Basic

📁 Source: Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean

Statistics

Metric	Count
Definitions	0
Theorems memLp_top_of_hasCompactSupport, eLpNormEssSup_map_measure, eLpNorm_map_measure, memLp_map_measure_iff, memLp_map_measure_iff, eLpNorm_compMeasurePreserving, ae_eq, comp_measurePreserving, comp_of_map, congr_enorm, congr_norm, eLpNorm_lt_top, eLpNorm_ne_top, enorm, left_of_add_measure, mono, mono', mono'_enorm, mono_measure, neg, norm, of_bound, of_discrete, of_enorm_bound, of_le, of_le_enorm, of_measure_le_smul, restrict, right_of_add_measure, smul_measure, zero, zero', ae_bdd_liminf_atTop_of_eLpNorm_bdd, ae_bdd_liminf_atTop_rpow_of_eLpNorm_bdd, ae_eq_zero_of_eLpNorm'_eq_zero, ae_le_eLpNormEssSup, eLpNorm'_congr_ae, eLpNorm'_congr_enorm_ae, eLpNorm'_congr_nnnorm_ae, eLpNorm'_congr_norm_ae, eLpNorm'_const, eLpNorm'_const', eLpNorm'_const_of_isProbabilityMeasure, eLpNorm'_enorm, eLpNorm'_enorm_rpow, eLpNorm'_eq_zero_iff, eLpNorm'_eq_zero_of_ae_eq_zero, eLpNorm'_eq_zero_of_ae_zero, eLpNorm'_eq_zero_of_ae_zero', eLpNorm'_exponent_zero, eLpNorm'_measure_zero_of_exponent_zero, eLpNorm'_measure_zero_of_neg, eLpNorm'_measure_zero_of_pos, eLpNorm'_mono_ae, eLpNorm'_mono_enorm_ae, eLpNorm'_mono_measure, eLpNorm'_mono_nnnorm_ae, eLpNorm'_neg, eLpNorm'_norm, eLpNorm'_norm_rpow, eLpNorm'_smul_measure, eLpNorm'_zero, eLpNorm'_zero', eLpNormEssSup_congr_ae, eLpNormEssSup_const, eLpNormEssSup_count, eLpNormEssSup_ennreal_smul_measure, eLpNormEssSup_eq_iSup, eLpNormEssSup_eq_zero_iff, eLpNormEssSup_le_of_ae_bound, eLpNormEssSup_le_of_ae_enorm_bound, eLpNormEssSup_le_of_ae_nnnorm_bound, eLpNormEssSup_lt_top_iff_isBoundedUnder, eLpNormEssSup_lt_top_of_ae_bound, eLpNormEssSup_lt_top_of_ae_enorm_bound, eLpNormEssSup_lt_top_of_ae_nnnorm_bound, eLpNormEssSup_map_measure, eLpNormEssSup_measure_zero, eLpNormEssSup_mono_enorm_ae, eLpNormEssSup_mono_measure, eLpNormEssSup_mono_nnnorm_ae, eLpNormEssSup_smul_measure, eLpNormEssSup_zero, eLpNorm_comp_measurePreserving, eLpNorm_congr_ae, eLpNorm_congr_enorm_ae, eLpNorm_congr_nnnorm_ae, eLpNorm_congr_norm_ae, eLpNorm_const, eLpNorm_const', eLpNorm_const_lt_top_iff, eLpNorm_const_lt_top_iff_enorm, eLpNorm_enorm, eLpNorm_enorm_rpow, eLpNorm_eq_zero_iff, eLpNorm_eq_zero_of_ae_zero, eLpNorm_exponent_zero, eLpNorm_indicator_sub_indicator, eLpNorm_le_add_measure_left, eLpNorm_le_add_measure_right, eLpNorm_le_of_ae_bound, eLpNorm_le_of_ae_enorm_bound, eLpNorm_le_of_ae_nnnorm_bound, eLpNorm_lt_top_iff_lintegral_rpow_enorm_lt_top, eLpNorm_lt_top_of_finite, eLpNorm_map_measure, eLpNorm_measure_zero, eLpNorm_mono, eLpNorm_mono_ae, eLpNorm_mono_ae', eLpNorm_mono_ae_real, eLpNorm_mono_enorm, eLpNorm_mono_enorm_ae, eLpNorm_mono_measure, eLpNorm_mono_nnnorm, eLpNorm_mono_nnnorm_ae, eLpNorm_mono_real, eLpNorm_neg, eLpNorm_norm, eLpNorm_norm_rpow, eLpNorm_ofReal, eLpNorm_of_isEmpty, eLpNorm_one_add_measure, eLpNorm_one_smul_measure, eLpNorm_restrict_eq_of_support_subset, eLpNorm_smul_measure_of_ne_top, eLpNorm_smul_measure_of_ne_top', eLpNorm_smul_measure_of_ne_zero, eLpNorm_smul_measure_of_ne_zero', eLpNorm_sub_comm, eLpNorm_zero, eLpNorm_zero', enorm_ae_le_eLpNormEssSup, lintegral_rpow_enorm_lt_top_of_eLpNorm'_lt_top, lintegral_rpow_enorm_lt_top_of_eLpNorm_lt_top, meas_eLpNormEssSup_lt, memLp_congr_ae, memLp_congr_enorm, memLp_congr_norm, memLp_const, memLp_const_enorm, memLp_const_iff, memLp_const_iff_enorm, memLp_enorm_iff, memLp_map_measure_iff, memLp_measure_zero, memLp_neg_iff, memLp_norm_iff, memLp_of_bounded, memLp_top_const, memLp_top_const_enorm, memLp_top_of_bound, memLp_top_of_bound_enorm, memLp_zero_iff_aestronglyMeasurable	154
Total	154

Continuous

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
memLp_top_of_hasCompactSupport 📖	mathematical	Continuous UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup HasCompactSupport NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup	MeasureTheory.MemLp ContinuousENorm.toENorm SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup SeminormedAddGroup.toContinuousENorm Top.top ENNReal instTopENNReal	—	bounded_above_of_compact_support MeasureTheory.memLp_top_of_bound MeasureTheory.StronglyMeasurable.aestronglyMeasurable stronglyMeasurable_of_hasCompactSupport PseudoEMetricSpace.pseudoMetrizableSpace Filter.Eventually.of_forall MeasureTheory.Measure.instOuterMeasureClass

MeasurableEmbedding

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
eLpNormEssSup_map_measure 📖	mathematical	MeasurableEmbedding	MeasureTheory.eLpNormEssSup ContinuousENorm.toENorm MeasureTheory.Measure.map	—	essSup_map_measure
eLpNorm_map_measure 📖	mathematical	MeasurableEmbedding	MeasureTheory.eLpNorm ContinuousENorm.toENorm MeasureTheory.Measure.map	—	MeasureTheory.eLpNorm_exponent_zero MeasureTheory.eLpNorm_exponent_top essSup_map_measure MeasureTheory.eLpNorm_eq_lintegral_rpow_enorm_toReal lintegral_map one_div
memLp_map_measure_iff 📖	mathematical	MeasurableEmbedding	MeasureTheory.MemLp ContinuousENorm.toENorm MeasureTheory.Measure.map	—	aestronglyMeasurable_map_iff eLpNorm_map_measure

MeasurableEquiv

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
memLp_map_measure_iff 📖	mathematical	—	MeasureTheory.MemLp ContinuousENorm.toENorm MeasureTheory.Measure.map DFunLike.coe MeasurableEquiv EquivLike.toFunLike instEquivLike	—	MeasurableEmbedding.memLp_map_measure_iff measurableEmbedding

MeasureTheory

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
ae_bdd_liminf_atTop_of_eLpNorm_bdd 📖	mathematical	Measurable ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder eLpNorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm ENNReal.ofNNReal	Filter.Eventually Preorder.toLT Filter.liminf ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder CompleteLinearOrder.toConditionallyCompleteLinearOrderBot ENNReal.instCompleteLinearOrder ENorm.enorm Filter.atTop Nat.instPreorder Top.top instTopENNReal ae Measure Measure.instFunLike Measure.instOuterMeasureClass	—	Measure.instOuterMeasureClass ae_lt_of_essSup_lt lt_of_le_of_lt eLpNorm_exponent_top ENNReal.lt_add_right ENNReal.coe_ne_top one_ne_zero NeZero.charZero_one ENNReal.instCharZero Filter.isBounded_le_of_top Filter.mp_mem ae_all_iff instCountableNat Filter.univ_mem' Filter.liminf_le_of_frequently_le' Filter.Frequently.of_forall Filter.atTop_neBot instIsDirectedOrder IsStrictOrderedRing.toIsOrderedRing Nat.instIsStrictOrderedRing instArchimedeanNat LT.lt.le ENNReal.add_lt_top ENNReal.coe_lt_top ENNReal.one_lt_top ae_bdd_liminf_atTop_rpow_of_eLpNorm_bdd ENNReal.toReal_pos OrderIso.liminf_apply Filter.isBounded_ge_of_bot Filter.isCobounded_ge_of_top ENNReal.rpow_one mul_inv_cancel₀ LT.lt.ne ENNReal.rpow_mul ENNReal.rpow_lt_top_of_nonneg inv_nonneg PosMulReflectLE.toPosMulReflectLT PosMulStrictMono.toPosMulReflectLE IsStrictOrderedRing.toPosMulStrictMono Real.instIsStrictOrderedRing
ae_bdd_liminf_atTop_rpow_of_eLpNorm_bdd 📖	mathematical	Measurable ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder eLpNorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm ENNReal.ofNNReal	Filter.Eventually Preorder.toLT Filter.liminf ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder CompleteLinearOrder.toConditionallyCompleteLinearOrderBot ENNReal.instCompleteLinearOrder Real ENNReal.instPowReal ENorm.enorm ENNReal.toReal Filter.atTop Nat.instPreorder Top.top instTopENNReal ae Measure Measure.instFunLike Measure.instOuterMeasureClass	—	Measure.instOuterMeasureClass ENNReal.rpow_zero Filter.univ_mem' Filter.liminf_const Filter.atTop_neBot instIsDirectedOrder IsStrictOrderedRing.toIsOrderedRing Nat.instIsStrictOrderedRing instArchimedeanNat ENNReal.one_lt_top ae_lt_top Measurable.liminf ENNReal.borelSpace ENNReal.instOrderTopology ENNReal.instSecondCountableTopology Measurable.pow_const ENNReal.hasMeasurablePow Measurable.coe_nnreal_ennreal Measurable.nnnorm LT.lt.ne lt_of_le_of_lt lintegral_liminf_le sSup_le LE.le.trans le_rfl ENNReal.rpow_inv_le_iff ENNReal.toReal_pos one_div eLpNorm_eq_lintegral_rpow_enorm_toReal Ne.lt_top ENNReal.rpow_ne_top_of_nonneg le_of_lt lt_of_le_of_ne' ENNReal.toReal_nonneg ENNReal.coe_ne_top
ae_eq_zero_of_eLpNorm'_eq_zero 📖	mathematical	Real Real.instLE Real.instZero AEStronglyMeasurable eLpNorm' ContinuousENorm.toENorm ESeminormedAddMonoid.toContinuousENorm ENormedAddMonoid.toESeminormedAddMonoid ENNReal instZeroENNReal	Filter.EventuallyEq ae Measure Measure.instFunLike Measure.instOuterMeasureClass Pi.instZero AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass ESeminormedAddMonoid.toAddMonoid	—	Filter.Eventually.mono Measure.instOuterMeasureClass one_div lintegral_eq_zero_iff' AEMeasurable.pow_const ENNReal.hasMeasurablePow AEStronglyMeasurable.enorm PosMulReflectLE.toPosMulReflectLT PosMulStrictMono.toPosMulReflectLE IsStrictOrderedRing.toPosMulStrictMono Real.instIsStrictOrderedRing IsOrderedRing.toPosMulMono Real.instIsOrderedRing LE.le.not_gt LT.lt.not_gt
ae_le_eLpNormEssSup 📖	mathematical	—	Filter.Eventually ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder ENorm.enorm eLpNormEssSup ae Measure Measure.instFunLike Measure.instOuterMeasureClass	—	ae_le_essSup TopologicalSpace.PseudoMetrizableSpace.firstCountableTopology TopologicalSpace.MetrizableSpace.toPseudoMetrizableSpace ENNReal.instMetrizableSpace ENNReal.instOrderTopology Filter.isBounded_le_of_top Measure.instOuterMeasureClass
eLpNorm'_congr_ae 📖	mathematical	Filter.EventuallyEq ae Measure Measure.instFunLike Measure.instOuterMeasureClass	eLpNorm'	—	Measure.instOuterMeasureClass eLpNorm'_congr_enorm_ae Filter.EventuallyEq.fun_comp
eLpNorm'_congr_enorm_ae 📖	mathematical	Filter.Eventually ENNReal ENorm.enorm ae Measure Measure.instFunLike Measure.instOuterMeasureClass	eLpNorm'	—	Measure.instOuterMeasureClass Filter.Eventually.mono lintegral_congr_ae
eLpNorm'_congr_nnnorm_ae 📖	mathematical	Filter.Eventually NNReal NNNorm.nnnorm SeminormedAddGroup.toNNNorm SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup ae Measure Measure.instFunLike Measure.instOuterMeasureClass	eLpNorm' ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddGroup.toContinuousENorm	—	Measure.instOuterMeasureClass Filter.Eventually.mono lintegral_congr_ae
eLpNorm'_congr_norm_ae 📖	mathematical	Filter.Eventually Real Norm.norm NormedAddCommGroup.toNorm ae Measure Measure.instFunLike Measure.instOuterMeasureClass	eLpNorm' ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm	—	Measure.instOuterMeasureClass eLpNorm'_congr_nnnorm_ae Filter.Eventually.mono NNReal.eq
eLpNorm'_const 📖	mathematical	Real Real.instLT Real.instZero	eLpNorm' ENNReal Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring CommSemiring.toSemiring ENNReal.instCommSemiring ENorm.enorm ENNReal.instPowReal DFunLike.coe Measure Set Measure.instFunLike Set.univ DivInvMonoid.toDiv Real.instDivInvMonoid Real.instOne	—	eLpNorm'_eq_lintegral_enorm lintegral_const ENNReal.mul_rpow_of_nonneg one_div PosMulReflectLE.toPosMulReflectLT PosMulStrictMono.toPosMulReflectLE IsStrictOrderedRing.toPosMulStrictMono Real.instIsStrictOrderedRing LT.lt.le ENNReal.rpow_mul mul_inv_cancel₀ ne_of_lt ENNReal.rpow_one
eLpNorm'_const' 📖	mathematical	—	eLpNorm' ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm ENNReal Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring CommSemiring.toSemiring ENNReal.instCommSemiring ENorm.enorm Real ENNReal.instPowReal DFunLike.coe Measure Set Measure.instFunLike Set.univ DivInvMonoid.toDiv Real.instDivInvMonoid Real.instOne	—	eLpNorm'_eq_lintegral_enorm lintegral_const ENNReal.mul_rpow_of_ne_top ENNReal.rpow_ne_top_of_nonneg' lt_of_le_of_ne' zero_le ENNReal.instCanonicallyOrderedAdd enorm_ne_top measure_ne_top ENNReal.rpow_mul one_div mul_inv_cancel₀ ENNReal.rpow_one
eLpNorm'_const_of_isProbabilityMeasure 📖	mathematical	Real Real.instLT Real.instZero	eLpNorm' ENorm.enorm	—	eLpNorm'_const IsProbabilityMeasure.measure_univ one_div ENNReal.one_rpow mul_one
eLpNorm'_enorm 📖	mathematical	—	eLpNorm' ENNReal instENormENNReal ENorm.enorm	—	one_div
eLpNorm'_enorm_rpow 📖	mathematical	Real Real.instLT Real.instZero	eLpNorm' ENNReal instENormENNReal ENNReal.instPowReal ENorm.enorm Real.instMul	—	one_div mul_assoc inv_mul_cancel₀ LT.lt.ne mul_one mul_comm
eLpNorm'_eq_zero_iff 📖	mathematical	Real Real.instLT Real.instZero AEStronglyMeasurable	eLpNorm' ContinuousENorm.toENorm ESeminormedAddMonoid.toContinuousENorm ENormedAddMonoid.toESeminormedAddMonoid ENNReal instZeroENNReal Filter.EventuallyEq ae Measure Measure.instFunLike Measure.instOuterMeasureClass Pi.instZero AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass ESeminormedAddMonoid.toAddMonoid	—	Measure.instOuterMeasureClass ae_eq_zero_of_eLpNorm'_eq_zero le_of_lt eLpNorm'_eq_zero_of_ae_zero
eLpNorm'_eq_zero_of_ae_eq_zero 📖	mathematical	Real Real.instLT Real.instZero Filter.Eventually ENNReal ENorm.enorm ContinuousENorm.toENorm ESeminormedAddMonoid.toContinuousENorm instZeroENNReal ae Measure Measure.instFunLike Measure.instOuterMeasureClass	eLpNorm'	—	Measure.instOuterMeasureClass eLpNorm'_zero eLpNorm'_congr_enorm_ae enorm_zero
eLpNorm'_eq_zero_of_ae_zero 📖	mathematical	Real Real.instLT Real.instZero Filter.EventuallyEq ae Measure Measure.instFunLike Measure.instOuterMeasureClass Pi.instZero AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass ESeminormedAddMonoid.toAddMonoid	eLpNorm' ContinuousENorm.toENorm ESeminormedAddMonoid.toContinuousENorm ENNReal instZeroENNReal	—	Measure.instOuterMeasureClass eLpNorm'_congr_ae eLpNorm'_zero
eLpNorm'_eq_zero_of_ae_zero' 📖	mathematical	Filter.EventuallyEq ae Measure Measure.instFunLike Measure.instOuterMeasureClass Pi.instZero AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass ESeminormedAddMonoid.toAddMonoid	eLpNorm' ContinuousENorm.toENorm ESeminormedAddMonoid.toContinuousENorm ENNReal instZeroENNReal	—	Measure.instOuterMeasureClass eLpNorm'_congr_ae eLpNorm'_zero'
eLpNorm'_exponent_zero 📖	mathematical	—	eLpNorm' Real Real.instZero ENNReal AddMonoidWithOne.toOne AddCommMonoidWithOne.toAddMonoidWithOne instAddCommMonoidWithOneENNReal	—	eLpNorm'.eq_1 div_zero ENNReal.rpow_zero
eLpNorm'_measure_zero_of_exponent_zero 📖	mathematical	—	eLpNorm' ContinuousENorm.toENorm ESeminormedAddMonoid.toContinuousENorm Real Real.instZero Measure Measure.instZero ENNReal AddMonoidWithOne.toOne AddCommMonoidWithOne.toAddMonoidWithOne instAddCommMonoidWithOneENNReal	—	ENNReal.rpow_zero lintegral_const MulZeroClass.mul_zero div_zero
eLpNorm'_measure_zero_of_neg 📖	mathematical	Real Real.instLT Real.instZero	eLpNorm' ContinuousENorm.toENorm ESeminormedAddMonoid.toContinuousENorm Measure Measure.instZero Top.top ENNReal instTopENNReal	—	lintegral_zero_measure one_div ENNReal.zero_rpow_of_neg IsOrderedRing.toPosMulMono Real.instIsOrderedRing
eLpNorm'_measure_zero_of_pos 📖	mathematical	Real Real.instLT Real.instZero	eLpNorm' ContinuousENorm.toENorm ESeminormedAddMonoid.toContinuousENorm Measure Measure.instZero ENNReal instZeroENNReal	—	lintegral_zero_measure one_div ENNReal.zero_rpow_of_pos PosMulReflectLE.toPosMulReflectLT PosMulStrictMono.toPosMulReflectLE IsStrictOrderedRing.toPosMulStrictMono Real.instIsStrictOrderedRing
eLpNorm'_mono_ae 📖	mathematical	Real Real.instLE Real.instZero Filter.Eventually Norm.norm NormedAddCommGroup.toNorm ae Measure Measure.instFunLike Measure.instOuterMeasureClass	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder eLpNorm' ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm	—	Measure.instOuterMeasureClass eLpNorm'_mono_enorm_ae
eLpNorm'_mono_enorm_ae 📖	mathematical	Real Real.instLE Real.instZero Filter.Eventually ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder ENorm.enorm ae Measure Measure.instFunLike Measure.instOuterMeasureClass	eLpNorm'	—	Measure.instOuterMeasureClass ENNReal.rpow_le_rpow lintegral_mono_ae Filter.Eventually.mono Mathlib.Meta.Positivity.div_nonneg_of_pos_of_nonneg PosMulReflectLE.toPosMulReflectLT PosMulStrictMono.toPosMulReflectLE IsStrictOrderedRing.toPosMulStrictMono Real.instIsStrictOrderedRing Mathlib.Meta.Positivity.pos_of_isNat Real.instIsOrderedRing Real.instNontrivial Mathlib.Meta.NormNum.isNat_ofNat Nat.cast_one
eLpNorm'_mono_measure 📖	mathematical	Measure Preorder.toLE PartialOrder.toPreorder Measure.instPartialOrder Real Real.instLE Real.instZero	ENNReal ENNReal.instPartialOrder eLpNorm' ContinuousENorm.toENorm	—	ENNReal.rpow_le_rpow lintegral_mono_fn' le_refl Mathlib.Meta.Positivity.div_nonneg_of_pos_of_nonneg PosMulReflectLE.toPosMulReflectLT PosMulStrictMono.toPosMulReflectLE IsStrictOrderedRing.toPosMulStrictMono Real.instIsStrictOrderedRing Mathlib.Meta.Positivity.pos_of_isNat Real.instIsOrderedRing Real.instNontrivial Mathlib.Meta.NormNum.isNat_ofNat Nat.cast_one
eLpNorm'_mono_nnnorm_ae 📖	mathematical	Real Real.instLE Real.instZero Filter.Eventually NNReal Preorder.toLE PartialOrder.toPreorder instPartialOrderNNReal NNNorm.nnnorm SeminormedAddGroup.toNNNorm SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup ae Measure Measure.instFunLike Measure.instOuterMeasureClass	ENNReal ENNReal.instPartialOrder eLpNorm' ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddGroup.toContinuousENorm	—	Measure.instOuterMeasureClass ENNReal.rpow_le_rpow lintegral_mono_ae Filter.Eventually.mono ENNReal.coe_le_coe_of_le Mathlib.Meta.Positivity.div_nonneg_of_pos_of_nonneg PosMulReflectLE.toPosMulReflectLT PosMulStrictMono.toPosMulReflectLE IsStrictOrderedRing.toPosMulStrictMono Real.instIsStrictOrderedRing Mathlib.Meta.Positivity.pos_of_isNat Real.instIsOrderedRing Real.instNontrivial Mathlib.Meta.NormNum.isNat_ofNat Nat.cast_one
eLpNorm'_neg 📖	mathematical	—	eLpNorm' ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm Pi.instNeg NegZeroClass.toNeg SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup	—	enorm_neg one_div
eLpNorm'_norm 📖	mathematical	—	eLpNorm' Real ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing SeminormedAddGroup.toContinuousENorm Norm.norm NormedAddCommGroup.toNorm NormedAddCommGroup.toSeminormedAddCommGroup	—	enorm_norm one_div
eLpNorm'_norm_rpow 📖	mathematical	Real Real.instLT Real.instZero	eLpNorm' ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing SeminormedAddGroup.toContinuousENorm Real.instPow Norm.norm NormedAddCommGroup.toNorm ENNReal ENNReal.instPowReal NormedAddCommGroup.toSeminormedAddCommGroup Real.instMul	—	one_div mul_assoc inv_mul_cancel₀ LT.lt.ne mul_one abs_eq_self Real.instIsOrderedAddMonoid Real.rpow_nonneg norm_nonneg mul_comm ENNReal.ofReal_rpow_of_nonneg LT.lt.le ENNReal.rpow_mul
eLpNorm'_smul_measure 📖	mathematical	Real Real.instLE Real.instZero	eLpNorm' ContinuousENorm.toENorm ENNReal Measure Measure.instSMul Algebra.toSMul ENNReal.instCommSemiring CommSemiring.toSemiring Algebra.id IsScalarTower.right Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring ENNReal.instPowReal DivInvMonoid.toDiv Real.instDivInvMonoid Real.instOne	—	IsScalarTower.right lintegral_smul_measure one_div ENNReal.mul_rpow_of_nonneg PosMulReflectLE.toPosMulReflectLT PosMulStrictMono.toPosMulReflectLE IsStrictOrderedRing.toPosMulStrictMono Real.instIsStrictOrderedRing
eLpNorm'_zero 📖	mathematical	Real Real.instLT Real.instZero	eLpNorm' ContinuousENorm.toENorm ESeminormedAddMonoid.toContinuousENorm Pi.instZero AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass ESeminormedAddMonoid.toAddMonoid ENNReal instZeroENNReal	—	enorm_zero ENNReal.zero_rpow_of_pos lintegral_const MulZeroClass.zero_mul one_div PosMulReflectLE.toPosMulReflectLT PosMulStrictMono.toPosMulReflectLE IsStrictOrderedRing.toPosMulStrictMono Real.instIsStrictOrderedRing
eLpNorm'_zero' 📖	mathematical	—	eLpNorm' ContinuousENorm.toENorm ESeminormedAddMonoid.toContinuousENorm Pi.instZero AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass ESeminormedAddMonoid.toAddMonoid ENNReal instZeroENNReal	—	le_or_gt eLpNorm'_zero lt_of_le_of_ne enorm_zero ENNReal.zero_rpow_of_neg lintegral_const ENNReal.top_mul one_div ENNReal.top_rpow_of_neg IsOrderedRing.toPosMulMono Real.instIsOrderedRing
eLpNormEssSup_congr_ae 📖	mathematical	Filter.EventuallyEq ae Measure Measure.instFunLike Measure.instOuterMeasureClass	eLpNormEssSup	—	Measure.instOuterMeasureClass essSup_congr_ae Filter.EventuallyEq.fun_comp
eLpNormEssSup_const 📖	mathematical	—	eLpNormEssSup ENorm.enorm	—	eLpNormEssSup_eq_essSup_enorm essSup_const
eLpNormEssSup_count 📖	mathematical	—	eLpNormEssSup Measure.count iSup ENNReal ConditionallyCompletePartialOrderSup.toSupSet ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup ConditionallyCompleteLattice.toConditionallyCompletePartialOrder ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder CompleteLinearOrder.toConditionallyCompleteLinearOrderBot ENNReal.instCompleteLinearOrder ENorm.enorm	—	essSup_count
eLpNormEssSup_ennreal_smul_measure 📖	mathematical	—	eLpNormEssSup ENNReal Measure Measure.instSMul Algebra.toSMul ENNReal.instCommSemiring CommSemiring.toSemiring Algebra.id IsScalarTower.right	—	IsScalarTower.right essSup_ennreal_smul_measure
eLpNormEssSup_eq_iSup 📖	mathematical	—	eLpNormEssSup iSup ENNReal ConditionallyCompletePartialOrderSup.toSupSet ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup ConditionallyCompleteLattice.toConditionallyCompletePartialOrder ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder CompleteLinearOrder.toConditionallyCompleteLinearOrderBot ENNReal.instCompleteLinearOrder ENorm.enorm	—	essSup_eq_iSup
eLpNormEssSup_eq_zero_iff 📖	mathematical	—	eLpNormEssSup ContinuousENorm.toENorm ESeminormedAddMonoid.toContinuousENorm ENormedAddMonoid.toESeminormedAddMonoid ENNReal instZeroENNReal Filter.EventuallyEq ae Measure Measure.instFunLike Measure.instOuterMeasureClass Pi.instZero AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass ESeminormedAddMonoid.toAddMonoid	—	Measure.instOuterMeasureClass
eLpNormEssSup_le_of_ae_bound 📖	mathematical	Filter.Eventually Real Real.instLE Norm.norm NormedAddCommGroup.toNorm ae Measure Measure.instFunLike Measure.instOuterMeasureClass	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder eLpNormEssSup ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm ENNReal.ofReal	—	Measure.instOuterMeasureClass eLpNormEssSup_le_of_ae_nnnorm_bound Filter.Eventually.mono LE.le.trans Real.le_coe_toNNReal
eLpNormEssSup_le_of_ae_enorm_bound 📖	mathematical	Filter.Eventually ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder ENorm.enorm ae Measure Measure.instFunLike Measure.instOuterMeasureClass	eLpNormEssSup	—	Measure.instOuterMeasureClass essSup_le_of_ae_le
eLpNormEssSup_le_of_ae_nnnorm_bound 📖	mathematical	Filter.Eventually NNReal Preorder.toLE PartialOrder.toPreorder instPartialOrderNNReal NNNorm.nnnorm SeminormedAddGroup.toNNNorm SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup ae Measure Measure.instFunLike Measure.instOuterMeasureClass	ENNReal ENNReal.instPartialOrder eLpNormEssSup ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddGroup.toContinuousENorm ENNReal.ofNNReal	—	Measure.instOuterMeasureClass essSup_le_of_ae_le Filter.Eventually.mono ENNReal.coe_le_coe
eLpNormEssSup_lt_top_iff_isBoundedUnder 📖	mathematical	—	ENNReal Preorder.toLT PartialOrder.toPreorder ENNReal.instPartialOrder eLpNormEssSup ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm Top.top instTopENNReal Filter.IsBoundedUnder NNReal Preorder.toLE instPartialOrderNNReal ae Measure Measure.instFunLike Measure.instOuterMeasureClass NNNorm.nnnorm SeminormedAddGroup.toNNNorm	—	Measure.instOuterMeasureClass ENNReal.coe_toNNReal LT.lt.ne ae_le_eLpNormEssSup eLpNormEssSup_lt_top_of_ae_nnnorm_bound
eLpNormEssSup_lt_top_of_ae_bound 📖	mathematical	Filter.Eventually Real Real.instLE Norm.norm NormedAddCommGroup.toNorm ae Measure Measure.instFunLike Measure.instOuterMeasureClass	ENNReal Preorder.toLT PartialOrder.toPreorder ENNReal.instPartialOrder eLpNormEssSup ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm Top.top instTopENNReal	—	Measure.instOuterMeasureClass LE.le.trans_lt eLpNormEssSup_le_of_ae_bound ENNReal.ofReal_lt_top
eLpNormEssSup_lt_top_of_ae_enorm_bound 📖	mathematical	Filter.Eventually ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder ENorm.enorm ENNReal.ofNNReal ae Measure Measure.instFunLike Measure.instOuterMeasureClass	Preorder.toLT eLpNormEssSup Top.top instTopENNReal	—	Measure.instOuterMeasureClass LE.le.trans_lt eLpNormEssSup_le_of_ae_enorm_bound ENNReal.coe_lt_top
eLpNormEssSup_lt_top_of_ae_nnnorm_bound 📖	mathematical	Filter.Eventually NNReal Preorder.toLE PartialOrder.toPreorder instPartialOrderNNReal NNNorm.nnnorm SeminormedAddGroup.toNNNorm SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup ae Measure Measure.instFunLike Measure.instOuterMeasureClass	ENNReal Preorder.toLT ENNReal.instPartialOrder eLpNormEssSup ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddGroup.toContinuousENorm Top.top instTopENNReal	—	Measure.instOuterMeasureClass LE.le.trans_lt eLpNormEssSup_le_of_ae_nnnorm_bound ENNReal.coe_lt_top
eLpNormEssSup_map_measure 📖	mathematical	AEStronglyMeasurable Measure.map AEMeasurable	eLpNormEssSup ContinuousENorm.toENorm	—	essSup_map_measure ENNReal.instSecondCountableTopology OrderTopology.to_orderClosedTopology ENNReal.instOrderTopology BorelSpace.opensMeasurable ENNReal.borelSpace AEStronglyMeasurable.enorm
eLpNormEssSup_measure_zero 📖	mathematical	—	eLpNormEssSup Measure Measure.instZero ENNReal instZeroENNReal	—	essSup_measure_zero bot_eq_zero' ENNReal.instCanonicallyOrderedAdd
eLpNormEssSup_mono_enorm_ae 📖	mathematical	Filter.Eventually ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder ENorm.enorm ae Measure Measure.instFunLike Measure.instOuterMeasureClass	eLpNormEssSup	—	Measure.instOuterMeasureClass essSup_mono_ae
eLpNormEssSup_mono_measure 📖	mathematical	Measure.AbsolutelyContinuous	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder eLpNormEssSup ContinuousENorm.toENorm	—	essSup_mono_measure
eLpNormEssSup_mono_nnnorm_ae 📖	mathematical	Filter.Eventually NNReal Preorder.toLE PartialOrder.toPreorder instPartialOrderNNReal NNNorm.nnnorm SeminormedAddGroup.toNNNorm SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup ae Measure Measure.instFunLike Measure.instOuterMeasureClass	ENNReal ENNReal.instPartialOrder eLpNormEssSup ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddGroup.toContinuousENorm	—	Measure.instOuterMeasureClass essSup_mono_ae Filter.Eventually.mono ENNReal.coe_le_coe
eLpNormEssSup_smul_measure 📖	mathematical	—	eLpNormEssSup Measure Measure.instSMul SMulZeroClass.toSMul ENNReal AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass ESeminormedAddMonoid.toAddMonoid ENNReal.instTopologicalSpace ESeminormedAddCommMonoid.toESeminormedAddMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid instENormedAddCommMonoidENNReal DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero Module.toDistribMulAction ENNReal.instAddCommMonoid	—	essSup_smul_measure
eLpNormEssSup_zero 📖	mathematical	—	eLpNormEssSup ContinuousENorm.toENorm ESeminormedAddMonoid.toContinuousENorm Pi.instZero AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass ESeminormedAddMonoid.toAddMonoid ENNReal instZeroENNReal	—	enorm_zero ENNReal.instCanonicallyOrderedAdd essSup_const_bot
eLpNorm_comp_measurePreserving 📖	mathematical	AEStronglyMeasurable MeasurePreserving	eLpNorm ContinuousENorm.toENorm	—	eLpNorm_map_measure MeasurePreserving.map_eq MeasurePreserving.aemeasurable
eLpNorm_congr_ae 📖	mathematical	Filter.EventuallyEq ae Measure Measure.instFunLike Measure.instOuterMeasureClass	eLpNorm	—	Measure.instOuterMeasureClass eLpNorm_congr_enorm_ae Filter.Eventually.mono
eLpNorm_congr_enorm_ae 📖	mathematical	Filter.Eventually ENNReal ENorm.enorm ae Measure Measure.instFunLike Measure.instOuterMeasureClass	eLpNorm	—	Measure.instOuterMeasureClass le_antisymm eLpNorm_mono_enorm_ae Filter.EventuallyEq.le Filter.EventuallyEq.symm
eLpNorm_congr_nnnorm_ae 📖	mathematical	Filter.Eventually NNReal NNNorm.nnnorm SeminormedAddGroup.toNNNorm SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup ae Measure Measure.instFunLike Measure.instOuterMeasureClass	eLpNorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddGroup.toContinuousENorm	—	Measure.instOuterMeasureClass le_antisymm eLpNorm_mono_nnnorm_ae Filter.EventuallyEq.le Filter.EventuallyEq.symm
eLpNorm_congr_norm_ae 📖	mathematical	Filter.Eventually Real Norm.norm NormedAddCommGroup.toNorm ae Measure Measure.instFunLike Measure.instOuterMeasureClass	eLpNorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm	—	Measure.instOuterMeasureClass eLpNorm_congr_nnnorm_ae Filter.Eventually.mono NNReal.eq
eLpNorm_const 📖	mathematical	—	eLpNorm ENNReal Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring CommSemiring.toSemiring ENNReal.instCommSemiring ENorm.enorm Real ENNReal.instPowReal DFunLike.coe Measure Set Measure.instFunLike Set.univ DivInvMonoid.toDiv Real.instDivInvMonoid Real.instOne ENNReal.toReal	—	eLpNorm_exponent_top eLpNormEssSup_const div_zero ENNReal.rpow_zero mul_one eLpNorm_eq_eLpNorm' eLpNorm'_const ENNReal.toReal_pos one_div
eLpNorm_const' 📖	mathematical	—	eLpNorm ENNReal Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring CommSemiring.toSemiring ENNReal.instCommSemiring ENorm.enorm Real ENNReal.instPowReal DFunLike.coe Measure Set Measure.instFunLike Set.univ DivInvMonoid.toDiv Real.instDivInvMonoid Real.instOne ENNReal.toReal	—	eLpNorm_eq_eLpNorm' eLpNorm'_const ENNReal.toReal_pos one_div
eLpNorm_const_lt_top_iff 📖	mathematical	—	ENNReal Preorder.toLT PartialOrder.toPreorder ENNReal.instPartialOrder eLpNorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm Top.top instTopENNReal NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup DFunLike.coe Measure Set Measure.instFunLike Set.univ	—	eLpNorm_const_lt_top_iff_enorm enorm_ne_top
eLpNorm_const_lt_top_iff_enorm 📖	mathematical	—	ENNReal Preorder.toLT PartialOrder.toPreorder ENNReal.instPartialOrder eLpNorm ContinuousENorm.toENorm ESeminormedAddMonoid.toContinuousENorm Top.top instTopENNReal ENorm.enorm instZeroENNReal DFunLike.coe Measure Set Measure.instFunLike Set.univ	—	ENNReal.toReal_pos eLpNorm_measure_zero eLpNorm_lt_top_iff_lintegral_rpow_enorm_lt_top ENNReal.zero_rpow_of_pos lintegral_const MulZeroClass.zero_mul eLpNorm_const' eq_or_ne one_div ENNReal.top_rpow_of_pos PosMulReflectLE.toPosMulReflectLT PosMulStrictMono.toPosMulReflectLE IsStrictOrderedRing.toPosMulStrictMono Real.instIsStrictOrderedRing ENNReal.mul_top ENNReal.mul_lt_top_iff Ne.lt_top IsOrderedRing.toPosMulMono Real.instIsOrderedRing ENNReal.rpow_ne_top_of_nonneg le_of_lt inv_pos_of_pos
eLpNorm_enorm 📖	mathematical	—	eLpNorm ENNReal instENormENNReal ENorm.enorm	—	eLpNorm_congr_enorm_ae Filter.Eventually.of_forall Measure.instOuterMeasureClass enorm_enorm
eLpNorm_enorm_rpow 📖	mathematical	Real Real.instLT Real.instZero	eLpNorm ENNReal instENormENNReal ENNReal.instPowReal ENorm.enorm Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring CommSemiring.toSemiring ENNReal.instCommSemiring ENNReal.ofReal	—	eLpNorm_exponent_zero MulZeroClass.zero_mul ENNReal.zero_rpow_of_pos eLpNorm_exponent_top ENNReal.top_mul' LT.lt.not_ge ENNReal.strictMono_rpow_of_pos ENNReal.rpow_left_bijective LT.lt.ne OrderIso.essSup_apply eLpNorm_eq_eLpNorm' mul_ne_zero ENNReal.instNoZeroDivisors ENNReal.ofReal_eq_zero not_le ENNReal.mul_ne_top ENNReal.ofReal_ne_top ENNReal.toReal_mul ENNReal.toReal_ofReal LT.lt.le eLpNorm'_enorm_rpow
eLpNorm_eq_zero_iff 📖	mathematical	AEStronglyMeasurable	eLpNorm ContinuousENorm.toENorm ESeminormedAddMonoid.toContinuousENorm ENormedAddMonoid.toESeminormedAddMonoid ENNReal instZeroENNReal Filter.EventuallyEq ae Measure Measure.instFunLike Measure.instOuterMeasureClass Pi.instZero AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass ESeminormedAddMonoid.toAddMonoid	—	Measure.instOuterMeasureClass eLpNorm_exponent_top eLpNormEssSup_eq_zero_iff eLpNorm_eq_eLpNorm' eLpNorm'_eq_zero_iff ENNReal.toReal_pos
eLpNorm_eq_zero_of_ae_zero 📖	mathematical	Filter.EventuallyEq ae Measure Measure.instFunLike Measure.instOuterMeasureClass Pi.instZero AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass ESeminormedAddMonoid.toAddMonoid	eLpNorm ContinuousENorm.toENorm ESeminormedAddMonoid.toContinuousENorm ENNReal instZeroENNReal	—	Measure.instOuterMeasureClass eLpNorm_zero eLpNorm_congr_ae
eLpNorm_exponent_zero 📖	mathematical	—	eLpNorm ENNReal instZeroENNReal	—	—
eLpNorm_indicator_sub_indicator 📖	mathematical	—	eLpNorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm Pi.instSub SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup Set.indicator NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup symmDiff Set SemilatticeSup.toMax Lattice.toSemilatticeSup ConditionallyCompleteLattice.toLattice CompleteLattice.toConditionallyCompleteLattice CompleteBooleanAlgebra.toCompleteLattice CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra Set.instCompleteAtomicBooleanAlgebra Set.instSDiff	—	eLpNorm_congr_norm_ae ae_of_all Measure.instOuterMeasureClass Set.apply_indicator_symmDiff norm_neg
eLpNorm_le_add_measure_left 📖	mathematical	—	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder eLpNorm ContinuousENorm.toENorm Measure Measure.instAdd	—	eLpNorm_mono_measure Measure.le_add_left le_refl
eLpNorm_le_add_measure_right 📖	mathematical	—	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder eLpNorm ContinuousENorm.toENorm Measure Measure.instAdd	—	eLpNorm_mono_measure Measure.le_add_right le_refl
eLpNorm_le_of_ae_bound 📖	mathematical	Filter.Eventually Real Real.instLE Norm.norm NormedAddCommGroup.toNorm ae Measure Measure.instFunLike Measure.instOuterMeasureClass	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder eLpNorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring CommSemiring.toSemiring ENNReal.instCommSemiring ENNReal.instPowReal DFunLike.coe Set Set.univ Real.instInv ENNReal.toReal ENNReal.ofReal	—	Measure.instOuterMeasureClass mul_comm eLpNorm_le_of_ae_nnnorm_bound Filter.Eventually.mono LE.le.trans Real.le_coe_toNNReal
eLpNorm_le_of_ae_enorm_bound 📖	mathematical	Filter.Eventually ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder ENorm.enorm ContinuousENorm.toENorm ESeminormedAddMonoid.toContinuousENorm ae Measure Measure.instFunLike Measure.instOuterMeasureClass	eLpNorm Algebra.toSMul ENNReal.instCommSemiring CommSemiring.toSemiring Algebra.id Real ENNReal.instPowReal DFunLike.coe Set Set.univ Real.instInv ENNReal.toReal	—	Measure.instOuterMeasureClass eq_zero_or_neZero eLpNorm_measure_zero ENNReal.instCanonicallyOrderedAdd eLpNorm_exponent_zero inv_zero ENNReal.rpow_zero mul_one Filter.Eventually.mono LE.le.trans le_refl LE.le.trans_eq eLpNorm_mono_enorm_ae eLpNorm_const one_div enorm_eq_self smul_eq_mul
eLpNorm_le_of_ae_nnnorm_bound 📖	mathematical	Filter.Eventually NNReal Preorder.toLE PartialOrder.toPreorder instPartialOrderNNReal NNNorm.nnnorm SeminormedAddGroup.toNNNorm SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup ae Measure Measure.instFunLike Measure.instOuterMeasureClass	ENNReal ENNReal.instPartialOrder eLpNorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddGroup.toContinuousENorm Algebra.toSMul instCommSemiringNNReal CommSemiring.toSemiring ENNReal.instCommSemiring ENNReal.instAlgebraNNReal Algebra.id Real ENNReal.instPowReal DFunLike.coe Set Set.univ Real.instInv ENNReal.toReal	—	Measure.instOuterMeasureClass eq_zero_or_neZero eLpNorm_measure_zero ENNReal.instCanonicallyOrderedAdd eLpNorm_exponent_zero inv_zero ENNReal.rpow_zero Filter.Eventually.mono LE.le.trans_eq NNReal.nnnorm_eq eLpNorm_mono_ae eLpNorm_const NNReal.enorm_eq one_div ENNReal.smul_def smul_eq_mul
eLpNorm_lt_top_iff_lintegral_rpow_enorm_lt_top 📖	mathematical	—	ENNReal Preorder.toLT PartialOrder.toPreorder ENNReal.instPartialOrder eLpNorm Top.top instTopENNReal lintegral Real ENNReal.instPowReal ENorm.enorm ENNReal.toReal	—	lintegral_rpow_enorm_lt_top_of_eLpNorm_lt_top ENNReal.toReal_pos div_pos PosMulReflectLE.toPosMulReflectLT PosMulStrictMono.toPosMulReflectLE IsStrictOrderedRing.toPosMulStrictMono Real.instIsStrictOrderedRing zero_lt_one Real.instZeroLEOneClass NeZero.charZero_one RCLike.charZero_rclike eLpNorm_eq_lintegral_rpow_enorm_toReal one_div ENNReal.rpow_lt_top_of_nonneg le_of_lt ne_of_lt
eLpNorm_lt_top_of_finite 📖	mathematical	—	ENNReal Preorder.toLT PartialOrder.toPreorder ENNReal.instPartialOrder eLpNorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm Top.top instTopENNReal	—	eq_or_ne eLpNorm_exponent_zero Measure.instOuterMeasureClass eLpNorm_exponent_top Filter.IsBoundedUnder.le_of_finite instIsDirectedOrder NNReal.instIsOrderedRing NNReal.instArchimedean eLpNorm_lt_top_iff_lintegral_rpow_enorm_lt_top IsFiniteMeasure.lintegral_lt_top_of_bounded_to_ennreal ENNReal.coe_rpow_of_nonneg ENNReal.toReal_nonneg Finite.exists_le
eLpNorm_map_measure 📖	mathematical	AEStronglyMeasurable Measure.map AEMeasurable	eLpNorm ContinuousENorm.toENorm	—	eLpNorm_exponent_zero eLpNorm_exponent_top eLpNormEssSup_map_measure eLpNorm_eq_lintegral_rpow_enorm_toReal lintegral_map' AEMeasurable.pow_const ENNReal.hasMeasurablePow AEStronglyMeasurable.enorm
eLpNorm_measure_zero 📖	mathematical	—	eLpNorm Measure Measure.instZero ENNReal instZeroENNReal	—	eLpNorm_exponent_zero eLpNorm_exponent_top eLpNormEssSup_measure_zero eLpNorm_eq_eLpNorm' lintegral_zero_measure one_div ENNReal.zero_rpow_of_pos PosMulReflectLE.toPosMulReflectLT PosMulStrictMono.toPosMulReflectLE IsStrictOrderedRing.toPosMulStrictMono Real.instIsStrictOrderedRing ENNReal.toReal_pos
eLpNorm_mono 📖	mathematical	Real Real.instLE Norm.norm NormedAddCommGroup.toNorm	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder eLpNorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm	—	eLpNorm_mono_ae Filter.Eventually.of_forall Measure.instOuterMeasureClass
eLpNorm_mono_ae 📖	mathematical	Filter.Eventually Real Real.instLE Norm.norm NormedAddCommGroup.toNorm ae Measure Measure.instFunLike Measure.instOuterMeasureClass	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder eLpNorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm	—	Measure.instOuterMeasureClass eLpNorm_mono_enorm_ae
eLpNorm_mono_ae' 📖	mathematical	Filter.Eventually ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder ENorm.enorm ae Measure Measure.instFunLike Measure.instOuterMeasureClass	eLpNorm	—	Measure.instOuterMeasureClass eLpNorm_mono_enorm_ae
eLpNorm_mono_ae_real 📖	mathematical	Filter.Eventually Real Real.instLE Norm.norm NormedAddCommGroup.toNorm ae Measure Measure.instFunLike Measure.instOuterMeasureClass	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder eLpNorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing	—	Measure.instOuterMeasureClass eLpNorm_mono_ae Filter.Eventually.mono LE.le.trans le_abs_self Eq.le Real.norm_eq_abs
eLpNorm_mono_enorm 📖	mathematical	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder ENorm.enorm	eLpNorm	—	eLpNorm_mono_enorm_ae Filter.Eventually.of_forall Measure.instOuterMeasureClass
eLpNorm_mono_enorm_ae 📖	mathematical	Filter.Eventually ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder ENorm.enorm ae Measure Measure.instFunLike Measure.instOuterMeasureClass	eLpNorm	—	Measure.instOuterMeasureClass le_rfl essSup_mono_ae eLpNorm'_mono_enorm_ae ENNReal.toReal_nonneg
eLpNorm_mono_measure 📖	mathematical	Measure Preorder.toLE PartialOrder.toPreorder Measure.instPartialOrder	ENNReal ENNReal.instPartialOrder eLpNorm ContinuousENorm.toENorm	—	eLpNorm_exponent_zero eLpNorm_exponent_top eLpNormEssSup_mono_measure Measure.absolutelyContinuous_of_le eLpNorm_eq_eLpNorm' eLpNorm'_mono_measure ENNReal.toReal_nonneg
eLpNorm_mono_nnnorm 📖	mathematical	NNReal Preorder.toLE PartialOrder.toPreorder instPartialOrderNNReal NNNorm.nnnorm SeminormedAddGroup.toNNNorm SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup	ENNReal ENNReal.instPartialOrder eLpNorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddGroup.toContinuousENorm	—	eLpNorm_mono_nnnorm_ae Filter.Eventually.of_forall Measure.instOuterMeasureClass
eLpNorm_mono_nnnorm_ae 📖	mathematical	Filter.Eventually NNReal Preorder.toLE PartialOrder.toPreorder instPartialOrderNNReal NNNorm.nnnorm SeminormedAddGroup.toNNNorm SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup ae Measure Measure.instFunLike Measure.instOuterMeasureClass	ENNReal ENNReal.instPartialOrder eLpNorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddGroup.toContinuousENorm	—	Measure.instOuterMeasureClass le_rfl essSup_mono_ae Filter.Eventually.mono ENNReal.coe_le_coe eLpNorm'_mono_nnnorm_ae ENNReal.toReal_nonneg
eLpNorm_mono_real 📖	mathematical	Real Real.instLE Norm.norm NormedAddCommGroup.toNorm	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder eLpNorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing	—	eLpNorm_mono_ae_real Filter.Eventually.of_forall Measure.instOuterMeasureClass
eLpNorm_neg 📖	mathematical	—	eLpNorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm Pi.instNeg NegZeroClass.toNeg SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup	—	eLpNorm_exponent_zero eLpNorm_exponent_top enorm_neg eLpNorm_eq_eLpNorm' eLpNorm'_neg
eLpNorm_norm 📖	mathematical	—	eLpNorm Real ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing SeminormedAddGroup.toContinuousENorm Norm.norm NormedAddCommGroup.toNorm NormedAddCommGroup.toSeminormedAddCommGroup	—	eLpNorm_congr_norm_ae Filter.Eventually.of_forall Measure.instOuterMeasureClass norm_norm
eLpNorm_norm_rpow 📖	mathematical	Real Real.instLT Real.instZero	eLpNorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing SeminormedAddGroup.toContinuousENorm Real.instPow Norm.norm NormedAddCommGroup.toNorm ENNReal ENNReal.instPowReal NormedAddCommGroup.toSeminormedAddCommGroup Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring CommSemiring.toSemiring ENNReal.instCommSemiring ENNReal.ofReal	—	eLpNorm_enorm_rpow ofReal_norm_eq_enorm ENNReal.ofReal_rpow_of_nonneg norm_nonneg le_of_lt eLpNorm_ofReal Filter.univ_mem' Measure.instOuterMeasureClass Real.rpow_nonneg
eLpNorm_ofReal 📖	mathematical	Filter.Eventually Real Real.instLE Real.instZero ae Measure Measure.instFunLike Measure.instOuterMeasureClass	eLpNorm ENNReal instENormENNReal ENNReal.ofReal ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing SeminormedAddGroup.toContinuousENorm	—	Measure.instOuterMeasureClass eLpNorm_congr_enorm_ae Filter.Eventually.mono Real.enorm_ofReal_of_nonneg
eLpNorm_of_isEmpty 📖	mathematical	—	eLpNorm ContinuousENorm.toENorm ESeminormedAddMonoid.toContinuousENorm ENNReal instZeroENNReal	—	Unique.instSubsingleton eLpNorm_zero
eLpNorm_one_add_measure 📖	mathematical	—	eLpNorm ENNReal AddMonoidWithOne.toOne AddCommMonoidWithOne.toAddMonoidWithOne instAddCommMonoidWithOneENNReal Measure Measure.instAdd Distrib.toAdd NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring CommSemiring.toSemiring ENNReal.instCommSemiring	—	eLpNorm_one_eq_lintegral_enorm lintegral_add_measure
eLpNorm_one_smul_measure 📖	mathematical	—	eLpNorm ContinuousENorm.toENorm ENNReal AddMonoidWithOne.toOne AddCommMonoidWithOne.toAddMonoidWithOne instAddCommMonoidWithOneENNReal Measure Measure.instSMul Algebra.toSMul ENNReal.instCommSemiring CommSemiring.toSemiring Algebra.id IsScalarTower.right Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring	—	IsScalarTower.right eLpNorm_smul_measure_of_ne_top div_one ENNReal.rpow_one
eLpNorm_restrict_eq_of_support_subset 📖	mathematical	Set Set.instHasSubset Function.support AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass ESeminormedAddMonoid.toAddMonoid ENormedAddMonoid.toESeminormedAddMonoid	eLpNorm ContinuousENorm.toENorm ESeminormedAddMonoid.toContinuousENorm Measure.restrict	—	eLpNorm_exponent_zero eLpNorm_exponent_top ENNReal.essSup_restrict_eq_of_support_subset enorm_ne_zero eLpNorm_eq_eLpNorm' setLIntegral_eq_of_support_subset ENNReal.toReal_pos
eLpNorm_smul_measure_of_ne_top 📖	mathematical	—	eLpNorm ContinuousENorm.toENorm ENNReal Measure Measure.instSMul Algebra.toSMul ENNReal.instCommSemiring CommSemiring.toSemiring Algebra.id IsScalarTower.right Real ENNReal.instPowReal ENNReal.toReal DivInvMonoid.toDiv ENNReal.instDivInvMonoid AddMonoidWithOne.toOne AddCommMonoidWithOne.toAddMonoidWithOne instAddCommMonoidWithOneENNReal	—	IsScalarTower.right eLpNorm_exponent_zero one_div ENNReal.inv_zero ENNReal.rpow_zero MulZeroClass.mul_zero
eLpNorm_smul_measure_of_ne_top' 📖	mathematical	—	eLpNorm ContinuousENorm.toENorm NNReal Measure Measure.instSMul Algebra.toSMul ENNReal instCommSemiringNNReal CommSemiring.toSemiring ENNReal.instCommSemiring ENNReal.instAlgebraNNReal Algebra.id IsScalarTower.right Real NNReal.instPowReal Real.instInv ENNReal.toReal	—	inv_nonneg_of_nonneg PosMulReflectLE.toPosMulReflectLT PosMulStrictMono.toPosMulReflectLE IsStrictOrderedRing.toPosMulStrictMono Real.instIsStrictOrderedRing ENNReal.toReal_nonneg IsScalarTower.right eLpNorm_smul_measure_of_ne_top one_div ENNReal.toReal_inv ENNReal.coe_rpow_of_nonneg
eLpNorm_smul_measure_of_ne_zero 📖	mathematical	—	eLpNorm ContinuousENorm.toENorm ENNReal Measure Measure.instSMul Algebra.toSMul ENNReal.instCommSemiring CommSemiring.toSemiring Algebra.id IsScalarTower.right Real ENNReal.instPowReal ENNReal.toReal DivInvMonoid.toDiv ENNReal.instDivInvMonoid AddMonoidWithOne.toOne AddCommMonoidWithOne.toAddMonoidWithOne instAddCommMonoidWithOneENNReal	—	IsScalarTower.right eLpNorm_exponent_zero one_div ENNReal.inv_zero ENNReal.rpow_zero MulZeroClass.mul_zero eLpNorm_exponent_top eLpNormEssSup_ennreal_smul_measure ENNReal.inv_top one_mul
eLpNorm_smul_measure_of_ne_zero' 📖	mathematical	—	eLpNorm ContinuousENorm.toENorm NNReal Measure Measure.instSMul Algebra.toSMul ENNReal instCommSemiringNNReal CommSemiring.toSemiring ENNReal.instCommSemiring ENNReal.instAlgebraNNReal Algebra.id IsScalarTower.right Real NNReal.instPowReal Real.instInv ENNReal.toReal	—	IsScalarTower.right eLpNorm_smul_measure_of_ne_zero ENNReal.coe_ne_zero one_div ENNReal.toReal_inv
eLpNorm_sub_comm 📖	mathematical	—	eLpNorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm Pi.instSub SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup	—	eLpNorm_neg neg_sub
eLpNorm_zero 📖	mathematical	—	eLpNorm ContinuousENorm.toENorm ESeminormedAddMonoid.toContinuousENorm Pi.instZero AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass ESeminormedAddMonoid.toAddMonoid ENNReal instZeroENNReal	—	eLpNorm_exponent_zero eLpNorm_exponent_top eLpNormEssSup_zero eLpNorm_eq_eLpNorm' eLpNorm'_zero ENNReal.toReal_pos
eLpNorm_zero' 📖	mathematical	—	eLpNorm ContinuousENorm.toENorm ESeminormedAddMonoid.toContinuousENorm AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass ESeminormedAddMonoid.toAddMonoid ENNReal instZeroENNReal	—	eLpNorm_zero
enorm_ae_le_eLpNormEssSup 📖	mathematical	—	Filter.Eventually ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder ENorm.enorm eLpNormEssSup ae Measure Measure.instFunLike Measure.instOuterMeasureClass	—	ENNReal.ae_le_essSup
lintegral_rpow_enorm_lt_top_of_eLpNorm'_lt_top 📖	mathematical	Real Real.instLT Real.instZero ENNReal Preorder.toLT PartialOrder.toPreorder ENNReal.instPartialOrder eLpNorm' Top.top instTopENNReal	lintegral ENNReal.instPowReal ENorm.enorm	—	lintegral_rpow_enorm_eq_rpow_eLpNorm' ENNReal.rpow_lt_top_of_nonneg le_of_lt ne_of_lt
lintegral_rpow_enorm_lt_top_of_eLpNorm_lt_top 📖	mathematical	ENNReal Preorder.toLT PartialOrder.toPreorder ENNReal.instPartialOrder eLpNorm Top.top instTopENNReal	lintegral Real ENNReal.instPowReal ENorm.enorm ENNReal.toReal	—	lintegral_rpow_enorm_lt_top_of_eLpNorm'_lt_top ENNReal.toReal_pos eLpNorm_eq_eLpNorm'
meas_eLpNormEssSup_lt 📖	mathematical	—	DFunLike.coe Measure Set ENNReal Measure.instFunLike setOf Preorder.toLT PartialOrder.toPreorder ENNReal.instPartialOrder eLpNormEssSup ENorm.enorm instZeroENNReal	—	meas_essSup_lt TopologicalSpace.PseudoMetrizableSpace.firstCountableTopology TopologicalSpace.MetrizableSpace.toPseudoMetrizableSpace ENNReal.instMetrizableSpace ENNReal.instOrderTopology Filter.isBounded_le_of_top Measure.instOuterMeasureClass
memLp_congr_ae 📖	mathematical	Filter.EventuallyEq ae Measure Measure.instFunLike Measure.instOuterMeasureClass	MemLp	—	Measure.instOuterMeasureClass aestronglyMeasurable_congr eLpNorm_congr_ae
memLp_congr_enorm 📖	mathematical	AEStronglyMeasurable Filter.Eventually ENNReal ENorm.enorm ContinuousENorm.toENorm ae Measure Measure.instFunLike Measure.instOuterMeasureClass	MemLp	—	Measure.instOuterMeasureClass MemLp.congr_enorm Filter.EventuallyEq.symm
memLp_congr_norm 📖	mathematical	AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup Filter.Eventually Real Norm.norm NormedAddCommGroup.toNorm ae Measure Measure.instFunLike Measure.instOuterMeasureClass	MemLp ContinuousENorm.toENorm SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup SeminormedAddGroup.toContinuousENorm	—	Measure.instOuterMeasureClass MemLp.congr_norm Filter.EventuallyEq.symm
memLp_const 📖	mathematical	—	MemLp ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toPseudoMetricSpace	—	memLp_const_enorm enorm_ne_top
memLp_const_enorm 📖	mathematical	—	MemLp ContinuousENorm.toENorm	—	aestronglyMeasurable_const eLpNorm_exponent_zero eLpNorm_measure_zero eLpNorm_const Ne.lt_top ENNReal.mul_ne_top ENNReal.rpow_ne_top_of_nonneg Mathlib.Meta.Positivity.div_nonneg_of_pos_of_nonneg PosMulReflectLE.toPosMulReflectLT PosMulStrictMono.toPosMulReflectLE IsStrictOrderedRing.toPosMulStrictMono Real.instIsStrictOrderedRing Mathlib.Meta.Positivity.pos_of_isNat Real.instIsOrderedRing Real.instNontrivial Mathlib.Meta.NormNum.isNat_ofNat Nat.cast_one ENNReal.toReal_nonneg measure_ne_top
memLp_const_iff 📖	mathematical	—	MemLp ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toPseudoMetricSpace NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup ENNReal Preorder.toLT PartialOrder.toPreorder ENNReal.instPartialOrder DFunLike.coe Measure Set Measure.instFunLike Set.univ Top.top instTopENNReal	—	memLp_const_iff_enorm enorm_ne_top
memLp_const_iff_enorm 📖	mathematical	—	MemLp ContinuousENorm.toENorm ESeminormedAddMonoid.toContinuousENorm ENorm.enorm ENNReal instZeroENNReal Preorder.toLT PartialOrder.toPreorder ENNReal.instPartialOrder DFunLike.coe Measure Set Measure.instFunLike Set.univ Top.top instTopENNReal	—	eLpNorm_const_lt_top_iff_enorm
memLp_enorm_iff 📖	mathematical	AEStronglyMeasurable	MemLp ENNReal instENormENNReal ENNReal.instTopologicalSpace ENorm.enorm ContinuousENorm.toENorm	—	eLpNorm_enorm MemLp.enorm
memLp_map_measure_iff 📖	mathematical	AEStronglyMeasurable Measure.map AEMeasurable	MemLp ContinuousENorm.toENorm	—	eLpNorm_map_measure AEStronglyMeasurable.comp_aemeasurable
memLp_measure_zero 📖	mathematical	—	MemLp ContinuousENorm.toENorm Measure Measure.instZero	—	eLpNorm_measure_zero
memLp_neg_iff 📖	mathematical	—	MemLp ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toPseudoMetricSpace Pi.instNeg NegZeroClass.toNeg SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup	—	MemLp.neg neg_neg
memLp_norm_iff 📖	mathematical	AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup	MemLp Real ContinuousENorm.toENorm SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing SeminormedAddGroup.toContinuousENorm Real.pseudoMetricSpace Norm.norm NormedAddCommGroup.toNorm	—	eLpNorm_norm MemLp.norm
memLp_of_bounded 📖	mathematical	Filter.Eventually Real Set Set.instMembership Set.Icc Real.instPreorder ae Measure Measure.instFunLike Measure.instOuterMeasureClass AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace	MemLp ContinuousENorm.toENorm SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing SeminormedAddGroup.toContinuousENorm	—	Measure.instOuterMeasureClass Filter.Eventually.mono MemLp.mono' memLp_const Filter.mp_mem Filter.univ_mem' abs_le_max_abs_abs Real.instIsOrderedAddMonoid
memLp_top_const 📖	mathematical	—	MemLp ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toPseudoMetricSpace Top.top ENNReal instTopENNReal	—	memLp_top_const_enorm enorm_ne_top
memLp_top_const_enorm 📖	mathematical	—	MemLp ContinuousENorm.toENorm Top.top ENNReal instTopENNReal	—	aestronglyMeasurable_const eLpNorm_measure_zero eLpNorm_const div_zero ENNReal.rpow_zero mul_one Ne.lt_top
memLp_top_of_bound 📖	mathematical	AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup Filter.Eventually Real Real.instLE Norm.norm NormedAddCommGroup.toNorm ae Measure Measure.instFunLike Measure.instOuterMeasureClass	MemLp ContinuousENorm.toENorm SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup SeminormedAddGroup.toContinuousENorm Top.top ENNReal instTopENNReal	—	Measure.instOuterMeasureClass eLpNorm_exponent_top eLpNormEssSup_lt_top_of_ae_bound
memLp_top_of_bound_enorm 📖	mathematical	AEStronglyMeasurable Filter.Eventually ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder ENorm.enorm ContinuousENorm.toENorm ENNReal.ofNNReal ae Measure Measure.instFunLike Measure.instOuterMeasureClass	MemLp Top.top instTopENNReal	—	Measure.instOuterMeasureClass eLpNorm_exponent_top eLpNormEssSup_lt_top_of_ae_enorm_bound
memLp_zero_iff_aestronglyMeasurable 📖	mathematical	—	MemLp ENNReal instZeroENNReal AEStronglyMeasurable	—	eLpNorm_exponent_zero

MeasureTheory.AEEqFun

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
eLpNorm_compMeasurePreserving 📖	mathematical	MeasureTheory.MeasurePreserving	MeasureTheory.eLpNorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm cast SeminormedAddCommGroup.toPseudoMetricSpace compMeasurePreserving	—	MeasureTheory.eLpNorm_congr_ae coeFn_compMeasurePreserving MeasureTheory.eLpNorm_comp_measurePreserving aestronglyMeasurable

MeasureTheory.MemLp

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
ae_eq 📖	—	Filter.EventuallyEq MeasureTheory.ae MeasureTheory.Measure MeasureTheory.Measure.instFunLike MeasureTheory.Measure.instOuterMeasureClass MeasureTheory.MemLp	—	—	MeasureTheory.Measure.instOuterMeasureClass MeasureTheory.memLp_congr_ae
comp_measurePreserving 📖	—	MeasureTheory.MemLp ContinuousENorm.toENorm MeasureTheory.MeasurePreserving	—	—	comp_of_map MeasureTheory.MeasurePreserving.map_eq MeasureTheory.MeasurePreserving.aemeasurable
comp_of_map 📖	—	MeasureTheory.MemLp ContinuousENorm.toENorm MeasureTheory.Measure.map AEMeasurable	—	—	MeasureTheory.memLp_map_measure_iff aestronglyMeasurable
congr_enorm 📖	—	MeasureTheory.MemLp ContinuousENorm.toENorm MeasureTheory.AEStronglyMeasurable Filter.Eventually ENNReal ENorm.enorm MeasureTheory.ae MeasureTheory.Measure MeasureTheory.Measure.instFunLike MeasureTheory.Measure.instOuterMeasureClass	—	—	MeasureTheory.Measure.instOuterMeasureClass of_le_enorm Filter.EventuallyEq.le Filter.EventuallyEq.symm
congr_norm 📖	—	MeasureTheory.MemLp ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toPseudoMetricSpace MeasureTheory.AEStronglyMeasurable Filter.Eventually Real Norm.norm NormedAddCommGroup.toNorm MeasureTheory.ae MeasureTheory.Measure MeasureTheory.Measure.instFunLike MeasureTheory.Measure.instOuterMeasureClass	—	—	MeasureTheory.Measure.instOuterMeasureClass mono Filter.EventuallyEq.le Filter.EventuallyEq.symm
eLpNorm_lt_top 📖	mathematical	MeasureTheory.MemLp	ENNReal Preorder.toLT PartialOrder.toPreorder ENNReal.instPartialOrder MeasureTheory.eLpNorm Top.top instTopENNReal	—	—
eLpNorm_ne_top 📖	—	MeasureTheory.MemLp	—	—	LT.lt.ne
enorm 📖	mathematical	MeasureTheory.MemLp ContinuousENorm.toENorm	ENNReal instENormENNReal ENNReal.instTopologicalSpace ENorm.enorm	—	AEMeasurable.aestronglyMeasurable TopologicalSpace.MetrizableSpace.toPseudoMetrizableSpace ENNReal.instMetrizableSpace BorelSpace.opensMeasurable ENNReal.borelSpace ENNReal.instSecondCountableTopology MeasureTheory.AEStronglyMeasurable.enorm aestronglyMeasurable MeasureTheory.eLpNorm_enorm eLpNorm_lt_top
left_of_add_measure 📖	—	MeasureTheory.MemLp ContinuousENorm.toENorm MeasureTheory.Measure MeasureTheory.Measure.instAdd	—	—	mono_measure MeasureTheory.Measure.le_add_right le_refl
mono 📖	—	MeasureTheory.MemLp ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toPseudoMetricSpace MeasureTheory.AEStronglyMeasurable Filter.Eventually Real Real.instLE Norm.norm NormedAddCommGroup.toNorm MeasureTheory.ae MeasureTheory.Measure MeasureTheory.Measure.instFunLike MeasureTheory.Measure.instOuterMeasureClass	—	—	of_le
mono' 📖	—	MeasureTheory.MemLp Real ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing SeminormedAddGroup.toContinuousENorm Real.pseudoMetricSpace MeasureTheory.AEStronglyMeasurable SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup Filter.Eventually Real.instLE Norm.norm NormedAddCommGroup.toNorm MeasureTheory.ae MeasureTheory.Measure MeasureTheory.Measure.instFunLike MeasureTheory.Measure.instOuterMeasureClass	—	—	MeasureTheory.Measure.instOuterMeasureClass of_le Filter.Eventually.mono le_trans le_abs_self
mono'_enorm 📖	—	MeasureTheory.MemLp ENNReal instENormENNReal ENNReal.instTopologicalSpace MeasureTheory.AEStronglyMeasurable Filter.Eventually Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder ENorm.enorm ContinuousENorm.toENorm MeasureTheory.ae MeasureTheory.Measure MeasureTheory.Measure.instFunLike MeasureTheory.Measure.instOuterMeasureClass	—	—	MeasureTheory.Measure.instOuterMeasureClass of_le_enorm Filter.Eventually.mono le_trans le_rfl
mono_measure 📖	—	MeasureTheory.Measure Preorder.toLE PartialOrder.toPreorder MeasureTheory.Measure.instPartialOrder MeasureTheory.MemLp ContinuousENorm.toENorm	—	—	MeasureTheory.AEStronglyMeasurable.mono_measure LE.le.trans_lt MeasureTheory.eLpNorm_mono_measure
neg 📖	mathematical	MeasureTheory.MemLp ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toPseudoMetricSpace	Pi.instNeg NegZeroClass.toNeg SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup	—	MeasureTheory.AEStronglyMeasurable.neg IsTopologicalAddGroup.toContinuousNeg SeminormedAddCommGroup.toIsTopologicalAddGroup MeasureTheory.eLpNorm_neg
norm 📖	mathematical	MeasureTheory.MemLp ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toPseudoMetricSpace	Real NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing Real.pseudoMetricSpace Norm.norm NormedAddCommGroup.toNorm	—	of_le MeasureTheory.AEStronglyMeasurable.norm aestronglyMeasurable Filter.Eventually.of_forall MeasureTheory.Measure.instOuterMeasureClass norm_norm
of_bound 📖	mathematical	MeasureTheory.AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup Filter.Eventually Real Real.instLE Norm.norm NormedAddCommGroup.toNorm MeasureTheory.ae MeasureTheory.Measure MeasureTheory.Measure.instFunLike MeasureTheory.Measure.instOuterMeasureClass	MeasureTheory.MemLp ContinuousENorm.toENorm SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup SeminormedAddGroup.toContinuousENorm	—	MeasureTheory.Measure.instOuterMeasureClass of_le MeasureTheory.memLp_const Filter.Eventually.mono le_trans le_abs_self
of_discrete 📖	mathematical	—	MeasureTheory.MemLp ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toPseudoMetricSpace	—	Finite.exists_le instIsDirectedOrder NNReal.instIsOrderedRing NNReal.instArchimedean of_bound MeasureTheory.AEStronglyMeasurable.of_discrete Finite.to_countable DiscreteMeasurableSpace.toMeasurableSingletonClass Filter.Eventually.of_forall MeasureTheory.Measure.instOuterMeasureClass
of_enorm_bound 📖	mathematical	MeasureTheory.AEStronglyMeasurable Filter.Eventually ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder ENorm.enorm ContinuousENorm.toENorm MeasureTheory.ae MeasureTheory.Measure MeasureTheory.Measure.instFunLike MeasureTheory.Measure.instOuterMeasureClass	MeasureTheory.MemLp	—	MeasureTheory.Measure.instOuterMeasureClass of_le_enorm MeasureTheory.memLp_const_enorm Filter.Eventually.mono enorm_eq_self
of_le 📖	—	MeasureTheory.MemLp ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toPseudoMetricSpace MeasureTheory.AEStronglyMeasurable Filter.Eventually Real Real.instLE Norm.norm NormedAddCommGroup.toNorm MeasureTheory.ae MeasureTheory.Measure MeasureTheory.Measure.instFunLike MeasureTheory.Measure.instOuterMeasureClass	—	—	MeasureTheory.Measure.instOuterMeasureClass LE.le.trans_lt MeasureTheory.eLpNorm_mono_ae Ne.lt_top eLpNorm_ne_top
of_le_enorm 📖	—	MeasureTheory.MemLp ContinuousENorm.toENorm MeasureTheory.AEStronglyMeasurable Filter.Eventually ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder ENorm.enorm MeasureTheory.ae MeasureTheory.Measure MeasureTheory.Measure.instFunLike MeasureTheory.Measure.instOuterMeasureClass	—	—	MeasureTheory.Measure.instOuterMeasureClass LE.le.trans_lt MeasureTheory.eLpNorm_mono_ae' Ne.lt_top eLpNorm_ne_top
of_measure_le_smul 📖	—	MeasureTheory.Measure Preorder.toLE PartialOrder.toPreorder MeasureTheory.Measure.instPartialOrder ENNReal MeasureTheory.Measure.instSMul Algebra.toSMul ENNReal.instCommSemiring CommSemiring.toSemiring Algebra.id IsScalarTower.right MeasureTheory.MemLp ContinuousENorm.toENorm	—	—	IsScalarTower.right MeasureTheory.AEStronglyMeasurable.mono_ac MeasureTheory.Measure.absolutelyContinuous_of_le_smul LE.le.trans_lt MeasureTheory.eLpNorm_mono_measure zero_smul MeasureTheory.eLpNorm_measure_zero MeasureTheory.eLpNorm_smul_measure_of_ne_zero smul_eq_mul ENNReal.mul_lt_top Ne.lt_top one_div ENNReal.toReal_inv IsOrderedRing.toPosMulMono Real.instIsOrderedRing PosMulReflectLE.toPosMulReflectLT PosMulStrictMono.toPosMulReflectLE IsStrictOrderedRing.toPosMulStrictMono Real.instIsStrictOrderedRing
restrict 📖	mathematical	MeasureTheory.MemLp ContinuousENorm.toENorm	MeasureTheory.Measure.restrict	—	mono_measure MeasureTheory.Measure.restrict_le_self
right_of_add_measure 📖	—	MeasureTheory.MemLp ContinuousENorm.toENorm MeasureTheory.Measure MeasureTheory.Measure.instAdd	—	—	mono_measure MeasureTheory.Measure.le_add_left le_refl
smul_measure 📖	mathematical	MeasureTheory.MemLp ContinuousENorm.toENorm	ENNReal MeasureTheory.Measure MeasureTheory.Measure.instSMul Algebra.toSMul ENNReal.instCommSemiring CommSemiring.toSemiring Algebra.id IsScalarTower.right	—	of_measure_le_smul IsScalarTower.right le_rfl
zero 📖	mathematical	—	MeasureTheory.MemLp ContinuousENorm.toENorm ESeminormedAddMonoid.toContinuousENorm Pi.instZero AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass ESeminormedAddMonoid.toAddMonoid	—	MeasureTheory.aestronglyMeasurable_zero MeasureTheory.eLpNorm_zero ENNReal.coe_lt_top
zero' 📖	mathematical	—	MeasureTheory.MemLp ContinuousENorm.toENorm ESeminormedAddMonoid.toContinuousENorm AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass ESeminormedAddMonoid.toAddMonoid	—	zero

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