolidNormReal norm_nonneg integral_mono_of_nonneg mul_assoc ContinuousLinearMap.le_of_opNorm₂_le_of_le le_rfl sub_sub_sub_cancel_right Integrable.comp_measurable Measurable.prodMk MeasurableSub₂.measurable_sub measurable_snd
convolution_assoc' 📖	mathematical	DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField DenselyNormedField.toNormedField RCLike.toDenselyNormedField RingHom.id Semiring.toNonAssocSemiring UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup ContinuousLinearMap.addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup ContinuousLinearMap.module smulCommClass_self CommRing.toCommMonoid EuclideanDomain.toCommRing Field.toEuclideanDomain DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup NormedSpace.toIsBoundedSMul Filter.Eventually ConvolutionExistsAt DenselyNormedField.toNontriviallyNormedField SubNegMonoid.toSub AddGroup.toSubNegMonoid ae Measure Measure.instFunLike Measure.instOuterMeasureClass Integrable SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Prod.instMeasurableSpace Measure.prod	convolution	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul Measure.instOuterMeasureClass integral_congr_ae Filter.Eventually.mono RingHomIsometric.ids ContinuousLinearMap.integral_comp_comm integral_integral_swap integral_sub_right_eq_self MeasurableAdd₂.toMeasurableAdd sub_sub_sub_cancel_right Measure.QuasiMeasurePreserving.ae quasiMeasurePreserving_sub_left_of_right_invariant
convolution_congr 📖	mathematical	Filter.EventuallyEq ae Measure Measure.instFunLike Measure.instOuterMeasureClass	convolution SubNegMonoid.toSub AddGroup.toSubNegMonoid	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul Measure.instOuterMeasureClass integral_congr_ae Filter.EventuallyEq.fun_comp Filter.EventuallyEq.prodMk Filter.EventuallyEq.comp_tendsto Measure.QuasiMeasurePreserving.tendsto_ae quasiMeasurePreserving_sub_left_of_right_invariant
convolution_def 📖	mathematical	—	convolution integral DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup ContinuousLinearMap.addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup ContinuousLinearMap.module smulCommClass_self CommRing.toCommMonoid EuclideanDomain.toCommRing Field.toEuclideanDomain DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup NormedSpace.toIsBoundedSMul	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul
convolution_eq_right' 📖	mathematical	Set Set.instHasSubset Function.support NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup Metric.ball SeminormedAddCommGroup.toPseudoMetricSpace SeminormedAddCommGroup.toAddCommGroup	convolution SubNegMonoid.toSub AddGroup.toSubNegMonoid SeminormedAddGroup.toAddGroup SeminormedAddCommGroup.toSeminormedAddGroup integral DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace NormedAddCommGroup.toSeminormedAddCommGroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toIsTopologicalAddGroup ContinuousLinearMap.addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup ContinuousLinearMap.module smulCommClass_self CommRing.toCommMonoid EuclideanDomain.toCommRing Field.toEuclideanDomain DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NormedSpace.toIsBoundedSMul	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul sub_eq_add_neg norm_neg add_mem_ball_iff_norm mem_ball_zero_iff Function.notMem_support ContinuousLinearMap.map_zero₂
convolution_eq_swap 📖	mathematical	—	convolution SubNegMonoid.toSub AddGroup.toSubNegMonoid AddCommGroup.toAddGroup integral DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup ContinuousLinearMap.addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup ContinuousLinearMap.module smulCommClass_self CommRing.toCommMonoid EuclideanDomain.toCommRing Field.toEuclideanDomain DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup NormedSpace.toIsBoundedSMul	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul RingHomIsometric.ids convolution_flip
convolution_flip 📖	mathematical	—	convolution SubNegMonoid.toSub AddGroup.toSubNegMonoid AddCommGroup.toAddGroup ContinuousLinearMap.flip NormedAddCommGroup.toSeminormedAddCommGroup RingHom.id Semiring.toNonAssocSemiring DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHomIsometric.ids SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul RingHomIsometric.ids integral_sub_left_eq_self sub_sub_self
convolution_integrand_bound_right_of_le_of_subset 📖	mathematical	Real Real.instLE Norm.norm NormedAddCommGroup.toNorm Set Set.instMembership Set.instHasSubset Set.add AddSemigroup.toAdd AddMonoid.toAddSemigroup SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid Set.neg NegZeroClass.toNeg SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid AddGroup.toSubtractionMonoid tsupport NegZeroClass.toZero SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup	DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup ContinuousLinearMap.addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup ContinuousLinearMap.module smulCommClass_self CommRing.toCommMonoid EuclideanDomain.toCommRing Field.toEuclideanDomain DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NormedSpace.toIsBoundedSMul SubNegMonoid.toSub Set.indicator Real.instZero Real.instMul ContinuousLinearMap.hasOpNorm ContinuousLinearMap.toSeminormedAddCommGroup RingHomIsometric.ids ContinuousLinearMap.toNormedSpace	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul Set.le_indicator RingHomIsometric.ids ContinuousLinearMap.le_of_opNorm₂_le_of_le le_rfl Set.neg_mem_neg subset_closure neg_sub sub_add_cancel Function.notMem_support ContinuousLinearMap.map_zero norm_zero
convolution_lsmul 📖	mathematical	—	convolution NonUnitalNormedRing.toNormedAddCommGroup NonUnitalNormedCommRing.toNonUnitalNormedRing NormedCommRing.toNonUnitalNormedCommRing NormedField.toNormedCommRing NontriviallyNormedField.toNormedField NormedField.toNormedSpace ContinuousLinearMap.lsmul NormedAddCommGroup.toSeminormedAddCommGroup SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedAlgebra.id NormedSpace.toModule NormedSpace.toIsBoundedSMul IsScalarTower.left MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField DistribMulAction.toMulAction AddCommMonoid.toAddMonoid ESeminormedAddCommMonoid.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid Module.toDistribMulAction integral SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass ESeminormedAddMonoid.toAddMonoid ESeminormedAddCommMonoid.toESeminormedAddMonoid DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul	—	NormedSpace.toIsBoundedSMul IsScalarTower.left
convolution_lsmul_swap 📖	mathematical	—	convolution NonUnitalNormedRing.toNormedAddCommGroup NonUnitalNormedCommRing.toNonUnitalNormedRing NormedCommRing.toNonUnitalNormedCommRing NormedField.toNormedCommRing NontriviallyNormedField.toNormedField NormedField.toNormedSpace SubNegMonoid.toSub AddGroup.toSubNegMonoid AddCommGroup.toAddGroup ContinuousLinearMap.lsmul NormedAddCommGroup.toSeminormedAddCommGroup SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedAlgebra.id NormedSpace.toModule NormedSpace.toIsBoundedSMul IsScalarTower.left MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField DistribMulAction.toMulAction AddCommMonoid.toAddMonoid ESeminormedAddCommMonoid.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid Module.toDistribMulAction integral SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass ESeminormedAddMonoid.toAddMonoid ESeminormedAddCommMonoid.toESeminormedAddMonoid DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul	—	convolution_eq_swap NormedSpace.toIsBoundedSMul IsScalarTower.left
convolution_mono_right 📖	mathematical	ConvolutionExistsAt Real Real.normedAddCommGroup DenselyNormedField.toNontriviallyNormedField Real.denselyNormedField InnerProductSpace.toNormedSpace Real.instRCLike NormedAddCommGroup.toSeminormedAddCommGroup RCLike.toInnerProductSpaceReal SubNegMonoid.toSub AddGroup.toSubNegMonoid ContinuousLinearMap.lsmul NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing SeminormedCommRing.toSeminormedRing RCLike.toNormedAlgebra NormedSpace.toModule Real.normedField Real.isBoundedSMul IsScalarTower.right Real.instCommSemiring CommSemiring.toSemiring Algebra.id Real.instLE Real.instZero	convolution	—	Real.isBoundedSMul IsScalarTower.right integral_mono Real.instIsOrderedAddMonoid IsStrictOrderedModule.toIsOrderedModule IsStrictOrderedRing.toIsStrictOrderedModule Real.instIsStrictOrderedRing instClosedIciTopology HasSolidNorm.orderClosedTopology instHasSolidNormReal mul_le_mul_of_nonneg_left IsOrderedRing.toPosMulMono Real.instIsOrderedRing
convolution_mono_right_of_nonneg 📖	mathematical	ConvolutionExistsAt Real Real.normedAddCommGroup DenselyNormedField.toNontriviallyNormedField Real.denselyNormedField InnerProductSpace.toNormedSpace Real.instRCLike NormedAddCommGroup.toSeminormedAddCommGroup RCLike.toInnerProductSpaceReal SubNegMonoid.toSub AddGroup.toSubNegMonoid ContinuousLinearMap.lsmul NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing SeminormedCommRing.toSeminormedRing RCLike.toNormedAlgebra NormedSpace.toModule Real.normedField Real.isBoundedSMul IsScalarTower.right Real.instCommSemiring CommSemiring.toSemiring Algebra.id Real.instLE Real.instZero	convolution	—	Real.isBoundedSMul IsScalarTower.right convolution_mono_right integral_undef integral_nonneg Real.instIsOrderedAddMonoid IsStrictOrderedModule.toIsOrderedModule IsStrictOrderedRing.toIsStrictOrderedModule Real.instIsStrictOrderedRing instClosedIciTopology HasSolidNorm.orderClosedTopology instHasSolidNormReal mul_nonneg IsOrderedRing.toPosMulMono Real.instIsOrderedRing
convolution_mul 📖	mathematical	—	convolution NonUnitalNormedRing.toNormedAddCommGroup NonUnitalNormedCommRing.toNonUnitalNormedRing NormedCommRing.toNonUnitalNormedCommRing NormedField.toNormedCommRing NontriviallyNormedField.toNormedField NormedField.toNormedSpace ContinuousLinearMap.mul NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing IsScalarTower.right Semifield.toCommSemiring Field.toSemifield NormedField.toField CommSemiring.toSemiring Algebra.id Algebra.to_smulCommClass integral Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing	—	IsScalarTower.right Algebra.to_smulCommClass
convolution_mul_swap 📖	mathematical	—	convolution NonUnitalNormedRing.toNormedAddCommGroup NonUnitalNormedCommRing.toNonUnitalNormedRing NormedCommRing.toNonUnitalNormedCommRing NormedField.toNormedCommRing NontriviallyNormedField.toNormedField NormedField.toNormedSpace SubNegMonoid.toSub AddGroup.toSubNegMonoid AddCommGroup.toAddGroup ContinuousLinearMap.mul NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing IsScalarTower.right Semifield.toCommSemiring Field.toSemifield NormedField.toField CommSemiring.toSemiring Algebra.id Algebra.to_smulCommClass integral Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing	—	convolution_eq_swap IsScalarTower.right Algebra.to_smulCommClass
convolution_neg_of_neg_eq 📖	mathematical	Filter.Eventually NegZeroClass.toNeg SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid ae Measure Measure.instFunLike Measure.instOuterMeasureClass	convolution SubNegMonoid.toSub AddGroup.toSubNegMonoid AddCommGroup.toAddGroup	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul Measure.instOuterMeasureClass integral_congr_ae Filter.mp_mem eventually_add_left_iff Filter.univ_mem' neg_add' integral_neg_eq_self neg_neg
convolution_precompR_apply 📖	mathematical	LocallyIntegrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup HasCompactSupport ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField DenselyNormedField.toNormedField RCLike.toDenselyNormedField RingHom.id Semiring.toNonAssocSemiring ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule ContinuousLinearMap.zero Continuous ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup	DFunLike.coe NontriviallyNormedField.toNormedField DenselyNormedField.toNontriviallyNormedField AddCommGroup.toAddCommMonoid ContinuousLinearMap.funLike convolution ContinuousLinearMap.toNormedAddCommGroup RingHomIsometric.ids SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing ContinuousLinearMap.toNormedSpace smulCommClass_self CommRing.toCommMonoid EuclideanDomain.toCommRing Field.toEuclideanDomain DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing Real Real.normedField IsScalarTower.to_smulCommClass' Real.instCommSemiring NormedAlgebra.toAlgebra RCLike.toNormedAlgebra Real.isScalarTower NormedAddCommGroup.toAddCommGroup IsBoundedSMul.continuousSMul Real.pseudoMetricSpace Real.instZero NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddCommGroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul Real.instMonoid Real.semiring NormedSpace.toIsBoundedSMul TopologicalSpace.t2Space_of_metrizableSpace EMetricSpace.metrizableSpace MetricSpace.toEMetricSpace NormedAddCommGroup.toMetricSpace SeminormedRing.toPseudoMetricSpace NormedRing.toRing NormedCommRing.toNormedRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing Algebra.toSMul NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal MonoidWithZero.toMonoid Semiring.toMonoidWithZero SubNegMonoid.toSub NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup ContinuousLinearMap.precompR	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul RingHomIsometric.ids HasCompactSupport.convolutionExists_right IsScalarTower.to_smulCommClass' Real.isScalarTower IsBoundedSMul.continuousSMul TopologicalSpace.t2Space_of_metrizableSpace EMetricSpace.metrizableSpace ContinuousLinearMap.integral_apply
convolution_smul 📖	mathematical	—	convolution SubNegMonoid.toSub AddGroup.toSubNegMonoid Function.hasSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass ESeminormedAddMonoid.toAddMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ESeminormedAddCommMonoid.toESeminormedAddMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField Module.toDistribMulAction ESeminormedAddCommMonoid.toAddCommMonoid NormedSpace.toModule	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul ContinuousLinearMap.map_smul NormedSpace.toNormSMulClass
convolution_tendsto_right 📖	mathematical	Filter.Eventually Real integral Real.normedAddCommGroup InnerProductSpace.toNormedSpace Real.instRCLike NormedAddCommGroup.toSeminormedAddCommGroup RCLike.toInnerProductSpaceReal Real.instOne Filter.Tendsto Set Function.support Real.instZero Filter.smallSets nhds UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid SeminormedAddCommGroup.toAddCommGroup AEStronglyMeasurable SProd.sprod Filter Filter.instSProd	convolution DenselyNormedField.toNontriviallyNormedField Real.denselyNormedField SubNegMonoid.toSub AddGroup.toSubNegMonoid SeminormedAddGroup.toAddGroup SeminormedAddCommGroup.toSeminormedAddGroup ContinuousLinearMap.lsmul SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing Real.normedCommRing RCLike.toNormedAlgebra NormedSpace.toModule Real.normedField NormedSpace.toIsBoundedSMul IsScalarTower.left MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField DistribMulAction.toMulAction AddCommMonoid.toAddMonoid ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid Module.toDistribMulAction	—	NormedSpace.toIsBoundedSMul IsScalarTower.left Metric.tendsto_nhds Nat.instAtLeastTwoHAddOfNat div_pos PosMulReflectLE.toPosMulReflectLT PosMulStrictMono.toPosMulReflectLE IsStrictOrderedRing.toPosMulStrictMono Real.instIsStrictOrderedRing Real.instIsOrderedRing Real.instNontrivial Filter.Tendsto.eventually Metric.ball_mem_nhds half_pos Filter.mp_mem Filter.univ_mem' LT.lt.trans half_lt_self LE.le.trans dist_triangle_right add_le_add IsOrderedAddMonoid.toAddLeftMono Real.instIsOrderedAddMonoid covariant_swap_add_of_covariant_add LT.lt.le LT.lt.trans_eq LE.le.trans_lt dist_triangle add_lt_add IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono AddLeftCancelSemigroup.toIsLeftCancelAdd IsRightCancelAdd.addRightStrictMono_of_addRightMono AddRightCancelSemigroup.toIsRightCancelAdd Membership.mem.out add_halves CharZero.NeZero.two RCLike.charZero_rclike dist_convolution_le add_pos add_lt_add_of_le_of_lt Mathlib.Tactic.Ring.of_eq Mathlib.Tactic.Ring.add_congr Mathlib.Tactic.Ring.div_congr Mathlib.Tactic.Ring.atom_pf Mathlib.Tactic.Ring.cast_pos Mathlib.Meta.NormNum.isNat_ofNat Mathlib.Meta.NormNum.instAtLeastTwo Mathlib.Tactic.Ring.div_pf Mathlib.Tactic.Ring.inv_single Mathlib.Meta.NormNum.IsNNRat.to_raw_eq Mathlib.Meta.NormNum.isNNRat_inv_pos Mathlib.Meta.NormNum.IsNat.to_isNNRat Mathlib.Meta.NormNum.IsNat.of_raw Mathlib.Tactic.Ring.add_mul Mathlib.Tactic.Ring.mul_add Mathlib.Tactic.Ring.mul_pf_left Mathlib.Tactic.Ring.one_mul Mathlib.Tactic.Ring.mul_zero Mathlib.Tactic.Ring.add_pf_add_zero Mathlib.Tactic.Ring.zero_mul Mathlib.Tactic.Ring.add_pf_add_overlap Mathlib.Tactic.Ring.add_overlap_pf Mathlib.Meta.NormNum.isNNRat_add Mathlib.Meta.NormNum.IsNNRat.of_raw Mathlib.Meta.NormNum.IsNNRat.den_nz Mathlib.Tactic.Ring.add_pf_zero_add Mathlib.Meta.NormNum.IsNat.to_raw_eq Mathlib.Meta.NormNum.IsNNRat.to_isNat
convolution_zero 📖	mathematical	—	convolution SubNegMonoid.toSub AddGroup.toSubNegMonoid Pi.instZero NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul ContinuousLinearMap.map_zero integral_zero
dist_convolution_le 📖	mathematical	Real Real.instLE Real.instZero Set Set.instHasSubset Function.support Metric.ball SeminormedAddCommGroup.toPseudoMetricSpace NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid SeminormedAddCommGroup.toAddCommGroup integral Real.normedAddCommGroup InnerProductSpace.toNormedSpace Real.instRCLike NormedAddCommGroup.toSeminormedAddCommGroup RCLike.toInnerProductSpaceReal Real.instOne AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Dist.dist PseudoMetricSpace.toDist	convolution DenselyNormedField.toNontriviallyNormedField Real.denselyNormedField SubNegMonoid.toSub AddGroup.toSubNegMonoid SeminormedAddGroup.toAddGroup SeminormedAddCommGroup.toSeminormedAddGroup ContinuousLinearMap.lsmul SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing Real.normedCommRing RCLike.toNormedAlgebra NormedSpace.toModule Real.normedField NormedSpace.toIsBoundedSMul IsScalarTower.left MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField DistribMulAction.toMulAction AddCommMonoid.toAddMonoid ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid Module.toDistribMulAction	—	integrable_of_integral_eq_one NormedSpace.toIsBoundedSMul IsScalarTower.left SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul integral_smul_const one_smul LE.le.trans RingHomIsometric.ids dist_convolution_le' Real.norm_of_nonneg mul_one LE.le.trans_eq mul_le_mul_of_nonneg_right IsOrderedRing.toMulPosMono Real.instIsOrderedRing ContinuousLinearMap.opNorm_lsmul_le one_mul
dist_convolution_le' 📖	mathematical	Real Real.instLE Real.instZero Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Set Set.instHasSubset Function.support NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup Metric.ball SeminormedAddCommGroup.toAddCommGroup AEStronglyMeasurable Dist.dist PseudoMetricSpace.toDist	convolution SubNegMonoid.toSub AddGroup.toSubNegMonoid SeminormedAddGroup.toAddGroup integral DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toIsTopologicalAddGroup ContinuousLinearMap.addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup ContinuousLinearMap.module smulCommClass_self CommRing.toCommMonoid EuclideanDomain.toCommRing Field.toEuclideanDomain DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NormedSpace.toIsBoundedSMul Real.instMul Norm.norm ContinuousLinearMap.hasOpNorm ContinuousLinearMap.toSeminormedAddCommGroup RingHomIsometric.ids ContinuousLinearMap.toNormedSpace Real.normedAddCommGroup InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal NormedAddCommGroup.toNorm	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul BddAbove.convolutionExistsAt ContinuousNeg.measurableNeg IsTopologicalAddGroup.toContinuousNeg ContinuousAdd.measurableMul₂ bddAbove_def norm_le_norm_add_const_of_dist_le mem_ball_iff_norm norm_sub_rev mem_ball_zero_iff IsOpen.measurableSet BorelSpace.opensMeasurable Metric.isOpen_ball Set.Subset.trans ContinuousLinearMap.map_zero₂ Integrable.integrableOn LE.le.trans ContinuousLinearMap.dist_le_opNorm RingHomIsometric.ids mul_le_mul_of_nonneg_left IsOrderedRing.toPosMulMono Real.instIsOrderedRing sub_eq_add_neg norm_neg add_mem_ball_iff_norm norm_nonneg Function.notMem_support ContinuousLinearMap.map_zero norm_zero MulZeroClass.zero_mul dist_self le_refl dist_eq_norm integral_sub ConvolutionExistsAt.integrable ContinuousLinearMap.integrable_comp norm_integral_le_of_norm_le Integrable.mul_const Integrable.norm Filter.Eventually.of_forall Measure.instOuterMeasureClass integral_mul_const mul_le_mul_of_nonneg_right IsOrderedRing.toMulPosMono ContinuousLinearMap.le_opNorm LE.le.trans_eq integral_mono Real.instIsOrderedAddMonoid IsStrictOrderedModule.toIsOrderedModule IsStrictOrderedRing.toIsStrictOrderedModule Real.instIsStrictOrderedRing instClosedIciTopology HasSolidNorm.orderClosedTopology instHasSolidNormReal Integrable.const_mul integral_const_mul
integrable_posConvolution 📖	mathematical	IntegrableOn Real Real.measurableSpace UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Set.Ioi Real.instPreorder Real.instZero	Integrable posConvolution	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul posConvolution_eq_convolution_indicator Integrable.integral_prod_left Integrable.convolution_integrand ContinuousAdd.measurableMul₂ Real.borelSpace instSecondCountableTopologyReal IsTopologicalSemiring.toContinuousAdd IsTopologicalRing.toIsTopologicalSemiring instIsTopologicalRingReal ContinuousNeg.measurableNeg IsTopologicalRing.toContinuousNeg integrable_indicator_iff measurableSet_Ioi BorelSpace.opensMeasurable instClosedIicTopology HasSolidNorm.orderClosedTopology instHasSolidNormReal Real.instIsOrderedAddMonoid
integral_convolution 📖	mathematical	Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup	integral convolution DenselyNormedField.toNontriviallyNormedField RCLike.toDenselyNormedField SubNegMonoid.toSub AddGroup.toSubNegMonoid DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField DenselyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup ContinuousLinearMap.addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup ContinuousLinearMap.module smulCommClass_self CommRing.toCommMonoid EuclideanDomain.toCommRing Field.toEuclideanDomain DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup NormedSpace.toIsBoundedSMul	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul integral_integral_swap Integrable.convolution_integrand ContinuousLinearMap.integral_comp_comm Integrable.comp_sub_right MeasurableAdd₂.toMeasurableAdd integral_sub_right_eq_self RingHomIsometric.ids
integral_posConvolution 📖	mathematical	IntegrableOn Real Real.measurableSpace UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Set.Ioi Real.instPreorder Real.instZero	integral Measure.restrict intervalIntegral DFunLike.coe ContinuousLinearMap Real.semiring RingHom.id Semiring.toNonAssocSemiring ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule Real.normedField ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup ContinuousLinearMap.addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup ContinuousLinearMap.module smulCommClass_self Real.instCommMonoid DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing Real.commRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul Real.pseudoMetricSpace NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup NormedSpace.toIsBoundedSMul Real.instSub	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul integral_indicator measurableSet_Ioi BorelSpace.opensMeasurable Real.borelSpace instClosedIicTopology HasSolidNorm.orderClosedTopology instHasSolidNormReal Real.instIsOrderedAddMonoid posConvolution_eq_convolution_indicator integral_convolution ContinuousAdd.measurableMul₂ instSecondCountableTopologyReal IsTopologicalSemiring.toContinuousAdd IsTopologicalRing.toIsTopologicalSemiring instIsTopologicalRingReal ContinuousNeg.measurableNeg IsTopologicalRing.toContinuousNeg integrable_indicator_iff
posConvolution_eq_convolution_indicator 📖	mathematical	—	posConvolution convolution Real DenselyNormedField.toNontriviallyNormedField Real.denselyNormedField Real.measurableSpace Real.instSub Set.indicator NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup Set.Ioi Real.instPreorder Real.instZero	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul convolution.eq_1 posConvolution.eq_1 Set.indicator.eq_1 intervalIntegral.integral_of_le le_of_lt integral_Ioc_eq_integral_Ioo integral_indicator measurableSet_Ioo BorelSpace.opensMeasurable Real.borelSpace HasSolidNorm.orderClosedTopology instHasSolidNormReal Real.instIsOrderedAddMonoid le_or_gt lt_or_ge Set.indicator_of_notMem Set.notMem_Ioo_of_le Set.notMem_Ioi map_zero AddMonoidHomClass.toZeroHomClass DistribMulActionSemiHomClass.toAddMonoidHomClass SemilinearMapClass.distribMulActionSemiHomClass ContinuousSemilinearMapClass.toSemilinearMapClass ContinuousLinearMap.continuousSemilinearMapClass ContinuousLinearMap.zero_apply Set.indicator_of_mem Set.mem_Ioi sub_pos IsRightCancelAdd.addRightStrictMono_of_addRightMono AddRightCancelSemigroup.toIsRightCancelAdd covariant_swap_add_of_covariant_add IsOrderedAddMonoid.toAddLeftMono Set.notMem_Ioo_of_ge sub_nonpos_of_le sub_nonpos LE.le.trans LT.lt.le Iff.not integral_zero
smul_convolution 📖	mathematical	—	convolution SubNegMonoid.toSub AddGroup.toSubNegMonoid Function.hasSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass ESeminormedAddMonoid.toAddMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ESeminormedAddCommMonoid.toESeminormedAddMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField Module.toDistribMulAction ESeminormedAddCommMonoid.toAddCommMonoid NormedSpace.toModule	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul ContinuousLinearMap.map_smul₂ NormedSpace.toNormSMulClass
support_convolution_subset 📖	mathematical	—	Set Set.instHasSubset Function.support NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup convolution SubNegMonoid.toSub AddGroup.toSubNegMonoid AddCommGroup.toAddGroup Set.add AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup AddCommGroup.toAddCommMonoid	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul HasSubset.Subset.trans Set.instIsTransSubset support_convolution_subset_swap Eq.subset Set.instReflSubset add_comm
support_convolution_subset_swap 📖	mathematical	—	Set Set.instHasSubset Function.support NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup convolution SubNegMonoid.toSub AddGroup.toSubNegMonoid Set.add AddSemigroup.toAdd AddMonoid.toAddSemigroup SubNegMonoid.toAddMonoid	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul convolution_def ContinuousLinearMap.map_zero ContinuousLinearMap.map_zero₂ sub_add_cancel integral_zero
zero_convolution 📖	mathematical	—	convolution SubNegMonoid.toSub AddGroup.toSubNegMonoid Pi.instZero NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul ContinuousLinearMap.map_zero₂ integral_zero

MeasureTheory.AEStronglyMeasurable

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
convolution_integrand 📖	mathematical	MeasureTheory.AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup	Prod.instMeasurableSpace DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup ContinuousLinearMap.addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup ContinuousLinearMap.module smulCommClass_self CommRing.toCommMonoid EuclideanDomain.toCommRing Field.toEuclideanDomain DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup NormedSpace.toIsBoundedSMul SubNegMonoid.toSub AddGroup.toSubNegMonoid MeasureTheory.Measure.prod	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul convolution_integrand' mono_ac MeasureTheory.Measure.QuasiMeasurePreserving.absolutelyContinuous MeasureTheory.quasiMeasurePreserving_sub_of_right_invariant
convolution_integrand' 📖	mathematical	MeasureTheory.AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup MeasureTheory.Measure.map Prod.instMeasurableSpace SubNegMonoid.toSub AddGroup.toSubNegMonoid MeasureTheory.Measure.prod	DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup ContinuousLinearMap.addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup ContinuousLinearMap.module smulCommClass_self CommRing.toCommMonoid EuclideanDomain.toCommRing Field.toEuclideanDomain DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup NormedSpace.toIsBoundedSMul	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul ContinuousLinearMap.aestronglyMeasurable_comp₂ comp_snd comp_measurable MeasurableSub₂.measurable_sub measurableDiv₂_of_add_neg
convolution_integrand_snd 📖	mathematical	MeasureTheory.AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup	DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup ContinuousLinearMap.addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup ContinuousLinearMap.module smulCommClass_self CommRing.toCommMonoid EuclideanDomain.toCommRing Field.toEuclideanDomain DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup NormedSpace.toIsBoundedSMul SubNegMonoid.toSub AddGroup.toSubNegMonoid	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul convolution_integrand_snd' MeasurableAdd₂.toMeasurableAdd mono_ac MeasureTheory.Measure.QuasiMeasurePreserving.absolutelyContinuous MeasureTheory.quasiMeasurePreserving_sub_left_of_right_invariant
convolution_integrand_snd' 📖	mathematical	MeasureTheory.AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup MeasureTheory.Measure.map SubNegMonoid.toSub AddGroup.toSubNegMonoid	DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup ContinuousLinearMap.addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup ContinuousLinearMap.module smulCommClass_self CommRing.toCommMonoid EuclideanDomain.toCommRing Field.toEuclideanDomain DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup NormedSpace.toIsBoundedSMul	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul ContinuousLinearMap.aestronglyMeasurable_comp₂ comp_measurable Measurable.const_sub measurableSub_of_add_neg measurable_id
convolution_integrand_swap_snd 📖	mathematical	MeasureTheory.AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup	DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup ContinuousLinearMap.addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup ContinuousLinearMap.module smulCommClass_self CommRing.toCommMonoid EuclideanDomain.toCommRing Field.toEuclideanDomain DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup NormedSpace.toIsBoundedSMul SubNegMonoid.toSub AddGroup.toSubNegMonoid	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul convolution_integrand_swap_snd' MeasurableAdd₂.toMeasurableAdd mono_ac MeasureTheory.Measure.QuasiMeasurePreserving.absolutelyContinuous MeasureTheory.quasiMeasurePreserving_sub_left_of_right_invariant
convolution_integrand_swap_snd' 📖	mathematical	MeasureTheory.AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup MeasureTheory.Measure.map SubNegMonoid.toSub AddGroup.toSubNegMonoid	DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup ContinuousLinearMap.addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup ContinuousLinearMap.module smulCommClass_self CommRing.toCommMonoid EuclideanDomain.toCommRing Field.toEuclideanDomain DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup NormedSpace.toIsBoundedSMul	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul ContinuousLinearMap.aestronglyMeasurable_comp₂ comp_measurable Measurable.const_sub measurableSub_of_add_neg measurable_id

MeasureTheory.ConvolutionExists

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
add_distrib 📖	mathematical	MeasureTheory.ConvolutionExists SubNegMonoid.toSub AddGroup.toSubNegMonoid	MeasureTheory.convolution Pi.instAdd AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup ESeminormedAddCommMonoid.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul MeasureTheory.ConvolutionExistsAt.add_distrib
distrib_add 📖	mathematical	MeasureTheory.ConvolutionExists SubNegMonoid.toSub AddGroup.toSubNegMonoid	MeasureTheory.convolution Pi.instAdd AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup ESeminormedAddCommMonoid.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul MeasureTheory.ConvolutionExistsAt.distrib_add

MeasureTheory.ConvolutionExistsAt

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
add_distrib 📖	mathematical	MeasureTheory.ConvolutionExistsAt SubNegMonoid.toSub AddGroup.toSubNegMonoid	MeasureTheory.convolution Pi.instAdd AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup ESeminormedAddCommMonoid.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul ContinuousLinearMap.map_add₂ MeasureTheory.integral_add
distrib_add 📖	mathematical	MeasureTheory.ConvolutionExistsAt SubNegMonoid.toSub AddGroup.toSubNegMonoid	MeasureTheory.convolution Pi.instAdd AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup ESeminormedAddCommMonoid.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul ContinuousLinearMap.map_add MeasureTheory.integral_add
integrable 📖	mathematical	MeasureTheory.ConvolutionExistsAt	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup ContinuousLinearMap.addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup ContinuousLinearMap.module smulCommClass_self CommRing.toCommMonoid EuclideanDomain.toCommRing Field.toEuclideanDomain DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup NormedSpace.toIsBoundedSMul	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul
integrable_swap 📖	mathematical	MeasureTheory.ConvolutionExistsAt SubNegMonoid.toSub AddGroup.toSubNegMonoid AddCommGroup.toAddGroup	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup ContinuousLinearMap.addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup ContinuousLinearMap.module smulCommClass_self CommRing.toCommMonoid EuclideanDomain.toCommRing Field.toEuclideanDomain DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup NormedSpace.toIsBoundedSMul	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul sub_sub_self MeasureTheory.Integrable.comp_sub_left
of_norm 📖	—	MeasureTheory.ConvolutionExistsAt Real Real.normedAddCommGroup DenselyNormedField.toNontriviallyNormedField Real.denselyNormedField InnerProductSpace.toNormedSpace Real.instRCLike NormedAddCommGroup.toSeminormedAddCommGroup RCLike.toInnerProductSpaceReal SubNegMonoid.toSub AddGroup.toSubNegMonoid Norm.norm NormedAddCommGroup.toNorm ContinuousLinearMap.mul NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing NonUnitalSeminormedRing.toSeminormedAddCommGroup IsScalarTower.right Real.instCommSemiring CommSemiring.toSemiring Algebra.id Algebra.to_smulCommClass MeasureTheory.AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace	—	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul IsScalarTower.right Algebra.to_smulCommClass of_norm' MeasurableAdd₂.toMeasurableAdd MeasureTheory.AEStronglyMeasurable.mono_ac MeasureTheory.Measure.QuasiMeasurePreserving.absolutelyContinuous MeasureTheory.quasiMeasurePreserving_sub_left_of_right_invariant
of_norm' 📖	—	MeasureTheory.ConvolutionExistsAt Real Real.normedAddCommGroup DenselyNormedField.toNontriviallyNormedField Real.denselyNormedField InnerProductSpace.toNormedSpace Real.instRCLike NormedAddCommGroup.toSeminormedAddCommGroup RCLike.toInnerProductSpaceReal SubNegMonoid.toSub AddGroup.toSubNegMonoid Norm.norm NormedAddCommGroup.toNorm ContinuousLinearMap.mul NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing NonUnitalSeminormedRing.toSeminormedAddCommGroup IsScalarTower.right Real.instCommSemiring CommSemiring.toSemiring Algebra.id Algebra.to_smulCommClass MeasureTheory.AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace MeasureTheory.Measure.map	—	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul IsScalarTower.right Algebra.to_smulCommClass MeasureTheory.Integrable.mono' RingHomIsometric.ids MeasureTheory.Integrable.const_mul MeasureTheory.AEStronglyMeasurable.convolution_integrand_snd' Filter.Eventually.of_forall MeasureTheory.Measure.instOuterMeasureClass IsTopologicalSemiring.toContinuousAdd IsTopologicalRing.toIsTopologicalSemiring NonUnitalSeminormedRing.toIsTopologicalRing ContinuousLinearMap.mul_apply' mul_assoc ContinuousLinearMap.le_opNorm₂

MeasureTheory.Integrable

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
ae_convolution_exists 📖	mathematical	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup	Filter.Eventually MeasureTheory.ConvolutionExistsAt SubNegMonoid.toSub AddGroup.toSubNegMonoid MeasureTheory.ae MeasureTheory.Measure MeasureTheory.Measure.instFunLike MeasureTheory.Measure.instOuterMeasureClass	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul MeasureTheory.Measure.instOuterMeasureClass MeasureTheory.integrable_prod_iff MeasureTheory.AEStronglyMeasurable.convolution_integrand aestronglyMeasurable convolution_integrand
convolution_integrand 📖	mathematical	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup	Prod.instMeasurableSpace DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule ContinuousLinearMap.funLike ContinuousLinearMap.topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup ContinuousLinearMap.addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup ContinuousLinearMap.module smulCommClass_self CommRing.toCommMonoid EuclideanDomain.toCommRing Field.toEuclideanDomain DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup NormedSpace.toIsBoundedSMul SubNegMonoid.toSub AddGroup.toSubNegMonoid MeasureTheory.Measure.prod	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul MeasureTheory.AEStronglyMeasurable.convolution_integrand aestronglyMeasurable MeasureTheory.AEStronglyMeasurable.integral_prod_right' MeasureTheory.AEStronglyMeasurable.norm MeasureTheory.AEStronglyMeasurable.prod_swap MeasureTheory.Measure.instOuterMeasureClass MeasureTheory.integrable_prod_iff' Filter.Eventually.of_forall ContinuousLinearMap.integrable_comp RingHomIsometric.ids comp_sub_right MeasurableAdd₂.toMeasurableAdd mono' MeasureTheory.integral_sub_right_eq_self mul_const const_mul norm Real.norm_of_nonneg MeasureTheory.integral_nonneg Real.instIsOrderedAddMonoid IsStrictOrderedModule.toIsOrderedModule IsStrictOrderedRing.toIsStrictOrderedModule Real.instIsStrictOrderedRing instClosedIciTopology HasSolidNorm.orderClosedTopology instHasSolidNormReal norm_nonneg MeasureTheory.integral_mono_of_nonneg ContinuousLinearMap.le_opNorm₂
integrable_convolution 📖	mathematical	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup	MeasureTheory.convolution SubNegMonoid.toSub AddGroup.toSubNegMonoid	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul integral_prod_left convolution_integrand

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