DivergenceTheorem

📁 Source: Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean

Statistics

Metric	Count
Definitions	0
Theorems integral2_divergence_prod_of_hasFDerivAt, integral2_divergence_prod_of_hasFDerivAt_off_countable, integral_divergence_of_hasFDerivAt_off_countable, integral_divergence_of_hasFDerivAt_off_countable', integral_divergence_of_hasFDerivAt_off_countable_of_equiv, integral_divergence_prod_Icc_of_hasFDerivAt_of_le, integral_divergence_prod_Icc_of_hasFDerivAt_off_countable_of_le, integral_eq_of_hasDerivAt_off_countable, integral_eq_of_hasDerivAt_off_countable_of_le	9
Total	9

MeasureTheory

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
integral2_divergence_prod_of_hasFDerivAt 📖	mathematical	ContinuousOn Real instTopologicalSpaceProd UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SProd.sprod Set Set.instSProd Set.uIcc Real.lattice HasFDerivAt DenselyNormedField.toNontriviallyNormedField Real.denselyNormedField Prod.instAddCommGroup Real.instAddCommGroup Prod.instModule Real.semiring Real.instAddCommMonoid NormedSpace.toModule Real.normedField NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal SeminormedAddCommGroup.toAddCommGroup IntegrableOn MeasureSpace.toMeasurableSpace Measure.prod.measureSpace Real.measureSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid DFunLike.coe ContinuousLinearMap RingHom.id Semiring.toNonAssocSemiring Prod.instAddCommMonoid ContinuousLinearMap.funLike Real.instOne Real.instZero MeasureSpace.volume	intervalIntegral SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup	—	integral2_divergence_prod_of_hasFDerivAt_off_countable Set.countable_empty
integral2_divergence_prod_of_hasFDerivAt_off_countable 📖	mathematical	ContinuousOn Real instTopologicalSpaceProd UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SProd.sprod Set Set.instSProd Set.uIcc Real.lattice HasFDerivAt DenselyNormedField.toNontriviallyNormedField Real.denselyNormedField Prod.instAddCommGroup Real.instAddCommGroup Prod.instModule Real.semiring Real.instAddCommMonoid NormedSpace.toModule Real.normedField NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal SeminormedAddCommGroup.toAddCommGroup IntegrableOn MeasureSpace.toMeasurableSpace Measure.prod.measureSpace Real.measureSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid DFunLike.coe ContinuousLinearMap RingHom.id Semiring.toNonAssocSemiring Prod.instAddCommMonoid ContinuousLinearMap.funLike Real.instOne Real.instZero MeasureSpace.volume	intervalIntegral SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup	—	intervalIntegral.integral_of_le setIntegral_congr_set Ioc_ae_eq_Icc Real.noAtoms_volume setIntegral_prod instSFiniteOfSigmaFinite Measure.IsAddHaarMeasure.sigmaFinite instIsAddHaarMeasureVolume FiniteDimensional.rclike_to_real Real.borelSpace SeparableWeaklyLocallyCompactAddGroup.sigmaCompactSpace instIsTopologicalAddGroupReal TopologicalSpace.SecondCountableTopology.to_separableSpace instSecondCountableTopologyReal instWeaklyLocallyCompactSpaceOfLocallyCompactSpace locallyCompact_of_proper instProperSpaceReal Set.uIcc_of_le Set.Icc_prod_Icc integral_divergence_prod_Icc_of_hasFDerivAt_off_countable_of_le min_eq_left max_eq_right intervalIntegral.integral_symm intervalIntegral.integral_neg max_comm min_comm Set.uIcc_comm le_of_not_ge Mathlib.Tactic.Abel.subst_into_negg Mathlib.Tactic.Abel.unfold_sub Mathlib.Tactic.Abel.subst_into_addg Mathlib.Tactic.Abel.term_atomg Mathlib.Tactic.Abel.term_neg neg_zero Mathlib.Tactic.Abel.term_add_constg zero_add Mathlib.Meta.NormNum.IsNat.to_eq Mathlib.Meta.NormNum.IsInt.to_isNat Mathlib.Meta.NormNum.isInt_neg Mathlib.Meta.NormNum.IsNat.to_isInt Mathlib.Meta.NormNum.isNat_ofNat Mathlib.Tactic.Abel.const_add_termg add_zero
integral_divergence_of_hasFDerivAt_off_countable 📖	mathematical	Pi.hasLe Real Real.instLE ContinuousOn Pi.topologicalSpace UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup Set.Icc Pi.preorder Real.instPreorder HasFDerivAt DenselyNormedField.toNontriviallyNormedField Real.denselyNormedField Pi.addCommGroup Real.instAddCommGroup Pi.Function.module Real.semiring Real.instAddCommMonoid NormedSpace.toModule Real.normedField NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal NormedAddCommGroup.toAddCommGroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid IntegrableOn MeasureSpace.toMeasurableSpace MeasureSpace.pi Fin.fintype Real.measureSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Finset.sum Finset.univ DFunLike.coe ContinuousLinearMap RingHom.id Semiring.toNonAssocSemiring Pi.addCommMonoid ContinuousLinearMap.funLike Pi.single Real.instZero Real.instOne MeasureSpace.volume	integral Measure.restrict SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup Fin.succAbove Fin.insertNth	—	em volume_pi setIntegral_congr_set Measure.univ_pi_Ioc_ae_eq_Icc Measure.IsAddHaarMeasure.sigmaFinite instIsAddHaarMeasureVolume FiniteDimensional.rclike_to_real Real.borelSpace SeparableWeaklyLocallyCompactAddGroup.sigmaCompactSpace instIsTopologicalAddGroupReal TopologicalSpace.SecondCountableTopology.to_separableSpace instSecondCountableTopologyReal instWeaklyLocallyCompactSpaceOfLocallyCompactSpace locallyCompact_of_proper instProperSpaceReal Real.noAtoms_volume Set.Ioc_eq_empty Eq.not_lt Set.univ_pi_eq_empty setIntegral_empty Finset.sum_eq_zero eq_or_ne sub_self Fin.exists_succAbove_eq Measure.instOuterMeasureClass ae_eq_empty Real.volume_Icc_pi Finset.prod_eq_zero Finset.mem_univ ENNReal.ofReal_zero LE.le.lt_of_ne
integral_divergence_of_hasFDerivAt_off_countable' 📖	mathematical	Pi.hasLe Real Real.instLE ContinuousOn Pi.topologicalSpace UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup Set.Icc Pi.preorder Real.instPreorder HasFDerivAt DenselyNormedField.toNontriviallyNormedField Real.denselyNormedField Pi.addCommGroup Real.instAddCommGroup Pi.Function.module Real.semiring Real.instAddCommMonoid NormedSpace.toModule Real.normedField NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal SeminormedAddCommGroup.toAddCommGroup IntegrableOn MeasureSpace.toMeasurableSpace MeasureSpace.pi Fin.fintype Real.measureSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Finset.sum ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid Finset.univ DFunLike.coe ContinuousLinearMap RingHom.id Semiring.toNonAssocSemiring Pi.addCommMonoid ContinuousLinearMap.funLike Pi.single Real.instZero Real.instOne MeasureSpace.volume	integral Measure.restrict SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup Fin.succAbove Fin.insertNth	—	integral_divergence_of_hasFDerivAt_off_countable continuousOn_pi hasFDerivAt_pi Finite.of_fintype
integral_divergence_of_hasFDerivAt_off_countable_of_equiv 📖	mathematical	Pi.hasLe Real Real.instLE DFunLike.coe ContinuousLinearEquiv Real.semiring RingHom.id Semiring.toNonAssocSemiring RingHomInvPair.ids UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid Pi.topologicalSpace Real.pseudoMetricSpace Pi.addCommMonoid Real.instAddCommMonoid NormedSpace.toModule Real.normedField Pi.Function.module NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal EquivLike.toFunLike ContinuousLinearEquiv.equivLike Preorder.toLE MeasurePreserving MeasureSpace.toMeasurableSpace MeasureSpace.pi Fin.fintype Real.measureSpace MeasureSpace.volume ContinuousOn Set.Icc HasFDerivAt DenselyNormedField.toNontriviallyNormedField Real.denselyNormedField SeminormedAddCommGroup.toAddCommGroup Finset.sum Finset.univ ContinuousLinearMap ContinuousLinearMap.funLike ContinuousLinearEquiv.symm Pi.single Real.instZero Real.instOne IntegrableOn SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup	integral Measure.restrict SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup Pi.preorder Real.instPreorder Fin.succAbove Fin.insertNth	—	RingHomInvPair.ids Homeomorph.measurableEmbedding Pi.borelSpace instCountableFin instSecondCountableTopologyReal Real.borelSpace Set.ext ContinuousLinearEquiv.symm_preimage_preimage MeasurePreserving.setIntegral_preimage_emb Finset.sum_congr ContinuousLinearEquiv.symm_apply_apply integral_divergence_of_hasFDerivAt_off_countable' RingHomCompTriple.ids Set.Countable.preimage ContinuousLinearEquiv.injective ContinuousOn.comp ContinuousLinearEquiv.continuousOn Eq.subset Set.instReflSubset HasFDerivAt.comp preimage_interior_subset_interior_preimage ContinuousLinearEquiv.continuous interior_pi_set Set.finite_univ Finite.of_fintype interior_Icc instOrderTopologyReal LinearOrderedSemiField.toDenselyOrdered PosMulReflectLE.toPosMulReflectLT PosMulStrictMono.toPosMulReflectLE IsStrictOrderedRing.toPosMulStrictMono Real.instIsStrictOrderedRing instNoMinOrderOfNontrivial Real.instIsOrderedRing Real.instNontrivial instNoMaxOrderOfNontrivial ContinuousLinearEquiv.apply_symm_apply ContinuousLinearEquiv.hasFDerivAt MeasurePreserving.integrableOn_comp_preimage
integral_divergence_prod_Icc_of_hasFDerivAt_of_le 📖	mathematical	Real Prod.instLE_mathlib Real.instLE ContinuousOn instTopologicalSpaceProd UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup Set.Icc Prod.instPreorder Real.instPreorder HasFDerivAt DenselyNormedField.toNontriviallyNormedField Real.denselyNormedField Prod.instAddCommGroup Real.instAddCommGroup Prod.instModule Real.semiring Real.instAddCommMonoid NormedSpace.toModule Real.normedField NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal SeminormedAddCommGroup.toAddCommGroup IntegrableOn MeasureSpace.toMeasurableSpace Measure.prod.measureSpace Real.measureSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid DFunLike.coe ContinuousLinearMap RingHom.id Semiring.toNonAssocSemiring Prod.instAddCommMonoid ContinuousLinearMap.funLike Real.instOne Real.instZero MeasureSpace.volume	integral Measure.restrict SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup intervalIntegral	—	integral_divergence_prod_Icc_of_hasFDerivAt_off_countable_of_le Set.diff_empty
integral_divergence_prod_Icc_of_hasFDerivAt_off_countable_of_le 📖	mathematical	Real Prod.instLE_mathlib Real.instLE ContinuousOn instTopologicalSpaceProd UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup Set.Icc Prod.instPreorder Real.instPreorder HasFDerivAt DenselyNormedField.toNontriviallyNormedField Real.denselyNormedField Prod.instAddCommGroup Real.instAddCommGroup Prod.instModule Real.semiring Real.instAddCommMonoid NormedSpace.toModule Real.normedField NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal SeminormedAddCommGroup.toAddCommGroup IntegrableOn MeasureSpace.toMeasurableSpace Measure.prod.measureSpace Real.measureSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid DFunLike.coe ContinuousLinearMap RingHom.id Semiring.toNonAssocSemiring Prod.instAddCommMonoid ContinuousLinearMap.funLike Real.instOne Real.instZero MeasureSpace.volume	integral Measure.restrict SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup intervalIntegral	—	RingHomInvPair.ids integral_divergence_of_hasFDerivAt_off_countable_of_equiv Prod.borelSpace Real.borelSpace secondCountableTopologyEither_of_right instSecondCountableTopologyReal OrderIso.le_iff_le MeasurePreserving.symm volume_preserving_finTwoArrow Measure.IsAddHaarMeasure.sigmaFinite instIsAddHaarMeasureVolume FiniteDimensional.rclike_to_real SeparableWeaklyLocallyCompactAddGroup.sigmaCompactSpace instIsTopologicalAddGroupReal TopologicalSpace.SecondCountableTopology.to_separableSpace instWeaklyLocallyCompactSpaceOfLocallyCompactSpace locallyCompact_of_proper instProperSpaceReal interior_Icc instOrderTopologyReal LinearOrderedSemiField.toDenselyOrdered PosMulReflectLE.toPosMulReflectLT PosMulStrictMono.toPosMulReflectLE IsStrictOrderedRing.toPosMulStrictMono Real.instIsStrictOrderedRing instNoMinOrderOfNontrivial Real.instIsOrderedRing Real.instNontrivial instNoMaxOrderOfNontrivial interior_prod_eq Set.Icc_prod_eq Fin.sum_univ_two OrderIso.preimage_Icc MeasurePreserving.setIntegral_preimage_emb volume_preserving_funUnique MeasurableEquiv.measurableEmbedding Finset.sum_congr intervalIntegral.integral_of_le setIntegral_congr_set Ioc_ae_eq_Icc Real.noAtoms_volume Mathlib.Tactic.Abel.subst_into_addg Mathlib.Tactic.Abel.unfold_sub Mathlib.Tactic.Abel.term_atomg Mathlib.Tactic.Abel.subst_into_negg Mathlib.Tactic.Abel.term_neg neg_zero Mathlib.Tactic.Abel.term_add_constg zero_add Mathlib.Tactic.Abel.const_add_termg add_zero
integral_eq_of_hasDerivAt_off_countable 📖	mathematical	ContinuousOn Real UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup Set.uIcc Real.lattice HasDerivAt DenselyNormedField.toNontriviallyNormedField Real.denselyNormedField NormedAddCommGroup.toAddCommGroup NormedSpace.toModule Real.normedField IsBoundedSMul.continuousSMul 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 MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField Module.toDistribMulAction AddCommGroup.toAddCommMonoid NormedSpace.toIsBoundedSMul IntervalIntegrable NormedAddGroup.toENormedAddMonoid NormedAddCommGroup.toNormedAddGroup MeasureSpace.volume Real.measureSpace	intervalIntegral SubNegMonoid.toSub NormedAddGroup.toAddGroup	—	IsBoundedSMul.continuousSMul NormedSpace.toIsBoundedSMul le_total integral_eq_of_hasDerivAt_off_countable_of_le Set.uIcc_of_le min_eq_left max_eq_right intervalIntegral.integral_symm neg_eq_iff_eq_neg neg_sub Set.uIcc_of_ge min_eq_right max_eq_left IntervalIntegrable.symm
integral_eq_of_hasDerivAt_off_countable_of_le 📖	mathematical	Real Real.instLE ContinuousOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup Set.Icc Real.instPreorder HasDerivAt DenselyNormedField.toNontriviallyNormedField Real.denselyNormedField NormedAddCommGroup.toAddCommGroup NormedSpace.toModule Real.normedField IsBoundedSMul.continuousSMul 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 MonoidWithZero.toMonoid Semiring.toMonoidWithZero DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField Module.toDistribMulAction AddCommGroup.toAddCommMonoid NormedSpace.toIsBoundedSMul IntervalIntegrable NormedAddGroup.toENormedAddMonoid NormedAddCommGroup.toNormedAddGroup MeasureSpace.volume Real.measureSpace	intervalIntegral SubNegMonoid.toSub NormedAddGroup.toAddGroup	—	IsBoundedSMul.continuousSMul NormedSpace.toIsBoundedSMul RingHomInvPair.ids IsScalarTower.left intervalIntegral.integral_of_le setIntegral_congr_set Ioc_ae_eq_Icc Real.noAtoms_volume integral_divergence_of_hasFDerivAt_off_countable_of_equiv Real.borelSpace OrderIso.le_iff_le MeasurePreserving.symm volume_preserving_funUnique instOrderTopologyReal LinearOrderedSemiField.toDenselyOrdered PosMulReflectLE.toPosMulReflectLT PosMulStrictMono.toPosMulReflectLE IsStrictOrderedRing.toPosMulStrictMono Real.instIsStrictOrderedRing instNoMinOrderOfNontrivial Real.instIsOrderedRing Real.instNontrivial instNoMaxOrderOfNontrivial Fin.sum_univ_one Pi.single_eq_same one_smul IntegrableOn.congr_set_ae intervalIntegrable_iff_integrableOn_Ioc_of_le PseudoEMetricSpace.pseudoMetrizableSpace Filter.EventuallyEq.symm Measure.instOuterMeasureClass Fin.isEmpty' Finset.sum_congr Finset.univ_unique IsScalarTower.right Measure.pi_of_empty Unique.instSubsingleton Set.Icc_self Measure.restrict_singleton Measure.dirac_apply' Pi.instMeasurableSingletonClass instCountableFin instMeasurableSingletonClassOfMeasurableEq instMeasurableEqOfSecondCountableTopologyOfT2Space BorelSpace.opensMeasurable instSecondCountableTopologyReal TopologicalSpace.t2Space_of_metrizableSpace EMetricSpace.metrizableSpace Set.indicator_of_mem integral_dirac' Finset.sum_sub_distrib Finset.sum_singleton Fin.insertNth_apply_same

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