Set

📁 Source: Mathlib/MeasureTheory/Integral/Bochner/Set.lean

Statistics

Metric	Count
Definitions LpToLpRestrictCLM	1
Theorems integral_pos_of_hasCompactSupport_nonneg_nonzero, setIntegral_map, measure_le_integral, simpleFunc_mul, simpleFunc_mul', LpToLpRestrictCLM_coeFn, Lp_toLp_restrict_add, Lp_toLp_restrict_smul, setIntegral_image_emb, setIntegral_preimage_emb, continuous_setIntegral, hasSum_integral_iUnion, hasSum_integral_iUnion_ae, integrableOn_iUnion_of_summable_integral_norm, integrableOn_iUnion_of_summable_norm_restrict, integrable_of_summable_norm_restrict, integral_Icc_eq_integral_Ico, integral_Icc_eq_integral_Ico', integral_Icc_eq_integral_Ioc, integral_Icc_eq_integral_Ioc', integral_Icc_eq_integral_Ioo, integral_Icc_eq_integral_Ioo', integral_Ici_eq_integral_Ioi, integral_Ici_eq_integral_Ioi', integral_Ico_eq_integral_Ioo, integral_Ico_eq_integral_Ioo', integral_Iic_eq_integral_Iio, integral_Iic_eq_integral_Iio', integral_Ioc_eq_integral_Ioo, integral_Ioc_eq_integral_Ioo', integral_add_compl, integral_add_compl₀, integral_biUnion_eq_sum_powerset, integral_biUnion_finset, integral_diff, integral_finset_biUnion, integral_fintype_iUnion, integral_iUnion, integral_iUnion_ae, integral_iUnion_fintype, integral_indicator, integral_indicatorConstLp, integral_indicator_const, integral_indicator_one, integral_indicator₀, integral_integral_indicator, integral_inter_add_diff, integral_inter_add_diff₀, integral_le_measure, integral_norm_eq_pos_sub_neg, integral_piecewise, integral_union_ae, integral_union_eq_left_of_ae, integral_union_eq_left_of_ae_aux, integral_union_eq_left_of_forall, integral_union_eq_left_of_forall₀, measureReal_biUnion_eq_sum_powerset, norm_Lp_toLp_restrict_le, norm_integral_sub_setIntegral_le, norm_setIntegral_le_of_norm_le_const, norm_setIntegral_le_of_norm_le_const_ae, norm_setIntegral_le_of_norm_le_const_ae', ofReal_setIntegral_one, ofReal_setIntegral_one_of_measure_ne_top, setIntegral_compl, setIntegral_compl₀, setIntegral_congr_ae, setIntegral_congr_ae₀, setIntegral_congr_fun, setIntegral_congr_fun₀, setIntegral_congr_set, setIntegral_const, setIntegral_empty, setIntegral_eq_integral_of_ae_compl_eq_zero, setIntegral_eq_integral_of_forall_compl_eq_zero, setIntegral_eq_of_subset_of_ae_diff_eq_zero, setIntegral_eq_of_subset_of_ae_diff_eq_zero_aux, setIntegral_eq_of_subset_of_forall_diff_eq_zero, setIntegral_eq_zero_iff_of_nonneg_ae, setIntegral_eq_zero_of_ae_eq_zero, setIntegral_eq_zero_of_forall_eq_zero, setIntegral_ge_of_const_le, setIntegral_ge_of_const_le_real, setIntegral_gt_gt, setIntegral_indicator, setIntegral_indicatorConstLp, setIntegral_le_integral, setIntegral_le_nonneg, setIntegral_map, setIntegral_map_equiv, setIntegral_mono, setIntegral_mono_ae, setIntegral_mono_ae_restrict, setIntegral_mono_on, setIntegral_mono_on_ae, setIntegral_mono_on_ae₀, setIntegral_mono_on₀, setIntegral_mono_set, setIntegral_neg_eq_setIntegral_nonpos, setIntegral_nonneg, setIntegral_nonneg_ae, setIntegral_nonneg_of_ae, setIntegral_nonneg_of_ae_restrict, setIntegral_nonpos, setIntegral_nonpos_ae, setIntegral_nonpos_le, setIntegral_nonpos_of_ae, setIntegral_nonpos_of_ae_restrict, setIntegral_one_eq_measureReal, setIntegral_pos_iff_support_of_nonneg_ae, setIntegral_support, setIntegral_trim, setIntegral_tsupport, setIntegral_union, setIntegral_univ, tendsto_setIntegral_of_antitone, tendsto_setIntegral_of_monotone, setIntegral_map, continuousOn_integral_bilinear_of_locally_integrable_of_compact_support, continuousOn_integral_of_compact_support, continuous_parametric_integral_of_continuous, measure_le_lintegral_thickenedIndicator, measure_le_lintegral_thickenedIndicatorAux	123
Total	124

Continuous

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
integral_pos_of_hasCompactSupport_nonneg_nonzero 📖	mathematical	Continuous Real UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace HasCompactSupport Real.instZero Pi.hasLe Real.instLE Pi.instZero	Real.instLT MeasureTheory.integral Real.normedAddCommGroup NormedField.toNormedSpace Real.normedField	—	MeasureTheory.integral_pos_of_integrable_nonneg_nonzero integrable_of_hasCompactSupport

MeasurableEmbedding

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
setIntegral_map 📖	mathematical	MeasurableEmbedding	MeasureTheory.integral MeasureTheory.Measure.restrict MeasureTheory.Measure.map Set.preimage	—	restrict_map integral_map

MeasureTheory

Definitions

Name	Category	Theorems
LpToLpRestrictCLM 📖	CompOp	1 math math: LpToLpRestrictCLM_coeFn

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
LpToLpRestrictCLM_coeFn 📖	mathematical	—	Filter.EventuallyEq ae Measure Measure.instFunLike Measure.instOuterMeasureClass Measure.restrict AEEqFun.cast UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup AEEqFun AddSubgroup AEEqFun.instAddGroup NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup SetLike.instMembership AddSubgroup.instSetLike Lp DFunLike.coe ContinuousLinearMap Ring.toSemiring NormedRing.toRing RingHom.id Semiring.toNonAssocSemiring SeminormedAddCommGroup.toIsTopologicalAddGroup Lp.instNormedAddCommGroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid Lp.instModule ContinuousLinearMap.funLike LpToLpRestrictCLM	—	SeminormedAddCommGroup.toIsTopologicalAddGroup MemLp.coeFn_toLp MemLp.restrict Lp.memLp
Lp_toLp_restrict_add 📖	mathematical	—	MemLp.toLp Measure.restrict AEEqFun.cast UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup AEEqFun AddSubgroup AEEqFun.instAddGroup NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup SetLike.instMembership AddSubgroup.instSetLike Lp SeminormedAddCommGroup.toIsTopologicalAddGroup AddSubgroup.add MemLp.restrict SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Lp.memLp	—	SeminormedAddCommGroup.toIsTopologicalAddGroup Lp.ext MemLp.restrict Lp.memLp Filter.Eventually.mp Measure.instOuterMeasureClass ae_restrict_of_ae Lp.coeFn_add MemLp.coeFn_toLp Filter.Eventually.mono Pi.add_apply
Lp_toLp_restrict_smul 📖	mathematical	—	MemLp.toLp Measure.restrict AEEqFun.cast UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup AEEqFun AddSubgroup AEEqFun.instAddGroup NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup SetLike.instMembership AddSubgroup.instSetLike Lp SeminormedAddCommGroup.toIsTopologicalAddGroup SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass SubNegMonoid.toAddMonoid AddGroup.toSubNegMonoid AddSubgroup.toAddGroup DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero Ring.toSemiring NormedRing.toRing Module.toDistribMulAction AddCommGroup.toAddCommMonoid AddSubgroup.toAddCommGroup AEEqFun.instAddCommGroup NormedAddCommGroup.toAddCommGroup Lp.instModule MemLp.restrict SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Lp.memLp	—	SeminormedAddCommGroup.toIsTopologicalAddGroup Lp.ext MemLp.restrict Lp.memLp Filter.Eventually.mp Measure.instOuterMeasureClass ae_restrict_of_ae Lp.coeFn_smul MemLp.coeFn_toLp Filter.Eventually.mono
continuous_setIntegral 📖	mathematical	—	Continuous AEEqFun UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup AddSubgroup AEEqFun.instAddGroup NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup SeminormedAddCommGroup.toIsTopologicalAddGroup SetLike.instMembership AddSubgroup.instSetLike Lp ENNReal AddMonoidWithOne.toOne AddCommMonoidWithOne.toAddMonoidWithOne instAddCommMonoidWithOneENNReal Lp.instNormedAddCommGroup fact_one_le_one_ennreal integral Measure.restrict AEEqFun.cast	—	SeminormedAddCommGroup.toIsTopologicalAddGroup le_rfl NormedSpace.toIsBoundedSMul integral_congr_ae LpToLpRestrictCLM_coeFn fact_one_le_one_ennreal Continuous.comp continuous_integral ContinuousLinearMap.continuous
hasSum_integral_iUnion 📖	mathematical	MeasurableSet Pairwise Function.onFun Set Disjoint OmegaCompletePartialOrder.toPartialOrder CompleteLattice.instOmegaCompletePartialOrder CompleteBooleanAlgebra.toCompleteLattice CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra Set.instCompleteAtomicBooleanAlgebra HeytingAlgebra.toOrderBot Order.Frame.toHeytingAlgebra CompleteDistribLattice.toFrame CompleteBooleanAlgebra.toCompleteDistribLattice IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Set.iUnion	HasSum ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid integral Measure.restrict SummationFilter.unconditional	—	hasSum_integral_iUnion_ae MeasurableSet.nullMeasurableSet Pairwise.mono Disjoint.aedisjoint
hasSum_integral_iUnion_ae 📖	mathematical	NullMeasurableSet Pairwise Function.onFun Set AEDisjoint IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Set.iUnion	HasSum ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid integral Measure.restrict SummationFilter.unconditional	—	Measure.restrict_iUnion_ae hasSum_integral_measure
integrableOn_iUnion_of_summable_integral_norm 📖	mathematical	IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Summable Real Real.instAddCommMonoid Real.pseudoMetricSpace integral Real.normedAddCommGroup NormedField.toNormedSpace Real.normedField Measure.restrict Norm.norm NormedAddCommGroup.toNorm SummationFilter.unconditional	Set.iUnion	—	AEStronglyMeasurable.iUnion PseudoEMetricSpace.pseudoMetrizableSpace LE.le.trans_lt lintegral_iUnion_le lintegral_coe_eq_integral Integrable.norm tsum_congr IsOrderedAddMonoid.toAddLeftMono Real.instIsOrderedAddMonoid integral_nonneg IsStrictOrderedModule.toIsOrderedModule IsStrictOrderedRing.toIsStrictOrderedModule Real.instIsStrictOrderedRing instClosedIciTopology HasSolidNorm.orderClosedTopology instHasSolidNormReal NNReal.coe_nonneg NNReal.summable_coe ENNReal.tsum_coe_eq Summable.hasSum ENNReal.coe_nnreal_eq ENNReal.ofReal_lt_top
integrableOn_iUnion_of_summable_norm_restrict 📖	mathematical	Summable Real Real.instAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Real.instMul Norm.norm ContinuousMap Set.Elem SetLike.coe TopologicalSpace.Compacts TopologicalSpace.Compacts.instSetLike instTopologicalSpaceSubtype Set Set.instMembership SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ContinuousMap.instNorm ContinuousMap.instCompactSpaceElemCoeCompacts ContinuousMap.restrict Measure.real SummationFilter.unconditional	IntegrableOn SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup DFunLike.coe ContinuousMap.instFunLike Set.iUnion	—	ContinuousMap.instCompactSpaceElemCoeCompacts integrableOn_iUnion_of_summable_integral_norm ContinuousOn.integrableOn_compact BorelSpace.opensMeasurable isFiniteMeasureOnCompacts_of_isLocallyFiniteMeasure TopologicalSpace.Compacts.isCompact Continuous.continuousOn ContinuousMapClass.map_continuous ContinuousMap.instContinuousMapClass Summable.of_nonneg_of_le integral_nonneg Real.instIsOrderedAddMonoid IsStrictOrderedModule.toIsOrderedModule IsStrictOrderedRing.toIsStrictOrderedModule Real.instIsStrictOrderedRing instClosedIciTopology HasSolidNorm.orderClosedTopology instHasSolidNormReal norm_nonneg Real.norm_of_nonneg norm_setIntegral_le_of_norm_le_const IsCompact.measure_lt_top ContinuousMap.norm_coe_le_norm norm_norm
integrable_of_summable_norm_restrict 📖	mathematical	Summable Real Real.instAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Real.instMul Norm.norm ContinuousMap Set.Elem SetLike.coe TopologicalSpace.Compacts TopologicalSpace.Compacts.instSetLike instTopologicalSpaceSubtype Set Set.instMembership SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ContinuousMap.instNorm ContinuousMap.instCompactSpaceElemCoeCompacts ContinuousMap.restrict Measure.real SummationFilter.unconditional Set.iUnion Set.univ	Integrable SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup DFunLike.coe ContinuousMap.instFunLike	—	ContinuousMap.instCompactSpaceElemCoeCompacts integrableOn_iUnion_of_summable_norm_restrict
integral_Icc_eq_integral_Ico 📖	mathematical	—	integral Measure.restrict Set.Icc PartialOrder.toPreorder Set.Ico	—	integral_Icc_eq_integral_Ico' NoAtoms.measure_singleton
integral_Icc_eq_integral_Ico' 📖	mathematical	DFunLike.coe Measure Set ENNReal Measure.instFunLike Set.instSingletonSet instZeroENNReal	integral Measure.restrict Set.Icc PartialOrder.toPreorder Set.Ico	—	setIntegral_congr_set Filter.EventuallyEq.symm Measure.instOuterMeasureClass Ico_ae_eq_Icc'
integral_Icc_eq_integral_Ioc 📖	mathematical	—	integral Measure.restrict Set.Icc PartialOrder.toPreorder Set.Ioc	—	integral_Icc_eq_integral_Ioc' NoAtoms.measure_singleton
integral_Icc_eq_integral_Ioc' 📖	mathematical	DFunLike.coe Measure Set ENNReal Measure.instFunLike Set.instSingletonSet instZeroENNReal	integral Measure.restrict Set.Icc PartialOrder.toPreorder Set.Ioc	—	setIntegral_congr_set Filter.EventuallyEq.symm Measure.instOuterMeasureClass Ioc_ae_eq_Icc'
integral_Icc_eq_integral_Ioo 📖	mathematical	—	integral Measure.restrict Set.Icc PartialOrder.toPreorder Set.Ioo	—	integral_Icc_eq_integral_Ico integral_Ico_eq_integral_Ioo
integral_Icc_eq_integral_Ioo' 📖	mathematical	DFunLike.coe Measure Set ENNReal Measure.instFunLike Set.instSingletonSet instZeroENNReal	integral Measure.restrict Set.Icc PartialOrder.toPreorder Set.Ioo	—	setIntegral_congr_set Filter.EventuallyEq.symm Measure.instOuterMeasureClass Ioo_ae_eq_Icc'
integral_Ici_eq_integral_Ioi 📖	mathematical	—	integral Measure.restrict Set.Ici PartialOrder.toPreorder Set.Ioi	—	integral_Ici_eq_integral_Ioi' NoAtoms.measure_singleton
integral_Ici_eq_integral_Ioi' 📖	mathematical	DFunLike.coe Measure Set ENNReal Measure.instFunLike Set.instSingletonSet instZeroENNReal	integral Measure.restrict Set.Ici PartialOrder.toPreorder Set.Ioi	—	setIntegral_congr_set Filter.EventuallyEq.symm Measure.instOuterMeasureClass Ioi_ae_eq_Ici'
integral_Ico_eq_integral_Ioo 📖	mathematical	—	integral Measure.restrict Set.Ico PartialOrder.toPreorder Set.Ioo	—	integral_Ico_eq_integral_Ioo' NoAtoms.measure_singleton
integral_Ico_eq_integral_Ioo' 📖	mathematical	DFunLike.coe Measure Set ENNReal Measure.instFunLike Set.instSingletonSet instZeroENNReal	integral Measure.restrict Set.Ico PartialOrder.toPreorder Set.Ioo	—	setIntegral_congr_set Filter.EventuallyEq.symm Measure.instOuterMeasureClass Ioo_ae_eq_Ico'
integral_Iic_eq_integral_Iio 📖	mathematical	—	integral Measure.restrict Set.Iic PartialOrder.toPreorder Set.Iio	—	integral_Iic_eq_integral_Iio' NoAtoms.measure_singleton
integral_Iic_eq_integral_Iio' 📖	mathematical	DFunLike.coe Measure Set ENNReal Measure.instFunLike Set.instSingletonSet instZeroENNReal	integral Measure.restrict Set.Iic PartialOrder.toPreorder Set.Iio	—	setIntegral_congr_set Filter.EventuallyEq.symm Measure.instOuterMeasureClass Iio_ae_eq_Iic'
integral_Ioc_eq_integral_Ioo 📖	mathematical	—	integral Measure.restrict Set.Ioc PartialOrder.toPreorder Set.Ioo	—	integral_Ioc_eq_integral_Ioo' NoAtoms.measure_singleton
integral_Ioc_eq_integral_Ioo' 📖	mathematical	DFunLike.coe Measure Set ENNReal Measure.instFunLike Set.instSingletonSet instZeroENNReal	integral Measure.restrict Set.Ioc PartialOrder.toPreorder Set.Ioo	—	setIntegral_congr_set Filter.EventuallyEq.symm Measure.instOuterMeasureClass Ioo_ae_eq_Ioc'
integral_add_compl 📖	mathematical	MeasurableSet Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup	AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid integral Measure.restrict Compl.compl Set Set.instCompl	—	integral_add_compl₀ MeasurableSet.nullMeasurableSet
integral_add_compl₀ 📖	mathematical	NullMeasurableSet Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup	AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid integral Measure.restrict Compl.compl Set Set.instCompl	—	integral_union_ae Disjoint.aedisjoint disjoint_compl_right NullMeasurableSet.compl Integrable.integrableOn Set.union_compl_self setIntegral_univ
integral_biUnion_eq_sum_powerset 📖	mathematical	MeasurableSet IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup	integral Measure.restrict Set.iUnion Finset Finset.instMembership Finset.sum ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid Finset.filter Finset.Nonempty Finset.decidableNonempty Finset.powerset Real SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass ESeminormedAddMonoid.toAddMonoid ESeminormedAddCommMonoid.toESeminormedAddMonoid DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul Real.instMonoid Module.toDistribMulAction Real.semiring NormedSpace.toModule Real.normedField Monoid.toNatPow Real.instNeg Real.instOne Finset.card Set.iInter	—	Finset.sum_congr NormedSpace.toNormSMulClass smulCommClass_self integral_indicator Finset.measurableSet_biUnion Finset.measurableSet_biInter integral_finset_sum integrable_indicator_iff Integrable.smul NormedSpace.toIsBoundedSMul IntegrableOn.mono Set.biInter_subset_of_mem le_rfl Set.indicator_smul_apply Nat.cast_one Int.cast_smul_eq_zsmul Finset.indicator_biUnion_eq_sum_powerset
integral_biUnion_finset 📖	mathematical	MeasurableSet Set.Pairwise SetLike.coe Finset Finset.instSetLike Function.onFun Set Disjoint OmegaCompletePartialOrder.toPartialOrder CompleteLattice.instOmegaCompletePartialOrder CompleteBooleanAlgebra.toCompleteLattice CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra Set.instCompleteAtomicBooleanAlgebra HeytingAlgebra.toOrderBot Order.Frame.toHeytingAlgebra CompleteDistribLattice.toFrame CompleteBooleanAlgebra.toCompleteDistribLattice IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup	integral Measure.restrict Set.iUnion Finset.instMembership Finset.sum ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid	—	Finset.induction_on Set.iUnion_congr_Prop Set.iUnion_of_empty instIsEmptyFalse Set.iUnion_empty Measure.restrict_empty integral_zero_measure Finset.set_biUnion_insert setIntegral_union Finset.coe_insert Finset.measurableSet_biUnion integrableOn_finset_iUnion PseudoEMetricSpace.pseudoMetrizableSpace Finset.sum_insert
integral_diff 📖	mathematical	MeasurableSet IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Set Set.instHasSubset	integral Measure.restrict Set.instSDiff SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup	—	eq_sub_iff_add_eq setIntegral_union disjoint_sdiff_self_left IntegrableOn.mono_set Set.diff_subset Set.diff_union_of_subset
integral_finset_biUnion 📖	mathematical	MeasurableSet Set.Pairwise SetLike.coe Finset Finset.instSetLike Function.onFun Set Disjoint OmegaCompletePartialOrder.toPartialOrder CompleteLattice.instOmegaCompletePartialOrder CompleteBooleanAlgebra.toCompleteLattice CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra Set.instCompleteAtomicBooleanAlgebra HeytingAlgebra.toOrderBot Order.Frame.toHeytingAlgebra CompleteDistribLattice.toFrame CompleteBooleanAlgebra.toCompleteDistribLattice IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup	integral Measure.restrict Set.iUnion Finset.instMembership Finset.sum ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid	—	integral_biUnion_finset
integral_fintype_iUnion 📖	mathematical	MeasurableSet Pairwise Function.onFun Set Disjoint OmegaCompletePartialOrder.toPartialOrder CompleteLattice.instOmegaCompletePartialOrder CompleteBooleanAlgebra.toCompleteLattice CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra Set.instCompleteAtomicBooleanAlgebra HeytingAlgebra.toOrderBot Order.Frame.toHeytingAlgebra CompleteDistribLattice.toFrame CompleteBooleanAlgebra.toCompleteDistribLattice IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup	integral Measure.restrict Set.iUnion Finset.sum ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid Finset.univ	—	integral_iUnion_fintype
integral_iUnion 📖	mathematical	MeasurableSet Pairwise Function.onFun Set Disjoint OmegaCompletePartialOrder.toPartialOrder CompleteLattice.instOmegaCompletePartialOrder CompleteBooleanAlgebra.toCompleteLattice CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra Set.instCompleteAtomicBooleanAlgebra HeytingAlgebra.toOrderBot Order.Frame.toHeytingAlgebra CompleteDistribLattice.toFrame CompleteBooleanAlgebra.toCompleteDistribLattice IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Set.iUnion	integral Measure.restrict tsum ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid SummationFilter.unconditional	—	HasSum.tsum_eq TopologicalSpace.t2Space_of_metrizableSpace EMetricSpace.metrizableSpace SummationFilter.instNeBotUnconditional hasSum_integral_iUnion
integral_iUnion_ae 📖	mathematical	NullMeasurableSet Pairwise Function.onFun Set AEDisjoint IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Set.iUnion	integral Measure.restrict tsum ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid SummationFilter.unconditional	—	HasSum.tsum_eq TopologicalSpace.t2Space_of_metrizableSpace EMetricSpace.metrizableSpace SummationFilter.instNeBotUnconditional hasSum_integral_iUnion_ae
integral_iUnion_fintype 📖	mathematical	MeasurableSet Pairwise Function.onFun Set Disjoint OmegaCompletePartialOrder.toPartialOrder CompleteLattice.instOmegaCompletePartialOrder CompleteBooleanAlgebra.toCompleteLattice CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra Set.instCompleteAtomicBooleanAlgebra HeytingAlgebra.toOrderBot Order.Frame.toHeytingAlgebra CompleteDistribLattice.toFrame CompleteBooleanAlgebra.toCompleteDistribLattice IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup	integral Measure.restrict Set.iUnion Finset.sum ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid Finset.univ	—	Set.iUnion_congr_Prop Set.iUnion_true integral_biUnion_finset Finset.coe_univ
integral_indicator 📖	mathematical	MeasurableSet	integral Set.indicator NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup Measure.restrict	—	integral_add_compl IntegrableOn.integrable_indicator integral_congr_ae indicator_ae_eq_restrict indicator_ae_eq_restrict_compl integral_zero add_zero integral_undef integrable_indicator_iff
integral_indicatorConstLp 📖	mathematical	MeasurableSet	integral AEEqFun.cast UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup AEEqFun AddSubgroup AEEqFun.instAddGroup NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup SetLike.instMembership AddSubgroup.instSetLike Lp indicatorConstLp Real SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass ESeminormedAddMonoid.toAddMonoid ESeminormedAddCommMonoid.toESeminormedAddMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul Real.instMonoid Module.toDistribMulAction Real.semiring ESeminormedAddCommMonoid.toAddCommMonoid NormedSpace.toModule Real.normedField Measure.real	—	setIntegral_univ setIntegral_indicatorConstLp MeasurableSet.univ Set.inter_univ
integral_indicator_const 📖	mathematical	MeasurableSet	integral Set.indicator NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup Real 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 Real.instMonoid Module.toDistribMulAction Real.semiring ESeminormedAddCommMonoid.toAddCommMonoid NormedSpace.toModule Real.normedField Measure.real	—	integral_indicator setIntegral_const
integral_indicator_one 📖	mathematical	MeasurableSet	integral Real Real.normedAddCommGroup NormedField.toNormedSpace Real.normedField Set.indicator Real.instZero Pi.instOne Real.instOne Measure.real	—	integral_indicator_const Real.instCompleteSpace smul_eq_mul mul_one
integral_indicator₀ 📖	mathematical	NullMeasurableSet	integral Set.indicator NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup Measure.restrict	—	integral_congr_ae indicator_ae_eq_of_ae_eq_set NullMeasurableSet.toMeasurable_ae_eq integral_indicator measurableSet_toMeasurable Measure.restrict_congr_set
integral_integral_indicator 📖	mathematical	MeasurableSet	integral Set.indicator NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup Measure.restrict	—	integral_indicator integral_indicator₂
integral_inter_add_diff 📖	mathematical	MeasurableSet IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup	AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid integral Measure.restrict Set Set.instInter Set.instSDiff	—	integral_inter_add_diff₀ MeasurableSet.nullMeasurableSet
integral_inter_add_diff₀ 📖	mathematical	NullMeasurableSet IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup	AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid integral Measure.restrict Set Set.instInter Set.instSDiff	—	Measure.restrict_inter_add_diff₀ integral_add_measure Integrable.mono_measure Measure.restrict_mono Set.inter_subset_left le_rfl Set.diff_subset
integral_le_measure 📖	mathematical	Real Real.instLE Real.instOne Real.instZero	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder ENNReal.ofReal integral Real.normedAddCommGroup NormedField.toNormedSpace Real.normedField DFunLike.coe Measure Set Measure.instFunLike	—	Integrable.pos_part ENNReal.ofReal_le_ofReal integral_mono Real.instIsOrderedAddMonoid IsStrictOrderedModule.toIsOrderedModule IsStrictOrderedRing.toIsStrictOrderedModule Real.instIsStrictOrderedRing instClosedIciTopology HasSolidNorm.orderClosedTopology instHasSolidNormReal le_max_left LE.le.trans ofReal_integral_eq_lintegral_ofReal Filter.Eventually.of_forall Measure.instOuterMeasureClass le_max_right lintegral_le_meas ENNReal.ofReal_le_of_le_toReal Real.instZeroLEOneClass le_trans zero_le_one ENNReal.ofReal_max ENNReal.ofReal_zero sup_of_le_left ENNReal.instCanonicallyOrderedAdd integral_undef
integral_norm_eq_pos_sub_neg 📖	mathematical	Integrable Real UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing	integral Real.normedAddCommGroup NormedField.toNormedSpace Real.normedField Norm.norm Real.norm Real.instSub Measure.restrict setOf Real.instLE Real.instZero	—	AEStronglyMeasurable.nullMeasurableSet_le HasSolidNorm.orderClosedTopology instHasSolidNormReal Real.instIsOrderedAddMonoid PseudoEMetricSpace.pseudoMetrizableSpace aestronglyMeasurable_const integral_add_compl₀ Integrable.norm setIntegral_congr_fun₀ Real.norm_eq_abs abs_eq_self integral_neg NullMeasurableSet.compl abs_eq_neg_self le_of_not_gt Mathlib.Tactic.Linarith.lt_irrefl Mathlib.Tactic.Ring.of_eq Mathlib.Tactic.Ring.add_congr Mathlib.Tactic.Ring.sub_congr Mathlib.Tactic.Ring.atom_pf Mathlib.Tactic.Ring.cast_zero Mathlib.Meta.NormNum.isNat_ofNat Nat.cast_zero Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero Mathlib.Tactic.Ring.add_pf_add_zero Mathlib.Tactic.Ring.neg_add Mathlib.Tactic.Ring.neg_mul Mathlib.Tactic.Ring.neg_one_mul Mathlib.Meta.NormNum.IsInt.to_raw_eq Mathlib.Meta.NormNum.isInt_mul Mathlib.Meta.NormNum.IsInt.of_raw Mathlib.Meta.NormNum.IsNat.to_isInt Mathlib.Meta.NormNum.IsNat.of_raw Mathlib.Tactic.Ring.add_pf_zero_add Mathlib.Tactic.Ring.add_pf_add_overlap_zero Mathlib.Tactic.Ring.add_overlap_pf_zero Mathlib.Meta.NormNum.IsInt.to_isNat Mathlib.Meta.NormNum.isInt_add Mathlib.Tactic.Linarith.add_neg Real.instIsStrictOrderedRing Mathlib.Tactic.Linarith.sub_neg_of_lt Real.instIsOrderedRing lt_of_not_ge Set.notMem_setOf_iff Set.mem_compl_iff setIntegral_neg_eq_setIntegral_nonpos Set.compl_setOf
integral_piecewise 📖	mathematical	MeasurableSet IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Compl.compl Set Set.instCompl	integral Set.piecewise AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid Measure.restrict	—	Set.indicator_add_compl_eq_piecewise integral_add' IntegrableOn.integrable_indicator MeasurableSet.compl integral_indicator
integral_union_ae 📖	mathematical	AEDisjoint NullMeasurableSet IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup	integral Measure.restrict Set Set.instUnion AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid	—	Measure.restrict_union₀ integral_add_measure
integral_union_eq_left_of_ae 📖	mathematical	Filter.Eventually NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup ae Measure Measure.instFunLike Measure.instOuterMeasureClass Measure.restrict	integral Set Set.instUnion	—	Measure.instOuterMeasureClass IntegrableOn.congr_fun_ae integrableOn_zero Filter.EventuallyEq.symm integral_congr_ae integral_union_eq_left_of_ae_aux Filter.mp_mem ae_mono Measure.restrict_mono Set.subset_union_right le_rfl Filter.univ_mem' Set.subset_union_left integral_undef PseudoEMetricSpace.pseudoMetrizableSpace
integral_union_eq_left_of_ae_aux 📖	mathematical	Filter.Eventually NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup ae Measure Measure.instFunLike Measure.instOuterMeasureClass Measure.restrict StronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup IntegrableOn SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Set Set.instUnion	integral	—	Measure.instOuterMeasureClass StronglyMeasurable.measurable PseudoEMetricSpace.pseudoMetrizableSpace MeasurableSingletonClass.measurableSet_singleton OpensMeasurableSpace.toMeasurableSingletonClass BorelSpace.opensMeasurable T5Space.toT1Space T6Space.toT5Space instT6SpaceOfMetrizableSpace EMetricSpace.metrizableSpace IntegrableOn.mono Set.subset_union_left le_rfl setIntegral_eq_zero_of_forall_eq_zero integral_inter_add_diff zero_add Set.union_diff_distrib Set.union_comm setIntegral_congr_set union_ae_eq_right measure_mono_null Set.diff_subset measure_eq_zero_iff_ae_notMem Filter.mp_mem ae_imp_of_ae_restrict Filter.univ_mem'
integral_union_eq_left_of_forall 📖	mathematical	MeasurableSet NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup	integral Measure.restrict Set Set.instUnion	—	integral_union_eq_left_of_forall₀ MeasurableSet.nullMeasurableSet
integral_union_eq_left_of_forall₀ 📖	mathematical	NullMeasurableSet NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup	integral Measure.restrict Set Set.instUnion	—	integral_union_eq_left_of_ae Measure.instOuterMeasureClass ae_restrict_iff'₀ Filter.Eventually.of_forall
measureReal_biUnion_eq_sum_powerset 📖	mathematical	MeasurableSet	Measure.real Set.iUnion Finset Finset.instMembership Finset.sum Real Real.instAddCommMonoid Finset.filter Finset.Nonempty Finset.decidableNonempty Finset.powerset Real.instMul Monoid.toNatPow Real.instMonoid Real.instNeg Real.instOne Finset.card Set.iInter	—	Finset.sum_congr integral_biUnion_eq_sum_powerset enorm_one NormedDivisionRing.to_normOneClass NeZero.charZero_one ENNReal.instCharZero Ne.lt_top
norm_Lp_toLp_restrict_le 📖	mathematical	—	Real Real.instLE Norm.norm AEEqFun UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup Measure.restrict AddSubgroup AEEqFun.instAddGroup NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup SeminormedAddCommGroup.toIsTopologicalAddGroup SetLike.instMembership AddSubgroup.instSetLike Lp Lp.instNorm MemLp.toLp AEEqFun.cast MemLp.restrict SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Lp.memLp	—	SeminormedAddCommGroup.toIsTopologicalAddGroup MemLp.restrict Lp.memLp Lp.norm_def eLpNorm_congr_ae MemLp.coeFn_toLp ENNReal.toReal_mono Lp.eLpNorm_ne_top eLpNorm_mono_measure Measure.restrict_le_self
norm_integral_sub_setIntegral_le 📖	mathematical	Filter.Eventually Real Real.instLE Norm.norm NormedAddCommGroup.toNorm ae Measure Measure.instFunLike Measure.instOuterMeasureClass MeasurableSet Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup	SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup integral Measure.restrict Real.instMul Measure.real Compl.compl Set Set.instCompl	—	Measure.instOuterMeasureClass sub_eq_iff_eq_add add_comm integral_add_compl integral_mono_ae Real.instIsOrderedAddMonoid IsStrictOrderedModule.toIsOrderedModule IsStrictOrderedRing.toIsStrictOrderedModule Real.instIsStrictOrderedRing instClosedIciTopology HasSolidNorm.orderClosedTopology instHasSolidNormReal Integrable.restrict Integrable.norm integrable_const isFiniteMeasureRestrict ae_restrict_of_ae setIntegral_const Real.instCompleteSpace smul_eq_mul le_trans norm_integral_le_integral_norm
norm_setIntegral_le_of_norm_le_const 📖	mathematical	ENNReal Preorder.toLT PartialOrder.toPreorder ENNReal.instPartialOrder DFunLike.coe Measure Set Measure.instFunLike Top.top instTopENNReal Real Real.instLE Norm.norm NormedAddCommGroup.toNorm	integral Measure.restrict Real.instMul Measure.real	—	norm_setIntegral_le_of_norm_le_const_ae' Filter.Eventually.of_forall Measure.instOuterMeasureClass
norm_setIntegral_le_of_norm_le_const_ae 📖	mathematical	ENNReal Preorder.toLT PartialOrder.toPreorder ENNReal.instPartialOrder DFunLike.coe Measure Set Measure.instFunLike Top.top instTopENNReal Filter.Eventually Real Real.instLE Norm.norm NormedAddCommGroup.toNorm ae Measure.instOuterMeasureClass Measure.restrict	integral Real.instMul Measure.real	—	Measure.instOuterMeasureClass measureReal_restrict_apply Set.univ_inter norm_integral_le_of_norm_le_const Measure.restrict_apply_univ
norm_setIntegral_le_of_norm_le_const_ae' 📖	mathematical	ENNReal Preorder.toLT PartialOrder.toPreorder ENNReal.instPartialOrder DFunLike.coe Measure Set Measure.instFunLike Top.top instTopENNReal Filter.Eventually ae Measure.instOuterMeasureClass	Real Real.instLE Norm.norm NormedAddCommGroup.toNorm integral Measure.restrict Real.instMul Measure.real	—	Measure.instOuterMeasureClass norm_setIntegral_le_of_norm_le_const_ae Filter.mp_mem Filter.univ_mem' StronglyMeasurable.measurable PseudoEMetricSpace.pseudoMetrizableSpace Real.borelSpace StronglyMeasurable.norm measurableSet_Iic BorelSpace.opensMeasurable instClosedIicTopology HasSolidNorm.orderClosedTopology instHasSolidNormReal Real.instIsOrderedAddMonoid ae_restrict_iff integral_non_aestronglyMeasurable frequently_ae_mem_iff Mathlib.Tactic.Contrapose.contrapose₄ Measure.restrict_eq_zero Filter.Frequently.exists Filter.Frequently.and_eventually LE.le.trans norm_nonneg norm_zero mul_nonneg IsOrderedRing.toPosMulMono Real.instIsOrderedRing measureReal_nonneg
ofReal_setIntegral_one 📖	mathematical	—	ENNReal.ofReal integral Real Real.normedAddCommGroup NormedField.toNormedSpace Real.normedField Measure.restrict Real.instOne DFunLike.coe Measure Set ENNReal Measure.instFunLike	—	ofReal_setIntegral_one_of_measure_ne_top measure_ne_top
ofReal_setIntegral_one_of_measure_ne_top 📖	mathematical	ENNReal DFunLike.coe Measure Set Measure.instFunLike Top.top instTopENNReal	ENNReal.ofReal integral Real Real.normedAddCommGroup NormedField.toNormedSpace Real.normedField Measure.restrict Real.instOne	—	NormOneClass.norm_one NormedDivisionRing.to_normOneClass CStarRing.norm_of_mem_unitary instCStarRingReal Real.instNontrivial SubmonoidClass.toOneMemClass Submonoid.instSubmonoidClass integral_const Real.instCompleteSpace Measure.restrict_apply Set.univ_inter mul_one ENNReal.ofReal_toReal lintegral_const one_mul setLIntegral_one
setIntegral_compl 📖	mathematical	MeasurableSet Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup	integral Measure.restrict Compl.compl Set Set.instCompl SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup	—	setIntegral_compl₀ MeasurableSet.nullMeasurableSet
setIntegral_compl₀ 📖	mathematical	NullMeasurableSet Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup	integral Measure.restrict Compl.compl Set Set.instCompl SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup	—	integral_add_compl₀ add_sub_cancel_left
setIntegral_congr_ae 📖	mathematical	MeasurableSet Filter.Eventually ae Measure Measure.instFunLike Measure.instOuterMeasureClass	integral Measure.restrict	—	Measure.instOuterMeasureClass integral_congr_ae ae_restrict_iff'
setIntegral_congr_ae₀ 📖	mathematical	NullMeasurableSet Filter.Eventually ae Measure Measure.instFunLike Measure.instOuterMeasureClass	integral Measure.restrict	—	Measure.instOuterMeasureClass integral_congr_ae ae_restrict_iff'₀
setIntegral_congr_fun 📖	mathematical	MeasurableSet Set.EqOn	integral Measure.restrict	—	setIntegral_congr_ae Filter.Eventually.of_forall Measure.instOuterMeasureClass
setIntegral_congr_fun₀ 📖	mathematical	NullMeasurableSet Set.EqOn	integral Measure.restrict	—	setIntegral_congr_ae₀ Filter.Eventually.of_forall Measure.instOuterMeasureClass
setIntegral_congr_set 📖	mathematical	Filter.EventuallyEq ae Measure Measure.instFunLike Measure.instOuterMeasureClass	integral Measure.restrict	—	Measure.instOuterMeasureClass Measure.restrict_congr_set
setIntegral_const 📖	mathematical	—	integral Measure.restrict Real 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 Real.instMonoid Module.toDistribMulAction Real.semiring ESeminormedAddCommMonoid.toAddCommMonoid NormedSpace.toModule Real.normedField Measure.real	—	integral_const measureReal_restrict_apply_univ
setIntegral_empty 📖	mathematical	—	integral Measure.restrict Set Set.instEmptyCollection NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup	—	Measure.restrict_empty integral_zero_measure
setIntegral_eq_integral_of_ae_compl_eq_zero 📖	mathematical	Filter.Eventually ae Measure Measure.instFunLike Measure.instOuterMeasureClass	integral Measure.restrict	—	Measure.instOuterMeasureClass setIntegral_univ setIntegral_eq_of_subset_of_ae_diff_eq_zero nullMeasurableSet_univ Set.subset_univ Filter.mp_mem Filter.univ_mem'
setIntegral_eq_integral_of_forall_compl_eq_zero 📖	mathematical	NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup	integral Measure.restrict	—	setIntegral_eq_integral_of_ae_compl_eq_zero Filter.Eventually.of_forall Measure.instOuterMeasureClass
setIntegral_eq_of_subset_of_ae_diff_eq_zero 📖	mathematical	NullMeasurableSet Set Set.instHasSubset Filter.Eventually ae Measure Measure.instFunLike Measure.instOuterMeasureClass	integral Measure.restrict	—	Measure.instOuterMeasureClass integral_congr_ae setIntegral_eq_of_subset_of_ae_diff_eq_zero_aux Filter.mp_mem ae_imp_of_ae_restrict Filter.univ_mem' Integrable.congr ae_restrict_of_ae_restrict_of_subset Filter.EventuallyEq.symm IntegrableOn.of_ae_diff_eq_zero PseudoEMetricSpace.pseudoMetrizableSpace integral_undef
setIntegral_eq_of_subset_of_ae_diff_eq_zero_aux 📖	mathematical	Set Set.instHasSubset Filter.Eventually ae Measure Measure.instFunLike Measure.instOuterMeasureClass StronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup IntegrableOn SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup	integral Measure.restrict	—	Measure.instOuterMeasureClass StronglyMeasurable.measurable PseudoEMetricSpace.pseudoMetrizableSpace MeasurableSingletonClass.measurableSet_singleton OpensMeasurableSpace.toMeasurableSingletonClass BorelSpace.opensMeasurable T5Space.toT1Space T6Space.toT5Space instT6SpaceOfMetrizableSpace EMetricSpace.metrizableSpace integral_inter_add_diff setIntegral_eq_zero_of_forall_eq_zero zero_add setIntegral_congr_set Filter.mp_mem Filter.univ_mem' IntegrableOn.mono le_rfl
setIntegral_eq_of_subset_of_forall_diff_eq_zero 📖	mathematical	MeasurableSet Set Set.instHasSubset NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup	integral Measure.restrict	—	setIntegral_eq_of_subset_of_ae_diff_eq_zero MeasurableSet.nullMeasurableSet Filter.Eventually.of_forall Measure.instOuterMeasureClass
setIntegral_eq_zero_iff_of_nonneg_ae 📖	mathematical	Filter.EventuallyLE Real Real.instLE ae Measure Measure.instFunLike Measure.instOuterMeasureClass Measure.restrict Pi.instZero Real.instZero IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing	integral Real.normedAddCommGroup NormedField.toNormedSpace Real.normedField Filter.EventuallyEq	—	Measure.instOuterMeasureClass integral_eq_zero_iff_of_nonneg_ae
setIntegral_eq_zero_of_ae_eq_zero 📖	mathematical	Filter.Eventually ae Measure Measure.instFunLike Measure.instOuterMeasureClass	integral Measure.restrict NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup	—	Measure.instOuterMeasureClass integral_eq_zero_of_ae Filter.EventuallyEq.eq_1 ae_restrict_iff StronglyMeasurable.measurableSet_eq_fun EMetricSpace.metrizableSpace stronglyMeasurable_zero Filter.mp_mem ae_imp_of_ae_restrict Filter.univ_mem' integral_congr_ae integral_undef Mathlib.Tactic.Contrapose.contrapose₄
setIntegral_eq_zero_of_forall_eq_zero 📖	mathematical	NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup	integral Measure.restrict	—	setIntegral_eq_zero_of_ae_eq_zero Filter.Eventually.of_forall Measure.instOuterMeasureClass
setIntegral_ge_of_const_le 📖	mathematical	MeasurableSet Preorder.toLE PartialOrder.toPreorder IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup	Real SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass ESeminormedAddMonoid.toAddMonoid ESeminormedAddCommMonoid.toESeminormedAddMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul Real.instMonoid Module.toDistribMulAction Real.semiring ESeminormedAddCommMonoid.toAddCommMonoid NormedSpace.toModule Real.normedField Measure.real integral Measure.restrict	—	setIntegral_const setIntegral_mono_on integrableOn_const enorm_ne_top
setIntegral_ge_of_const_le_real 📖	mathematical	MeasurableSet Real Real.instLE IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing	Real.instMul Measure.real integral Real.normedAddCommGroup NormedField.toNormedSpace Real.normedField Measure.restrict	—	mul_comm setIntegral_ge_of_const_le Real.instIsOrderedAddMonoid IsStrictOrderedModule.toIsOrderedModule IsStrictOrderedRing.toIsStrictOrderedModule Real.instIsStrictOrderedRing instClosedIciTopology HasSolidNorm.orderClosedTopology instHasSolidNormReal Real.instCompleteSpace
setIntegral_gt_gt 📖	mathematical	Real Real.instLE Real.instZero IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing setOf Real.instLT	Real.instMul Measure.real integral Real.normedAddCommGroup NormedField.toNormedSpace Real.normedField Measure.restrict	—	aestronglyMeasurable_const lt_of_le_of_lt setLIntegral_mono_ae AEStronglyMeasurable.enorm ae_of_all Measure.instOuterMeasureClass LE.le.trans le_of_lt Real.nnnorm_of_nonneg sub_pos IsRightCancelAdd.addRightStrictMono_of_addRightMono AddRightCancelSemigroup.toIsRightCancelAdd covariant_swap_add_of_covariant_add IsOrderedAddMonoid.toAddLeftMono Real.instIsOrderedAddMonoid smul_eq_mul setIntegral_const Real.instCompleteSpace integral_sub setIntegral_pos_iff_support_of_nonneg_ae Pi.zero_def Filter.EventuallyLE.eq_1 ae_restrict_iff₀ nullMeasurableSet_le BorelSpace.opensMeasurable Real.borelSpace instSecondCountableTopologyReal HasSolidNorm.orderClosedTopology instHasSolidNormReal aemeasurable_zero AEMeasurable.sub ContinuousSub.measurableSub₂ IsTopologicalAddGroup.to_continuousSub instIsTopologicalAddGroupReal AEStronglyMeasurable.aemeasurable PseudoEMetricSpace.pseudoMetrizableSpace aemeasurable_const Filter.Eventually.of_forall sub_nonneg Integrable.sub Set.inter_eq_self_of_subset_right ne_of_lt pos_iff_ne_zero ENNReal.instCanonicallyOrderedAdd
setIntegral_indicator 📖	mathematical	MeasurableSet	integral Measure.restrict Set.indicator NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup Set Set.instInter	—	integral_indicator Measure.restrict_restrict Set.inter_comm
setIntegral_indicatorConstLp 📖	mathematical	MeasurableSet	integral Measure.restrict AEEqFun.cast UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup AEEqFun AddSubgroup AEEqFun.instAddGroup NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup SetLike.instMembership AddSubgroup.instSetLike Lp indicatorConstLp Real SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass ESeminormedAddMonoid.toAddMonoid ESeminormedAddCommMonoid.toESeminormedAddMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul Real.instMonoid Module.toDistribMulAction Real.semiring ESeminormedAddCommMonoid.toAddCommMonoid NormedSpace.toModule Real.normedField Measure.real Set Set.instInter	—	setIntegral_congr_ae Filter.Eventually.mono Measure.instOuterMeasureClass indicatorConstLp_coeFn integral_indicator_const measureReal_restrict_apply
setIntegral_le_integral 📖	mathematical	Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Filter.EventuallyLE Preorder.toLE PartialOrder.toPreorder ae Measure Measure.instFunLike Measure.instOuterMeasureClass Pi.instZero NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup	integral Measure.restrict	—	Measure.instOuterMeasureClass integral_mono_measure Measure.restrict_le_self
setIntegral_le_nonneg 📖	mathematical	MeasurableSet StronglyMeasurable Real UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Integrable SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing	Real.instLE integral Real.normedAddCommGroup NormedField.toNormedSpace Real.normedField Measure.restrict setOf Real.instZero	—	integral_indicator StronglyMeasurable.measurableSet_le HasSolidNorm.orderClosedTopology instHasSolidNormReal Real.instIsOrderedAddMonoid PseudoEMetricSpace.pseudoMetrizableSpace stronglyMeasurable_const integral_mono IsStrictOrderedModule.toIsOrderedModule IsStrictOrderedRing.toIsStrictOrderedModule Real.instIsStrictOrderedRing instClosedIciTopology Integrable.indicator Set.indicator_le_indicator_nonneg
setIntegral_map 📖	mathematical	MeasurableSet AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup Measure.map AEMeasurable	integral Measure.restrict Set.preimage	—	Measure.restrict_map_of_aemeasurable integral_map AEMeasurable.mono_measure Measure.restrict_le_self AEStronglyMeasurable.mono_measure Measure.map_mono_of_aemeasurable
setIntegral_map_equiv 📖	mathematical	—	integral Measure.restrict Measure.map DFunLike.coe MeasurableEquiv EquivLike.toFunLike MeasurableEquiv.instEquivLike Set.preimage	—	MeasurableEmbedding.setIntegral_map MeasurableEquiv.measurableEmbedding
setIntegral_mono 📖	mathematical	IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Pi.hasLe Preorder.toLE PartialOrder.toPreorder	integral Measure.restrict	—	integral_mono
setIntegral_mono_ae 📖	mathematical	IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Filter.EventuallyLE Preorder.toLE PartialOrder.toPreorder ae Measure Measure.instFunLike Measure.instOuterMeasureClass	integral Measure.restrict	—	Measure.instOuterMeasureClass setIntegral_mono_ae_restrict ae_restrict_of_ae
setIntegral_mono_ae_restrict 📖	mathematical	IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Filter.EventuallyLE Preorder.toLE PartialOrder.toPreorder ae Measure Measure.instFunLike Measure.instOuterMeasureClass Measure.restrict	integral	—	Measure.instOuterMeasureClass integral_mono_ae integral_def
setIntegral_mono_on 📖	mathematical	IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup MeasurableSet Preorder.toLE PartialOrder.toPreorder	integral Measure.restrict	—	setIntegral_mono_ae_restrict Measure.instOuterMeasureClass ae_restrict_eq ae_of_all
setIntegral_mono_on_ae 📖	mathematical	IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup MeasurableSet Filter.Eventually ae Measure Measure.instFunLike Measure.instOuterMeasureClass	Preorder.toLE PartialOrder.toPreorder integral Measure.restrict	—	Measure.instOuterMeasureClass setIntegral_mono_ae_restrict Filter.EventuallyLE.eq_1 ae_restrict_iff'
setIntegral_mono_on_ae₀ 📖	mathematical	IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup NullMeasurableSet Filter.Eventually ae Measure Measure.instFunLike Measure.instOuterMeasureClass	Preorder.toLE PartialOrder.toPreorder integral Measure.restrict	—	Measure.instOuterMeasureClass setIntegral_congr_set Filter.EventuallyEq.symm NullMeasurableSet.toMeasurable_ae_eq setIntegral_mono_on_ae integrableOn_congr_set_ae measurableSet_toMeasurable Filter.mp_mem Filter.EventuallyEq.mem_iff Filter.univ_mem'
setIntegral_mono_on₀ 📖	mathematical	IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup NullMeasurableSet Preorder.toLE PartialOrder.toPreorder	integral Measure.restrict	—	setIntegral_mono_on_ae₀ Filter.Eventually.of_forall Measure.instOuterMeasureClass
setIntegral_mono_set 📖	mathematical	IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Filter.EventuallyLE Preorder.toLE PartialOrder.toPreorder ae Measure Measure.instFunLike Measure.instOuterMeasureClass Measure.restrict Pi.instZero NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup Prop.le	integral	—	Measure.instOuterMeasureClass integral_mono_measure Measure.restrict_mono_ae
setIntegral_neg_eq_setIntegral_nonpos 📖	mathematical	AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup	integral Measure.restrict setOf Preorder.toLT PartialOrder.toPreorder NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup Preorder.toLE	—	AEStronglyMeasurable.nullMeasurableSet_eq_fun EMetricSpace.metrizableSpace aestronglyMeasurable_zero integral_union_eq_left_of_ae Filter.mp_mem Measure.instOuterMeasureClass ae_restrict_mem₀ Filter.univ_mem'
setIntegral_nonneg 📖	mathematical	MeasurableSet Real Real.instLE Real.instZero	integral Real.normedAddCommGroup NormedField.toNormedSpace Real.normedField Measure.restrict	—	setIntegral_nonneg_of_ae_restrict Measure.instOuterMeasureClass ae_restrict_iff' ae_of_all
setIntegral_nonneg_ae 📖	mathematical	MeasurableSet Filter.Eventually ae Measure Measure.instFunLike Measure.instOuterMeasureClass	Real Real.instLE Real.instZero integral Real.normedAddCommGroup NormedField.toNormedSpace Real.normedField Measure.restrict	—	Measure.instOuterMeasureClass setIntegral_nonneg_of_ae_restrict Filter.EventuallyLE.eq_1 ae_restrict_iff'
setIntegral_nonneg_of_ae 📖	mathematical	Filter.EventuallyLE Real Real.instLE ae Measure Measure.instFunLike Measure.instOuterMeasureClass Pi.instZero Real.instZero	integral Real.normedAddCommGroup NormedField.toNormedSpace Real.normedField Measure.restrict	—	Measure.instOuterMeasureClass setIntegral_nonneg_of_ae_restrict ae_restrict_of_ae
setIntegral_nonneg_of_ae_restrict 📖	mathematical	Filter.EventuallyLE Real Real.instLE ae Measure Measure.instFunLike Measure.instOuterMeasureClass Measure.restrict Pi.instZero Real.instZero	integral Real.normedAddCommGroup NormedField.toNormedSpace Real.normedField	—	Measure.instOuterMeasureClass integral_nonneg_of_ae Real.instIsOrderedAddMonoid IsStrictOrderedModule.toIsOrderedModule IsStrictOrderedRing.toIsStrictOrderedModule Real.instIsStrictOrderedRing instClosedIciTopology HasSolidNorm.orderClosedTopology instHasSolidNormReal
setIntegral_nonpos 📖	mathematical	MeasurableSet Real Real.instLE Real.instZero	integral Real.normedAddCommGroup NormedField.toNormedSpace Real.normedField Measure.restrict	—	setIntegral_nonpos_ae ae_of_all Measure.instOuterMeasureClass
setIntegral_nonpos_ae 📖	mathematical	MeasurableSet Filter.Eventually ae Measure Measure.instFunLike Measure.instOuterMeasureClass	Real Real.instLE integral Real.normedAddCommGroup NormedField.toNormedSpace Real.normedField Measure.restrict Real.instZero	—	Measure.instOuterMeasureClass setIntegral_nonpos_of_ae_restrict Filter.EventuallyLE.eq_1 ae_restrict_iff'
setIntegral_nonpos_le 📖	mathematical	MeasurableSet StronglyMeasurable Real UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace Integrable SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing	Real.instLE integral Real.normedAddCommGroup NormedField.toNormedSpace Real.normedField Measure.restrict setOf Real.instZero	—	integral_indicator StronglyMeasurable.measurableSet_le HasSolidNorm.orderClosedTopology instHasSolidNormReal Real.instIsOrderedAddMonoid PseudoEMetricSpace.pseudoMetrizableSpace stronglyMeasurable_const integral_mono IsStrictOrderedModule.toIsOrderedModule IsStrictOrderedRing.toIsStrictOrderedModule Real.instIsStrictOrderedRing instClosedIciTopology Integrable.indicator Set.indicator_nonpos_le_indicator
setIntegral_nonpos_of_ae 📖	mathematical	Filter.EventuallyLE Real Real.instLE ae Measure Measure.instFunLike Measure.instOuterMeasureClass Pi.instZero Real.instZero	integral Real.normedAddCommGroup NormedField.toNormedSpace Real.normedField Measure.restrict	—	Measure.instOuterMeasureClass setIntegral_nonpos_of_ae_restrict ae_restrict_of_ae
setIntegral_nonpos_of_ae_restrict 📖	mathematical	Filter.EventuallyLE Real Real.instLE ae Measure Measure.instFunLike Measure.instOuterMeasureClass Measure.restrict Pi.instZero Real.instZero	integral Real.normedAddCommGroup NormedField.toNormedSpace Real.normedField	—	Measure.instOuterMeasureClass integral_nonpos_of_ae Real.instIsOrderedAddMonoid IsStrictOrderedModule.toIsOrderedModule IsStrictOrderedRing.toIsStrictOrderedModule Real.instIsStrictOrderedRing instClosedIciTopology HasSolidNorm.orderClosedTopology instHasSolidNormReal
setIntegral_one_eq_measureReal 📖	mathematical	—	integral Real Real.normedAddCommGroup NormedField.toNormedSpace Real.normedField Measure.restrict Real.instOne Measure.real	—	integral_const Real.instCompleteSpace measureReal_restrict_apply Set.univ_inter mul_one
setIntegral_pos_iff_support_of_nonneg_ae 📖	mathematical	Filter.EventuallyLE Real Real.instLE ae Measure Measure.instFunLike Measure.instOuterMeasureClass Measure.restrict Pi.instZero Real.instZero IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing	Real.instLT integral Real.normedAddCommGroup NormedField.toNormedSpace Real.normedField ENNReal Preorder.toLT PartialOrder.toPreorder ENNReal.instPartialOrder instZeroENNReal DFunLike.coe Set Set.instInter Function.support	—	Measure.instOuterMeasureClass integral_pos_iff_support_of_nonneg_ae Measure.restrict_apply₀ Function.support_eq_preimage AEMeasurable.nullMeasurable AEStronglyMeasurable.aemeasurable PseudoEMetricSpace.pseudoMetrizableSpace Real.borelSpace Integrable.aestronglyMeasurable MeasurableSet.compl MeasurableSingletonClass.measurableSet_singleton instMeasurableSingletonClassOfMeasurableEq instMeasurableEqOfSecondCountableTopologyOfT2Space BorelSpace.opensMeasurable instSecondCountableTopologyReal TopologicalSpace.t2Space_of_metrizableSpace EMetricSpace.metrizableSpace
setIntegral_support 📖	mathematical	—	integral Measure.restrict Function.support NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup	—	setIntegral_univ setIntegral_eq_of_subset_of_forall_diff_eq_zero MeasurableSet.univ Set.subset_univ Function.notMem_support Set.notMem_of_mem_diff
setIntegral_trim 📖	mathematical	MeasurableSpace MeasurableSpace.instLE StronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup MeasurableSet	integral Measure.restrict Measure.trim	—	integral_trim restrict_trim
setIntegral_tsupport 📖	mathematical	—	integral Measure.restrict tsupport NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup	—	setIntegral_univ setIntegral_eq_of_subset_of_forall_diff_eq_zero MeasurableSet.univ Set.subset_univ image_eq_zero_of_notMem_tsupport Set.notMem_of_mem_diff
setIntegral_union 📖	mathematical	Disjoint Set OmegaCompletePartialOrder.toPartialOrder CompleteLattice.instOmegaCompletePartialOrder CompleteBooleanAlgebra.toCompleteLattice CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra Set.instCompleteAtomicBooleanAlgebra HeytingAlgebra.toOrderBot Order.Frame.toHeytingAlgebra CompleteDistribLattice.toFrame CompleteBooleanAlgebra.toCompleteDistribLattice MeasurableSet IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup	integral Measure.restrict Set.instUnion AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid	—	integral_union_ae Disjoint.aedisjoint MeasurableSet.nullMeasurableSet
setIntegral_univ 📖	mathematical	—	integral Measure.restrict Set.univ	—	Measure.restrict_univ
tendsto_setIntegral_of_antitone 📖	mathematical	MeasurableSet Antitone Set PartialOrder.toPreorder OmegaCompletePartialOrder.toPartialOrder CompleteLattice.instOmegaCompletePartialOrder CompleteBooleanAlgebra.toCompleteLattice CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra Set.instCompleteAtomicBooleanAlgebra IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup	Filter.Tendsto integral Measure.restrict Filter.atTop nhds Set.iInter	—	Filter.Tendsto.of_neBot_imp Filter.atTop_neBot_iff Filter.Tendsto.sub IsTopologicalAddGroup.to_continuousSub SeminormedAddCommGroup.toIsTopologicalAddGroup tendsto_const_nhds tendsto_setIntegral_of_monotone Set.diff_subset_diff_right Set.diff_iInter IntegrableOn.mono_set Set.diff_subset integral_diff MeasurableSet.iInter_of_antitone Set.iInter_subset sub_sub_cancel Filter.Tendsto.congr' Filter.Eventually.mono Filter.eventually_ge_atTop
tendsto_setIntegral_of_monotone 📖	mathematical	MeasurableSet Monotone Set PartialOrder.toPreorder OmegaCompletePartialOrder.toPartialOrder CompleteLattice.instOmegaCompletePartialOrder CompleteBooleanAlgebra.toCompleteLattice CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra Set.instCompleteAtomicBooleanAlgebra IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Set.iUnion	Filter.Tendsto integral Measure.restrict Filter.atTop nhds	—	Filter.Tendsto.of_neBot_imp Filter.atTop_neBot_iff MeasurableSet.iUnion_of_monotone Set.subset_iUnion Filter.HasBasis.tendsto_right_iff Metric.nhds_basis_closedBall CanLift.prf NNReal.canLift LT.lt.le tendsto_measure_iUnion_atTop ENNReal.Icc_mem_nhds LT.lt.ne withDensity_apply LT.lt.ne' ENNReal.coe_pos Filter.mp_mem Filter.univ_mem' mem_closedBall_iff_norm' integral_diff coe_nnnorm NNReal.coe_le_coe ENNReal.coe_le_coe LE.le.trans enorm_integral_le_lintegral_enorm MeasurableSet.diff measure_diff MeasurableSet.nullMeasurableSet LE.le.trans_lt ENNReal.add_lt_top ENNReal.coe_lt_top tsub_le_iff_tsub_le ENNReal.instOrderedSub

MeasureTheory.Integrable

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
measure_le_integral 📖	mathematical	MeasureTheory.Integrable Real UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing Filter.EventuallyLE Real.instLE MeasureTheory.ae MeasureTheory.Measure MeasureTheory.Measure.instFunLike MeasureTheory.Measure.instOuterMeasureClass Pi.instZero Real.instZero Real.instOne	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder DFunLike.coe Set ENNReal.ofReal MeasureTheory.integral Real.normedAddCommGroup NormedField.toNormedSpace Real.normedField	—	MeasureTheory.Measure.instOuterMeasureClass MeasureTheory.ofReal_integral_eq_lintegral_ofReal MeasureTheory.meas_le_lintegral₀ Measurable.comp_aemeasurable Continuous.measurable BorelSpace.opensMeasurable Real.borelSpace ENNReal.borelSpace ENNReal.continuous_ofReal MeasureTheory.AEStronglyMeasurable.aemeasurable PseudoEMetricSpace.pseudoMetrizableSpace ENNReal.ofReal_one ENNReal.ofReal_le_ofReal
simpleFunc_mul 📖	mathematical	MeasureTheory.Integrable Real UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing	Pi.instMul Real.instMul DFunLike.coe MeasureTheory.SimpleFunc MeasureTheory.SimpleFunc.instFunLike	—	MeasureTheory.SimpleFunc.induction Set.piecewise_eq_indicator Set.indicator_of_mem Set.indicator_of_notMem MulZeroClass.zero_mul MeasureTheory.integrable_indicator_iff integrableOn smul Real.isBoundedSMul congr add IsTopologicalSemiring.toContinuousAdd IsTopologicalRing.toIsTopologicalSemiring instIsTopologicalRingReal MeasureTheory.Measure.instOuterMeasureClass MeasureTheory.SimpleFunc.coe_add add_mul Distrib.rightDistribClass Filter.EventuallyEq.refl
simpleFunc_mul' 📖	mathematical	MeasurableSpace MeasurableSpace.instLE MeasureTheory.Integrable Real UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing	Pi.instMul Real.instMul DFunLike.coe MeasureTheory.SimpleFunc MeasureTheory.SimpleFunc.instFunLike	—	MeasureTheory.SimpleFunc.coe_toLargerSpace_eq simpleFunc_mul

MeasureTheory.MeasurePreserving

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
setIntegral_image_emb 📖	mathematical	MeasureTheory.MeasurePreserving MeasurableEmbedding	MeasureTheory.integral MeasureTheory.Measure.restrict Set.image	—	integral_comp restrict_image_emb
setIntegral_preimage_emb 📖	mathematical	MeasureTheory.MeasurePreserving MeasurableEmbedding	MeasureTheory.integral MeasureTheory.Measure.restrict Set.preimage	—	integral_comp restrict_preimage_emb

Topology.IsClosedEmbedding

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
setIntegral_map 📖	mathematical	Topology.IsClosedEmbedding	MeasureTheory.integral MeasureTheory.Measure.restrict MeasureTheory.Measure.map Set.preimage	—	MeasurableEmbedding.setIntegral_map measurableEmbedding

(root)

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
continuousOn_integral_bilinear_of_locally_integrable_of_compact_support 📖	mathematical	IsCompact ContinuousOn instTopologicalSpaceProd UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SProd.sprod Set Set.instSProd Set.univ NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup MeasureTheory.IntegrableOn SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup	MeasureTheory.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.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	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul ContinuousOn.comp_continuous Continuous.prodMk_right Metric.continuousWithinAt_iff' RingHomIsometric.ids MeasureTheory.integral_mul_const exists_pos_mul_lt Real.instIsStrictOrderedRing IsCompact.mem_uniformity_of_prod ContinuousOn.mono Set.prod_mono_right Set.subset_univ Metric.dist_mem_uniformity dist_eq_norm ContinuousOn.prodMk continuousOn_const continuousOn_id' ContinuousOn.comp IsCompact.exists_bound_of_continuousOn LE.le.trans le_max_left norm_zero MeasureTheory.Integrable.mul_const MeasureTheory.Integrable.const_mul MeasureTheory.Integrable.norm MeasureTheory.Integrable.mono' ContinuousLinearMap.aestronglyMeasurable_comp₂ MeasureTheory.Integrable.aestronglyMeasurable MeasureTheory.StronglyMeasurable.aestronglyMeasurable Continuous.stronglyMeasurable_of_support_subset_isCompact PseudoEMetricSpace.pseudoMetrizableSpace Function.support_subset_iff' Filter.Eventually.of_forall MeasureTheory.Measure.instOuterMeasureClass ContinuousLinearMap.le_opNorm₂ mul_le_mul_of_nonneg_left IsOrderedRing.toPosMulMono Real.instIsOrderedRing mul_nonneg norm_nonneg Filter.mp_mem self_mem_nhdsWithin Filter.univ_mem' MeasureTheory.setIntegral_eq_integral_of_forall_compl_eq_zero map_zero AddMonoidHomClass.toZeroHomClass DistribMulActionSemiHomClass.toAddMonoidHomClass SemilinearMapClass.distribMulActionSemiHomClass ContinuousSemilinearMapClass.toSemilinearMapClass ContinuousLinearMap.continuousSemilinearMapClass MeasureTheory.integral_sub MeasureTheory.norm_integral_le_integral_norm MeasureTheory.integral_mono_of_nonneg Real.instIsOrderedAddMonoid IsStrictOrderedModule.toIsOrderedModule IsStrictOrderedRing.toIsStrictOrderedModule instClosedIciTopology HasSolidNorm.orderClosedTopology instHasSolidNormReal map_sub le_of_lt sub_self
continuousOn_integral_of_compact_support 📖	mathematical	IsCompact ContinuousOn instTopologicalSpaceProd UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SProd.sprod Set Set.instSProd Set.univ NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup	MeasureTheory.integral	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul IsScalarTower.left one_smul continuousOn_integral_bilinear_of_locally_integrable_of_compact_support MeasureTheory.integrableOn_const IsCompact.measure_ne_top enorm_ne_top
continuous_parametric_integral_of_continuous 📖	mathematical	Continuous instTopologicalSpaceProd UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup IsCompact	MeasureTheory.integral MeasureTheory.Measure.restrict	—	continuous_iff_continuousAt WeaklyLocallyCompactSpace.exists_compact_mem_nhds instWeaklyLocallyCompactSpaceOfLocallyCompactSpace IsCompact.bddAbove_image instClosedIciTopology HasSolidNorm.orderClosedTopology instHasSolidNormReal Real.instIsOrderedAddMonoid IsCompact.prod Continuous.continuousOn Continuous.norm MeasureTheory.continuousAt_of_dominated Filter.univ_mem' Continuous.aestronglyMeasurable PseudoEMetricSpace.pseudoMetrizableSpace Continuous.comp' Continuous.prodMk continuous_const continuous_id' Filter.mp_mem MeasureTheory.Measure.instOuterMeasureClass MeasureTheory.ae_restrict_iff IsClosed.measurableSet isClosed_le Set.mem_image_of_mem Set.mk_mem_prod MeasureTheory.integrableOn_const IsCompact.measure_ne_top isFiniteMeasureOnCompacts_of_isLocallyFiniteMeasure enorm_ne_top Continuous.continuousAt
measure_le_lintegral_thickenedIndicator 📖	mathematical	MeasurableSet Real Real.instLT Real.instZero	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder DFunLike.coe MeasureTheory.Measure Set MeasureTheory.Measure.instFunLike MeasureTheory.lintegral ENNReal.ofNNReal BoundedContinuousFunction NNReal UniformSpace.toTopologicalSpace PseudoEMetricSpace.toUniformSpace instPseudoMetricSpaceNNReal BoundedContinuousFunction.instFunLike thickenedIndicator	—	ENNReal.coe_toNNReal LT.lt.ne thickenedIndicatorAux_lt_top measure_le_lintegral_thickenedIndicatorAux
measure_le_lintegral_thickenedIndicatorAux 📖	mathematical	MeasurableSet	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder DFunLike.coe MeasureTheory.Measure Set MeasureTheory.Measure.instFunLike MeasureTheory.lintegral thickenedIndicatorAux	—	MeasureTheory.lintegral_indicator MeasureTheory.lintegral_one MeasureTheory.Measure.restrict_apply Set.univ_inter MeasureTheory.lintegral_mono indicator_le_thickenedIndicatorAux

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