Monotonicity

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

Statistics

Metric	Count
Definitions	0
Theorems eLpNorm_star, im, of_enorm_le_mul, of_le_mul, of_le_mul', of_nnnorm_le_mul, re, eLpNorm'_le_mul_eLpNorm'_of_ae_le_mul, eLpNorm'_le_nnreal_smul_eLpNorm'_of_ae_le_mul, eLpNorm'_le_nnreal_smul_eLpNorm'_of_ae_le_mul', eLpNormEssSup_le_nnreal_smul_eLpNormEssSup_of_ae_le_mul, eLpNormEssSup_le_nnreal_smul_eLpNormEssSup_of_ae_le_mul', eLpNorm_conj, eLpNorm_eq_zero_and_zero_of_ae_le_mul_neg, eLpNorm_le_mul_eLpNorm_of_ae_le_mul, eLpNorm_le_mul_eLpNorm_of_ae_le_mul', eLpNorm_le_mul_eLpNorm_of_ae_le_mul'', eLpNorm_le_nnreal_smul_eLpNorm_of_ae_le_mul, eLpNorm_le_nnreal_smul_eLpNorm_of_ae_le_mul', eLpNorm_star, le_eLpNorm_of_bddBelow	21
Total	21

MeasureTheory

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
eLpNorm'_le_mul_eLpNorm'_of_ae_le_mul 📖	mathematical	AEStronglyMeasurable Filter.Eventually ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder ENorm.enorm ContinuousENorm.toENorm ESeminormedAddMonoid.toContinuousENorm Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring CommSemiring.toSemiring ENNReal.instCommSemiring ae Measure Measure.instFunLike Measure.instOuterMeasureClass Real Real.instLT Real.instZero	eLpNorm'	—	Measure.instOuterMeasureClass Mathlib.Tactic.Linarith.lt_irrefl Mathlib.Tactic.Ring.of_eq Mathlib.Tactic.Ring.add_congr Mathlib.Tactic.Ring.sub_congr Mathlib.Tactic.Ring.cast_zero Mathlib.Meta.NormNum.isNat_ofNat Nat.cast_zero Mathlib.Tactic.Ring.atom_pf Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_add Mathlib.Tactic.Ring.neg_mul Mathlib.Tactic.Ring.neg_one_mul Mathlib.Meta.NormNum.IsInt.to_raw_eq Mathlib.Meta.NormNum.isInt_mul Mathlib.Meta.NormNum.IsInt.of_raw Mathlib.Meta.NormNum.IsNat.to_isInt Mathlib.Meta.NormNum.IsNat.of_raw Mathlib.Tactic.Ring.neg_zero Mathlib.Tactic.Ring.add_pf_zero_add Mathlib.Tactic.Ring.add_pf_add_zero Mathlib.Tactic.Ring.add_pf_add_overlap_zero Mathlib.Tactic.Ring.add_overlap_pf_zero Mathlib.Meta.NormNum.IsInt.to_isNat Mathlib.Meta.NormNum.isInt_add Mathlib.Tactic.Linarith.add_neg Real.instIsStrictOrderedRing Mathlib.Tactic.Linarith.sub_neg_of_lt Real.instIsOrderedRing Filter.Eventually.mono lintegral_eq_zero_iff' AEMeasurable.pow_const ENNReal.hasMeasurablePow AEStronglyMeasurable.enorm one_div PosMulReflectLE.toPosMulReflectLT PosMulStrictMono.toPosMulReflectLE IsStrictOrderedRing.toPosMulStrictMono IsOrderedRing.toPosMulMono Filter.Eventually.and MulZeroClass.mul_zero ENNReal.instCanonicallyOrderedAdd eLpNorm'_eq_zero_of_ae_eq_zero ENNReal.top_mul LT.lt.le ENNReal.rpow_le_rpow_iff ENNReal.mul_rpow_of_nonneg inv_mul_cancel₀ LT.lt.ne' ENNReal.rpow_one lintegral_const_mul' lintegral_mono_ae ENNReal.mul_rpow_eq_ite
eLpNorm'_le_nnreal_smul_eLpNorm'_of_ae_le_mul 📖	mathematical	Filter.Eventually NNReal Preorder.toLE PartialOrder.toPreorder instPartialOrderNNReal NNNorm.nnnorm SeminormedAddGroup.toNNNorm SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring instSemiringNNReal ae Measure Measure.instFunLike Measure.instOuterMeasureClass Real Real.instLT Real.instZero	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	—	Measure.instOuterMeasureClass ENNReal.rpow_le_rpow_iff ENNReal.smul_def smul_eq_mul ENNReal.mul_rpow_of_nonneg LT.lt.le one_div inv_mul_cancel₀ LT.lt.ne ENNReal.rpow_one ENNReal.coe_rpow_of_nonneg lintegral_const_mul' ENNReal.coe_ne_top lintegral_mono_ae NNReal.rpow_le_rpow_iff
eLpNorm'_le_nnreal_smul_eLpNorm'_of_ae_le_mul' 📖	mathematical	Filter.Eventually ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder ENorm.enorm ContinuousENorm.toENorm Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring CommSemiring.toSemiring ENNReal.instCommSemiring ENNReal.ofNNReal ae Measure Measure.instFunLike Measure.instOuterMeasureClass Real Real.instLT Real.instZero	eLpNorm' NNReal Algebra.toSMul instCommSemiringNNReal ENNReal.instAlgebraNNReal Algebra.id	—	Measure.instOuterMeasureClass ENNReal.rpow_le_rpow_iff ENNReal.smul_def smul_eq_mul ENNReal.mul_rpow_of_nonneg LT.lt.le one_div inv_mul_cancel₀ LT.lt.ne' ENNReal.rpow_one ENNReal.coe_rpow_of_nonneg lintegral_const_mul' ENNReal.coe_ne_top lintegral_mono_ae Mathlib.Tactic.Linarith.lt_irrefl Mathlib.Tactic.Ring.of_eq Mathlib.Tactic.Ring.add_congr Mathlib.Tactic.Ring.sub_congr Mathlib.Tactic.Ring.cast_zero Mathlib.Meta.NormNum.isNat_ofNat Nat.cast_zero Mathlib.Tactic.Ring.atom_pf Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_add Mathlib.Tactic.Ring.neg_mul Mathlib.Tactic.Ring.neg_one_mul Mathlib.Meta.NormNum.IsInt.to_raw_eq Mathlib.Meta.NormNum.isInt_mul Mathlib.Meta.NormNum.IsInt.of_raw Mathlib.Meta.NormNum.IsNat.to_isInt Mathlib.Meta.NormNum.IsNat.of_raw Mathlib.Tactic.Ring.neg_zero Mathlib.Tactic.Ring.add_pf_zero_add Mathlib.Tactic.Ring.add_pf_add_zero Mathlib.Tactic.Ring.add_pf_add_overlap_zero Mathlib.Tactic.Ring.add_overlap_pf_zero Mathlib.Meta.NormNum.IsInt.to_isNat Mathlib.Meta.NormNum.isInt_add Mathlib.Tactic.Linarith.add_neg Real.instIsStrictOrderedRing Mathlib.Tactic.Linarith.sub_neg_of_lt Real.instIsOrderedRing ENNReal.mul_rpow_eq_ite
eLpNormEssSup_le_nnreal_smul_eLpNormEssSup_of_ae_le_mul 📖	mathematical	Filter.Eventually NNReal Preorder.toLE PartialOrder.toPreorder instPartialOrderNNReal NNNorm.nnnorm SeminormedAddGroup.toNNNorm SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring instSemiringNNReal ae Measure Measure.instFunLike Measure.instOuterMeasureClass	ENNReal ENNReal.instPartialOrder eLpNormEssSup ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddGroup.toContinuousENorm Algebra.toSMul instCommSemiringNNReal CommSemiring.toSemiring ENNReal.instCommSemiring ENNReal.instAlgebraNNReal Algebra.id	—	Measure.instOuterMeasureClass essSup_mono_ae Filter.Eventually.mono ENNReal.coe_le_coe ENNReal.essSup_const_mul
eLpNormEssSup_le_nnreal_smul_eLpNormEssSup_of_ae_le_mul' 📖	mathematical	Filter.Eventually ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder ENorm.enorm ContinuousENorm.toENorm Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring CommSemiring.toSemiring ENNReal.instCommSemiring ae Measure Measure.instFunLike Measure.instOuterMeasureClass	eLpNormEssSup Algebra.toSMul Algebra.id	—	Measure.instOuterMeasureClass essSup_mono_ae ENNReal.essSup_const_mul
eLpNorm_conj 📖	mathematical	—	eLpNorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing DenselyNormedField.toNormedField RCLike.toDenselyNormedField SeminormedAddGroup.toContinuousENorm DFunLike.coe RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring Pi.commSemiring Semifield.toCommSemiring Field.toSemifield NormedField.toField RingHom.instFunLike starRingEnd Pi.starRing' NonUnitalNormedCommRing.toNonUnitalNormedRing NormedCommRing.toNonUnitalNormedCommRing RCLike.toStarRing	—	RCLike.norm_conj norm_norm
eLpNorm_eq_zero_and_zero_of_ae_le_mul_neg 📖	mathematical	Filter.Eventually Real Real.instLE Norm.norm NormedAddCommGroup.toNorm Real.instMul ae Measure Measure.instFunLike Measure.instOuterMeasureClass Real.instLT Real.instZero	eLpNorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm ENNReal instZeroENNReal	—	Measure.instOuterMeasureClass eLpNorm_congr_ae eLpNorm_zero Real.instIsStrictOrderedRing norm_nonneg
eLpNorm_le_mul_eLpNorm_of_ae_le_mul 📖	mathematical	Filter.Eventually Real Real.instLE Norm.norm NormedAddCommGroup.toNorm Real.instMul 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.ofReal	—	Measure.instOuterMeasureClass eLpNorm_le_nnreal_smul_eLpNorm_of_ae_le_mul Filter.Eventually.mono LE.le.trans mul_le_mul_of_nonneg_right IsOrderedRing.toMulPosMono Real.instIsOrderedRing Real.le_coe_toNNReal norm_nonneg
eLpNorm_le_mul_eLpNorm_of_ae_le_mul' 📖	mathematical	Filter.Eventually ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder ENorm.enorm ContinuousENorm.toENorm Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring CommSemiring.toSemiring ENNReal.instCommSemiring ENNReal.ofNNReal ae Measure Measure.instFunLike Measure.instOuterMeasureClass	eLpNorm	—	Measure.instOuterMeasureClass eLpNorm_le_nnreal_smul_eLpNorm_of_ae_le_mul'
eLpNorm_le_mul_eLpNorm_of_ae_le_mul'' 📖	mathematical	AEStronglyMeasurable Filter.Eventually ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder ENorm.enorm ContinuousENorm.toENorm ESeminormedAddMonoid.toContinuousENorm Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring CommSemiring.toSemiring ENNReal.instCommSemiring ae Measure Measure.instFunLike Measure.instOuterMeasureClass	eLpNorm	—	Measure.instOuterMeasureClass eLpNorm_exponent_zero MulZeroClass.mul_zero mul_ite eLpNormEssSup_le_nnreal_smul_eLpNormEssSup_of_ae_le_mul' eLpNorm'_le_mul_eLpNorm'_of_ae_le_mul ENNReal.toReal_pos
eLpNorm_le_nnreal_smul_eLpNorm_of_ae_le_mul 📖	mathematical	Filter.Eventually NNReal Preorder.toLE PartialOrder.toPreorder instPartialOrderNNReal NNNorm.nnnorm SeminormedAddGroup.toNNNorm SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring instSemiringNNReal 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	—	Measure.instOuterMeasureClass eLpNorm_exponent_zero smul_zero eLpNormEssSup_le_nnreal_smul_eLpNormEssSup_of_ae_le_mul eLpNorm_eq_eLpNorm' eLpNorm'_le_nnreal_smul_eLpNorm'_of_ae_le_mul ENNReal.toReal_pos
eLpNorm_le_nnreal_smul_eLpNorm_of_ae_le_mul' 📖	mathematical	Filter.Eventually ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder ENorm.enorm ContinuousENorm.toENorm Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring CommSemiring.toSemiring ENNReal.instCommSemiring ENNReal.ofNNReal ae Measure Measure.instFunLike Measure.instOuterMeasureClass	eLpNorm NNReal Algebra.toSMul instCommSemiringNNReal ENNReal.instAlgebraNNReal Algebra.id	—	Measure.instOuterMeasureClass eLpNorm_exponent_zero smul_zero eLpNormEssSup_le_nnreal_smul_eLpNormEssSup_of_ae_le_mul' eLpNorm_eq_eLpNorm' eLpNorm'_le_nnreal_smul_eLpNorm'_of_ae_le_mul' ENNReal.toReal_pos
eLpNorm_star 📖	mathematical	—	eLpNorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm Star.star Pi.instStarForall InvolutiveStar.toStar StarAddMonoid.toInvolutiveStar ESeminormedAddMonoid.toAddMonoid SeminormedAddCommGroup.toPseudoMetricSpace ESeminormedAddCommMonoid.toESeminormedAddMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid	—	eLpNorm_congr_norm_ae Filter.Eventually.of_forall Measure.instOuterMeasureClass norm_star
le_eLpNorm_of_bddBelow 📖	mathematical	MeasurableSet Filter.Eventually ae Measure Measure.instFunLike Measure.instOuterMeasureClass	ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder NNReal Algebra.toSMul instCommSemiringNNReal CommSemiring.toSemiring ENNReal.instCommSemiring ENNReal.instAlgebraNNReal Algebra.id Real ENNReal.instPowReal DFunLike.coe Set DivInvMonoid.toDiv Real.instDivInvMonoid Real.instOne ENNReal.toReal eLpNorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm	—	Measure.instOuterMeasureClass ENNReal.smul_def smul_eq_mul eLpNorm_eq_lintegral_rpow_enorm_toReal one_div ENNReal.le_rpow_inv_iff ENNReal.toReal_pos ENNReal.mul_rpow_of_nonneg ENNReal.toReal_nonneg ENNReal.rpow_mul inv_mul_cancel₀ LT.lt.ne ENNReal.rpow_one setLIntegral_const lintegral_indicator lintegral_mono_ae Filter.mp_mem Filter.univ_mem' Set.indicator_of_mem ENNReal.rpow_le_rpow coe_le_enorm Set.indicator_of_notMem ENNReal.instCanonicallyOrderedAdd

MeasureTheory.AEEqFun

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
eLpNorm_star 📖	mathematical	—	MeasureTheory.eLpNorm ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm cast SeminormedAddCommGroup.toPseudoMetricSpace Star.star MeasureTheory.AEEqFun instStarOfContinuousStar InvolutiveStar.toStar StarAddMonoid.toInvolutiveStar ESeminormedAddMonoid.toAddMonoid ESeminormedAddCommMonoid.toESeminormedAddMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedStarGroup.to_continuousStar	—	NormedStarGroup.to_continuousStar MeasureTheory.eLpNorm_congr_ae coeFn_star MeasureTheory.eLpNorm_star

MeasureTheory.MemLp

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
im 📖	mathematical	MeasureTheory.MemLp ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing DenselyNormedField.toNormedField RCLike.toDenselyNormedField SeminormedAddGroup.toContinuousENorm SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing	Real Real.normedCommRing Real.pseudoMetricSpace DFunLike.coe AddMonoidHom AddMonoid.toZero AddCommMonoid.toAddMonoid NonUnitalNonAssocSemiring.toAddCommMonoid NonUnitalSemiring.toNonUnitalNonAssocSemiring Semiring.toNonUnitalSemiring Ring.toSemiring CommRing.toRing Field.toCommRing NormedField.toField AddSemigroup.toAdd AddMonoid.toAddSemigroup AddZeroClass.toAddZero AddMonoid.toAddZeroClass Real.instAddMonoid AddMonoidHom.instFunLike RCLike.im	—	one_mul RCLike.norm_im_le_norm of_le_mul Continuous.comp_aestronglyMeasurable RCLike.continuous_im Filter.Eventually.of_forall MeasureTheory.Measure.instOuterMeasureClass
of_enorm_le_mul 📖	—	MeasureTheory.MemLp ContinuousENorm.toENorm MeasureTheory.AEStronglyMeasurable Filter.Eventually ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder ENorm.enorm Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring CommSemiring.toSemiring ENNReal.instCommSemiring ENNReal.ofNNReal MeasureTheory.ae MeasureTheory.Measure MeasureTheory.Measure.instFunLike MeasureTheory.Measure.instOuterMeasureClass	—	—	MeasureTheory.Measure.instOuterMeasureClass LE.le.trans_lt MeasureTheory.eLpNorm_le_nnreal_smul_eLpNorm_of_ae_le_mul' ENNReal.mul_lt_top ENNReal.coe_lt_top Ne.lt_top eLpNorm_ne_top
of_le_mul 📖	—	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 Real.instMul MeasureTheory.ae MeasureTheory.Measure MeasureTheory.Measure.instFunLike MeasureTheory.Measure.instOuterMeasureClass	—	—	MeasureTheory.Measure.instOuterMeasureClass LE.le.trans_lt MeasureTheory.eLpNorm_le_mul_eLpNorm_of_ae_le_mul ENNReal.mul_lt_top ENNReal.ofReal_lt_top Ne.lt_top eLpNorm_ne_top
of_le_mul' 📖	—	MeasureTheory.MemLp ContinuousENorm.toENorm MeasureTheory.AEStronglyMeasurable Filter.Eventually ENNReal Preorder.toLE PartialOrder.toPreorder ENNReal.instPartialOrder ENorm.enorm Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring CommSemiring.toSemiring ENNReal.instCommSemiring ENNReal.ofNNReal MeasureTheory.ae MeasureTheory.Measure MeasureTheory.Measure.instFunLike MeasureTheory.Measure.instOuterMeasureClass	—	—	MeasureTheory.Measure.instOuterMeasureClass LE.le.trans_lt MeasureTheory.eLpNorm_le_mul_eLpNorm_of_ae_le_mul' ENNReal.mul_lt_top ENNReal.coe_lt_top Ne.lt_top eLpNorm_ne_top
of_nnnorm_le_mul 📖	—	MeasureTheory.MemLp ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toPseudoMetricSpace MeasureTheory.AEStronglyMeasurable Filter.Eventually NNReal Preorder.toLE PartialOrder.toPreorder instPartialOrderNNReal NNNorm.nnnorm SeminormedAddGroup.toNNNorm Distrib.toMul NonUnitalNonAssocSemiring.toDistrib NonAssocSemiring.toNonUnitalNonAssocSemiring Semiring.toNonAssocSemiring instSemiringNNReal MeasureTheory.ae MeasureTheory.Measure MeasureTheory.Measure.instFunLike MeasureTheory.Measure.instOuterMeasureClass	—	—	MeasureTheory.Measure.instOuterMeasureClass LE.le.trans_lt MeasureTheory.eLpNorm_le_nnreal_smul_eLpNorm_of_ae_le_mul ENNReal.mul_lt_top ENNReal.coe_lt_top Ne.lt_top eLpNorm_ne_top
re 📖	mathematical	MeasureTheory.MemLp ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing DenselyNormedField.toNormedField RCLike.toDenselyNormedField SeminormedAddGroup.toContinuousENorm SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing	Real Real.normedCommRing Real.pseudoMetricSpace DFunLike.coe AddMonoidHom AddMonoid.toZero AddCommMonoid.toAddMonoid NonUnitalNonAssocSemiring.toAddCommMonoid NonUnitalSemiring.toNonUnitalNonAssocSemiring Semiring.toNonUnitalSemiring Ring.toSemiring CommRing.toRing Field.toCommRing NormedField.toField AddSemigroup.toAdd AddMonoid.toAddSemigroup AddZeroClass.toAddZero AddMonoid.toAddZeroClass Real.instAddMonoid AddMonoidHom.instFunLike RCLike.re	—	one_mul RCLike.norm_re_le_norm of_le_mul Continuous.comp_aestronglyMeasurable RCLike.continuous_re Filter.Eventually.of_forall MeasureTheory.Measure.instOuterMeasureClass

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