UniformIntegrable

📁 Source: Mathlib/MeasureTheory/Function/UniformIntegrable.lean

Statistics

Metric	Count
Definitions `UnifIntegrable`, `UniformIntegrable`	2
Theorems `eLpNormEssSup_indicator_norm_ge_eq_zero`, `eLpNorm_indicator_le`, `eLpNorm_indicator_le'`, `eLpNorm_indicator_le_of_meas`, `eLpNorm_indicator_norm_ge_le`, `eLpNorm_indicator_norm_ge_pos_le`, `integral_indicator_norm_ge_le`, `integral_indicator_norm_ge_nonneg_le`, `integral_indicator_norm_ge_nonneg_le_of_meas`, `add`, `ae_eq`, `indicator`, `neg`, `restrict`, `sub`, `unifIntegrable_of_ae_tendsto`, `unifIntegrable_of_tendstoInMeasure`, `ae_eq`, `aestronglyMeasurable`, `integrable_of_ae_tendsto`, `integrable_of_tendstoInMeasure`, `memLp`, `memLp_of_ae_tendsto`, `memLp_of_tendstoInMeasure`, `spec`, `spec'`, `unifIntegrable`, `uniformIntegrable_of_ae_tendsto`, `uniformIntegrable_of_tendstoInMeasure`, `eLpNorm_indicator_le_of_bound`, `tendstoInMeasure_iff_tendsto_Lp_finite`, `tendsto_Lp_finite_of_tendstoInMeasure`, `tendsto_Lp_finite_of_tendsto_ae`, `tendsto_Lp_finite_of_tendsto_ae_of_meas`, `tendsto_indicator_ge`, `unifIntegrable_congr_ae`, `unifIntegrable_const`, `unifIntegrable_fin`, `unifIntegrable_finite`, `unifIntegrable_of`, `unifIntegrable_of'`, `unifIntegrable_of_tendsto_Lp`, `unifIntegrable_of_tendsto_Lp_zero`, `unifIntegrable_subsingleton`, `unifIntegrable_zero_meas`, `uniformIntegrable_average`, `uniformIntegrable_average_real`, `uniformIntegrable_congr_ae`, `uniformIntegrable_const`, `uniformIntegrable_finite`, `uniformIntegrable_iff`, `uniformIntegrable_of`, `uniformIntegrable_of'`, `uniformIntegrable_subsingleton`, `uniformIntegrable_zero_meas`	55
Total	57

MeasureTheory

Definitions

Name	Category	Theorems
`UnifIntegrable` 📖	MathDef	13 math math: `unifIntegrable_zero_meas`, `unifIntegrable_subsingleton`, `tendstoInMeasure_iff_tendsto_Lp`, `unifIntegrable_of_tendsto_Lp_zero`, `unifIntegrable_fin`, `unifIntegrable_of'`, `unifIntegrable_const`, `unifIntegrable_congr_ae`, `UniformIntegrable.unifIntegrable`, `unifIntegrable_finite`, `tendstoInMeasure_iff_tendsto_Lp_finite`, `unifIntegrable_of`, `unifIntegrable_of_tendsto_Lp`
`UniformIntegrable` 📖	MathDef	12 math math: `uniformIntegrable_finite`, `uniformIntegrable_const`, `ProbabilityTheory.MemLp.uniformIntegrable_of_identDistrib_aux`, `uniformIntegrable_iff`, `uniformIntegrable_subsingleton`, `ProbabilityTheory.MemLp.uniformIntegrable_of_identDistrib`, `uniformIntegrable_zero_meas`, `uniformIntegrable_of`, `uniformIntegrable_of'`, `Integrable.uniformIntegrable_condExp_filtration`, `uniformIntegrable_congr_ae`, `Integrable.uniformIntegrable_condExp`

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`eLpNorm_indicator_le_of_bound` 📖	mathematical	`Real` `Real.instLT` `Real.instZero` `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` `Set.indicator` `NegZeroClass.toZero` `SubNegZeroMonoid.toNegZeroClass` `SubtractionMonoid.toSubNegZeroMonoid` `SubtractionCommMonoid.toSubtractionMonoid` `AddCommGroup.toDivisionAddCommMonoid` `NormedAddCommGroup.toAddCommGroup` `ENNReal.ofReal`	—	`zero_lt_one` `Real.instZeroLEOneClass` `NeZero.charZero_one` `RCLike.charZero_rclike` `Pi.zero_apply` `norm_le_zero_iff` `LT.lt.le` `lt_of_lt_of_le` `Set.indicator_zero'` `eLpNorm_zero` `ENNReal.instCanonicallyOrderedAdd` `Real.rpow_pos_of_pos` `div_pos` `PosMulReflectLE.toPosMulReflectLT` `PosMulStrictMono.toPosMulReflectLE` `IsStrictOrderedRing.toPosMulStrictMono` `Real.instIsStrictOrderedRing` `eLpNorm_exponent_zero` `eLpNorm_indicator_eq_eLpNorm_restrict` `Measure.instOuterMeasureClass` `Filter.univ_mem'` `le_trans` `eLpNorm_le_of_ae_bound` `Measure.restrict_apply` `MeasurableSet.univ` `Set.univ_inter` `ENNReal.le_div_iff_mul_le` `ENNReal.ofReal_ne_top` `ENNReal.rpow_inv_le_iff` `ENNReal.toReal_pos` `ENNReal.ofReal_rpow_of_pos` `ENNReal.rpow_le_rpow_iff` `ENNReal.ofReal_div_of_pos` `le_refl`
`tendstoInMeasure_iff_tendsto_Lp_finite` 📖	mathematical	`ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `MemLp` `ContinuousENorm.toENorm` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `NormedAddCommGroup.toSeminormedAddCommGroup` `SeminormedAddGroup.toContinuousENorm` `SeminormedAddCommGroup.toPseudoMetricSpace`	`TendstoInMeasure` `PseudoEMetricSpace.toEDist` `EMetricSpace.toPseudoEMetricSpace` `MetricSpace.toEMetricSpace` `NormedAddCommGroup.toMetricSpace` `Filter.atTop` `Nat.instPreorder` `UnifIntegrable` `Filter.Tendsto` `eLpNorm` `Pi.instSub` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `NormedAddGroup.toAddGroup` `NormedAddCommGroup.toNormedAddGroup` `nhds` `ENNReal.instTopologicalSpace` `instZeroENNReal`	—	`tendsto_Lp_finite_of_tendstoInMeasure` `tendstoInMeasure_of_tendsto_eLpNorm` `LT.lt.ne` `lt_of_lt_of_le` `zero_lt_one` `IsOrderedRing.toZeroLEOneClass` `ENNReal.instIsOrderedRing` `NeZero.charZero_one` `ENNReal.instCharZero` `MemLp.aestronglyMeasurable` `unifIntegrable_of_tendsto_Lp`
`tendsto_Lp_finite_of_tendstoInMeasure` 📖	mathematical	`ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `AEStronglyMeasurable` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddCommGroup.toPseudoMetricSpace` `NormedAddCommGroup.toSeminormedAddCommGroup` `MemLp` `ContinuousENorm.toENorm` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `SeminormedAddGroup.toContinuousENorm` `UnifIntegrable` `TendstoInMeasure` `PseudoEMetricSpace.toEDist` `EMetricSpace.toPseudoEMetricSpace` `MetricSpace.toEMetricSpace` `NormedAddCommGroup.toMetricSpace` `Filter.atTop` `Nat.instPreorder`	`Filter.Tendsto` `eLpNorm` `Pi.instSub` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `NormedAddGroup.toAddGroup` `NormedAddCommGroup.toNormedAddGroup` `nhds` `ENNReal.instTopologicalSpace` `instZeroENNReal`	—	`Filter.tendsto_of_subseq_tendsto` `instIsCountablyGenerated_atTop` `OrderTopology.of_linearLocallyFinite` `instDiscreteTopologyNat` `TopologicalSpace.SecondCountableTopology.to_separableSpace` `TopologicalSpace.instSecondCountableTopologyOfLindelofSpaceOfPseudoMetrizableSpace` `Countable.LindelofSpace` `instCountableNat` `PseudoEMetricSpace.pseudoMetrizableSpace` `Measure.instOuterMeasureClass` `TendstoInMeasure.exists_seq_tendsto_ae` `Filter.Tendsto.comp` `tendsto_Lp_finite_of_tendsto_ae`
`tendsto_Lp_finite_of_tendsto_ae` 📖	mathematical	`ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `AEStronglyMeasurable` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddCommGroup.toPseudoMetricSpace` `NormedAddCommGroup.toSeminormedAddCommGroup` `MemLp` `ContinuousENorm.toENorm` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `SeminormedAddGroup.toContinuousENorm` `UnifIntegrable` `Filter.Eventually` `Filter.Tendsto` `Filter.atTop` `Nat.instPreorder` `nhds` `ae` `Measure` `Measure.instFunLike` `Measure.instOuterMeasureClass`	`eLpNorm` `Pi.instSub` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `NormedAddGroup.toAddGroup` `NormedAddCommGroup.toNormedAddGroup` `ENNReal.instTopologicalSpace` `instZeroENNReal`	—	`Measure.instOuterMeasureClass` `eLpNorm_congr_ae` `Filter.EventuallyEq.sub` `tendsto_Lp_finite_of_tendsto_ae_of_meas` `MemLp.ae_eq` `UnifIntegrable.ae_eq` `ae_all_iff` `instCountableNat` `Filter.mp_mem` `Filter.univ_mem'`
`tendsto_Lp_finite_of_tendsto_ae_of_meas` 📖	mathematical	`ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `StronglyMeasurable` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddCommGroup.toPseudoMetricSpace` `NormedAddCommGroup.toSeminormedAddCommGroup` `MemLp` `ContinuousENorm.toENorm` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `SeminormedAddGroup.toContinuousENorm` `UnifIntegrable` `Filter.Eventually` `Filter.Tendsto` `Filter.atTop` `Nat.instPreorder` `nhds` `ae` `Measure` `Measure.instFunLike` `Measure.instOuterMeasureClass`	`eLpNorm` `Pi.instSub` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `NormedAddGroup.toAddGroup` `NormedAddCommGroup.toNormedAddGroup` `ENNReal.instTopologicalSpace` `instZeroENNReal`	—	`Measure.instOuterMeasureClass` `ENNReal.tendsto_atTop_zero` `top_le_iff` `eLpNorm_measure_zero` `ENNReal.instCanonicallyOrderedAdd` `Nat.instAtLeastTwoHAddOfNat` `div_pos` `PosMulReflectLE.toPosMulReflectLT` `PosMulStrictMono.toPosMulReflectLE` `IsStrictOrderedRing.toPosMulStrictMono` `Real.instIsStrictOrderedRing` `ENNReal.toReal_pos` `LT.lt.ne'` `LT.lt.ne` `Real.instIsOrderedRing` `Real.instNontrivial` `Mathlib.Meta.Positivity.div_nonneg_of_pos_of_nonneg` `Mathlib.Meta.Positivity.pos_of_isNat` `Mathlib.Meta.NormNum.isNat_ofNat` `Nat.cast_one` `ENNReal.toReal_nonneg` `Real.rpow_pos_of_pos` `measureUnivNNReal_pos` `MemLp.eLpNorm_indicator_le` `tendstoUniformlyOn_of_ae_tendsto'` `instCountableNat` `lt_min` `Filter.eventually_atTop` `instIsDirectedOrder` `IsStrictOrderedRing.toIsOrderedRing` `Nat.instIsStrictOrderedRing` `instArchimedeanNat` `Metric.tendstoUniformlyOn_iff` `mul_pos` `Set.indicator_self_add_compl` `le_imp_le_of_le_of_le` `eLpNorm_add_le` `StronglyMeasurable.aestronglyMeasurable` `StronglyMeasurable.indicator` `StronglyMeasurable.sub` `IsTopologicalAddGroup.to_continuousSub` `SeminormedAddCommGroup.toIsTopologicalAddGroup` `MeasurableSet.compl` `le_refl` `sub_eq_add_neg` `Set.indicator_add'` `Set.indicator_neg'` `add_le_add_left` `covariant_swap_add_of_covariant_add` `IsOrderedAddMonoid.toAddLeftMono` `ENNReal.instIsOrderedAddMonoid` `StronglyMeasurable.neg` `IsTopologicalAddGroup.toContinuousNeg` `le_trans` `ENNReal.ofReal_le_ofReal_iff` `LT.lt.le` `min_le_left` `min_le_right` `div_nonneg` `mul_nonneg` `IsOrderedRing.toPosMulMono` `le_of_lt` `Mathlib.Meta.NormNum.instAtLeastTwo` `Real.rpow_nonneg` `NNReal.coe_nonneg` `eLpNorm_indicator_sub_le_of_dist_bdd` `dist_comm` `div_mul_eq_div_mul_one_div` `ENNReal.ofReal_toReal` `measure_lt_top` `ENNReal.ofReal_rpow_of_nonneg` `ENNReal.ofReal_mul` `mul_assoc` `ENNReal.ofReal_le_ofReal` `mul_le_of_le_one_right` `mul_comm` `mul_one_div` `div_le_one` `Real.rpow_le_rpow` `ENNReal.toReal_le_of_le_ofReal` `ENNReal.ofReal_coe_nnreal` `coe_measureUnivNNReal` `measure_mono` `Set.subset_univ` `ENNReal.ofReal_div_of_pos` `ENNReal.ofReal_ofNat` `eLpNorm_neg` `ENNReal.add_thirds` `add_le_add`
`tendsto_indicator_ge` 📖	mathematical	—	`Filter.Tendsto` `Set.indicator` `NegZeroClass.toZero` `SubNegZeroMonoid.toNegZeroClass` `SubtractionMonoid.toSubNegZeroMonoid` `SubtractionCommMonoid.toSubtractionMonoid` `AddCommGroup.toDivisionAddCommMonoid` `NormedAddCommGroup.toAddCommGroup` `setOf` `Real` `Real.instLE` `Real.instNatCast` `NNReal.toReal` `NNNorm.nnnorm` `SeminormedAddGroup.toNNNorm` `SeminormedAddCommGroup.toSeminormedAddGroup` `NormedAddCommGroup.toSeminormedAddCommGroup` `Filter.atTop` `Nat.instPreorder` `nhds` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddCommGroup.toPseudoMetricSpace`	—	`tendsto_atTop_of_eventually_const` `Real.instIsStrictOrderedRing` `Set.indicator_of_notMem` `lt_of_le_of_lt` `Nat.le_ceil` `lt_of_lt_of_le` `lt_add_one` `Real.instZeroLEOneClass` `NeZero.charZero_one` `RCLike.charZero_rclike` `IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono` `AddLeftCancelSemigroup.toIsLeftCancelAdd` `IsOrderedAddMonoid.toAddLeftMono` `Real.instIsOrderedAddMonoid` `Nat.cast_one`
`unifIntegrable_congr_ae` 📖	mathematical	`Filter.EventuallyEq` `ae` `Measure` `Measure.instFunLike` `Measure.instOuterMeasureClass`	`UnifIntegrable`	—	`Measure.instOuterMeasureClass` `UnifIntegrable.ae_eq` `Filter.EventuallyEq.symm`
`unifIntegrable_const` 📖	mathematical	`ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `MemLp` `ContinuousENorm.toENorm` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `NormedAddCommGroup.toSeminormedAddCommGroup` `SeminormedAddGroup.toContinuousENorm` `SeminormedAddCommGroup.toPseudoMetricSpace`	`UnifIntegrable`	—	`MemLp.eLpNorm_indicator_le`
`unifIntegrable_fin` 📖	mathematical	`ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `MemLp` `ContinuousENorm.toENorm` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `NormedAddCommGroup.toSeminormedAddCommGroup` `SeminormedAddGroup.toContinuousENorm` `SeminormedAddCommGroup.toPseudoMetricSpace`	`UnifIntegrable`	—	`unifIntegrable_subsingleton` `MemLp.eLpNorm_indicator_le` `lt_min` `le_trans` `ENNReal.ofReal_le_ofReal` `min_le_left` `min_le_right` `le_antisymm`
`unifIntegrable_finite` 📖	mathematical	`ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `MemLp` `ContinuousENorm.toENorm` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `NormedAddCommGroup.toSeminormedAddCommGroup` `SeminormedAddGroup.toContinuousENorm` `SeminormedAddCommGroup.toPseudoMetricSpace`	`UnifIntegrable`	—	`Finite.exists_equiv_fin` `unifIntegrable_fin` `Equiv.symm_apply_apply`
`unifIntegrable_of` 📖	mathematical	`ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `AEStronglyMeasurable` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddCommGroup.toPseudoMetricSpace` `NormedAddCommGroup.toSeminormedAddCommGroup` `eLpNorm` `ContinuousENorm.toENorm` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `SeminormedAddGroup.toContinuousENorm` `Set.indicator` `NegZeroClass.toZero` `SubNegZeroMonoid.toNegZeroClass` `SubtractionMonoid.toSubNegZeroMonoid` `SubtractionCommMonoid.toSubtractionMonoid` `AddCommGroup.toDivisionAddCommMonoid` `NormedAddCommGroup.toAddCommGroup` `setOf` `NNReal` `instPartialOrderNNReal` `NNNorm.nnnorm` `SeminormedAddGroup.toNNNorm` `ENNReal.ofReal`	`UnifIntegrable`	—	`Measure.instOuterMeasureClass` `UnifIntegrable.ae_eq` `unifIntegrable_of'` `le_trans` `le_of_eq` `eLpNorm_congr_ae` `Filter.mp_mem` `Filter.univ_mem'` `Set.indicator_of_mem` `Set.mem_setOf` `Set.indicator_of_notMem` `lt_max_of_lt_right` `one_pos` `LinearOrderedCommMonoidWithZero.toZeroLeOneClass` `NeZero.charZero_one` `IsStrictOrderedRing.toCharZero` `NNReal.instIsStrictOrderedRing` `eLpNorm_mono` `norm_indicator_eq_indicator_norm` `le_imp_le_of_le_of_le` `Set.indicator_le_indicator_apply_of_subset` `Set.setOf_subset_setOf_of_imp` `le_max_left` `le_refl` `norm_nonneg` `Filter.EventuallyEq.symm`
`unifIntegrable_of'` 📖	mathematical	`ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `StronglyMeasurable` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddCommGroup.toPseudoMetricSpace` `NormedAddCommGroup.toSeminormedAddCommGroup` `NNReal` `Preorder.toLT` `instPartialOrderNNReal` `instZeroNNReal` `eLpNorm` `ContinuousENorm.toENorm` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `SeminormedAddGroup.toContinuousENorm` `Set.indicator` `NegZeroClass.toZero` `SubNegZeroMonoid.toNegZeroClass` `SubtractionMonoid.toSubNegZeroMonoid` `SubtractionCommMonoid.toSubtractionMonoid` `AddCommGroup.toDivisionAddCommMonoid` `NormedAddCommGroup.toAddCommGroup` `setOf` `NNNorm.nnnorm` `SeminormedAddGroup.toNNNorm` `ENNReal.ofReal`	`UnifIntegrable`	—	`LT.lt.ne` `lt_of_lt_of_le` `zero_lt_one` `IsOrderedRing.toZeroLEOneClass` `ENNReal.instIsOrderedRing` `NeZero.charZero_one` `ENNReal.instCharZero` `unifIntegrable_zero_meas` `Measure.measure_univ_eq_zero` `Nat.instAtLeastTwoHAddOfNat` `half_pos` `PosMulReflectLE.toPosMulReflectLT` `PosMulStrictMono.toPosMulReflectLE` `IsStrictOrderedRing.toPosMulStrictMono` `Real.instIsStrictOrderedRing` `Real.rpow_pos_of_pos` `div_pos` `mul_pos` `two_pos` `Real.instZeroLEOneClass` `RCLike.charZero_rclike` `IsOrderedAddMonoid.toAddLeftMono` `Real.instIsOrderedAddMonoid` `NNReal.coe_pos` `Measure.instOuterMeasureClass` `eLpNorm_eq_zero_iff` `StronglyMeasurable.aestronglyMeasurable` `StronglyMeasurable.indicator` `indicator_meas_zero` `ENNReal.instCanonicallyOrderedAdd` `le_trans` `Eq.le` `Set.indicator_union_of_disjoint` `Disjoint.inf_left'` `Disjoint.inf_right'` `disjoint_iff_inf_le` `Set.mem_setOf_eq` `eq_of_le_of_not_lt` `not_lt` `LT.lt.le` `Set.inter_union_distrib_left` `Set.ext` `Set.inter_univ` `eLpNorm_add_le` `MeasurableSet.inter` `StronglyMeasurable.measurableSet_le` `OrderTopology.to_orderClosedTopology` `NNReal.instOrderTopology` `PseudoEMetricSpace.pseudoMetrizableSpace` `stronglyMeasurable_const` `StronglyMeasurable.nnnorm` `StronglyMeasurable.measurableSet_lt` `add_le_add` `ENNReal.instIsOrderedAddMonoid` `covariant_swap_add_of_covariant_add` `eLpNorm_mono` `norm_indicator_le_of_subset` `Set.inter_subset_right` `Set.indicator_indicator` `eLpNorm_indicator_eq_eLpNorm_restrict` `Filter.univ_mem'` `norm_indicator_eq_indicator_norm` `Set.indicator_le'` `NNReal.coe_nonneg` `eLpNorm_le_of_ae_bound` `mul_comm` `Measure.restrict_apply'` `Set.univ_inter` `ENNReal.ofReal_coe_nnreal` `one_div` `le_refl` `le_imp_le_of_le_of_le` `add_le_add_left` `add_le_add_right` `mul_le_mul_right` `IsOrderedMonoid.toMulLeftMono` `ENNReal.instIsOrderedMonoid` `ENNReal.rpow_inv_le_iff` `ENNReal.toReal_pos` `ENNReal.ofReal_rpow_of_pos` `ENNReal.ofReal_mul` `div_div` `mul_div_cancel₀` `ENNReal.ofReal_add` `add_halves` `ZeroLEOneClass.neZero.two`
`unifIntegrable_of_tendsto_Lp` 📖	mathematical	`ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `MemLp` `ContinuousENorm.toENorm` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `NormedAddCommGroup.toSeminormedAddCommGroup` `SeminormedAddGroup.toContinuousENorm` `SeminormedAddCommGroup.toPseudoMetricSpace` `Filter.Tendsto` `eLpNorm` `Pi.instSub` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `NormedAddGroup.toAddGroup` `NormedAddCommGroup.toNormedAddGroup` `Filter.atTop` `Nat.instPreorder` `nhds` `ENNReal.instTopologicalSpace` `instZeroENNReal`	`UnifIntegrable`	—	`add_sub_cancel` `UnifIntegrable.add` `unifIntegrable_const` `unifIntegrable_of_tendsto_Lp_zero` `MemLp.sub` `MemLp.aestronglyMeasurable` `AEStronglyMeasurable.sub` `SeminormedAddCommGroup.toIsTopologicalAddGroup`
`unifIntegrable_of_tendsto_Lp_zero` 📖	mathematical	`ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `MemLp` `ContinuousENorm.toENorm` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `NormedAddCommGroup.toSeminormedAddCommGroup` `SeminormedAddGroup.toContinuousENorm` `SeminormedAddCommGroup.toPseudoMetricSpace` `Filter.Tendsto` `eLpNorm` `Filter.atTop` `Nat.instPreorder` `nhds` `ENNReal.instTopologicalSpace` `instZeroENNReal`	`UnifIntegrable`	—	`ENNReal.tendsto_atTop_zero` `unifIntegrable_fin` `LE.le.trans` `eLpNorm_indicator_le`
`unifIntegrable_subsingleton` 📖	mathematical	`ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `MemLp` `ContinuousENorm.toENorm` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `NormedAddCommGroup.toSeminormedAddCommGroup` `SeminormedAddGroup.toContinuousENorm` `SeminormedAddCommGroup.toPseudoMetricSpace`	`UnifIntegrable`	—	`MemLp.eLpNorm_indicator_le` `Lean.Meta.FastSubsingleton.elim` `zero_lt_one` `Real.instZeroLEOneClass` `NeZero.charZero_one` `RCLike.charZero_rclike`
`unifIntegrable_zero_meas` 📖	mathematical	—	`UnifIntegrable` `Measure` `Measure.instZero`	—	`one_pos` `Real.instZeroLEOneClass` `NeZero.charZero_one` `RCLike.charZero_rclike` `eLpNorm_measure_zero` `ENNReal.instCanonicallyOrderedAdd`
`uniformIntegrable_average` 📖	mathematical	`ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `UniformIntegrable`	`Real` `Function.hasSMul` `SMulZeroClass.toSMul` `AddZero.toZero` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `ESeminormedAddMonoid.toAddMonoid` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddCommGroup.toPseudoMetricSpace` `NormedAddCommGroup.toSeminormedAddCommGroup` `ESeminormedAddCommMonoid.toESeminormedAddMonoid` `ENormedAddCommMonoid.toESeminormedAddCommMonoid` `NormedAddCommGroup.toENormedAddCommMonoid` `DistribSMul.toSMulZeroClass` `DistribMulAction.toDistribSMul` `Real.instMonoid` `Module.toDistribMulAction` `Real.semiring` `ESeminormedAddCommMonoid.toAddCommMonoid` `NormedSpace.toModule` `Real.normedField` `Real.instInv` `Real.instNatCast` `Finset.sum` `Pi.addCommMonoid` `Finset.range`	—	`AEStronglyMeasurable.const_smul` `UniformContinuousConstSMul.to_continuousConstSMul` `IsBoundedSMul.toUniformContinuousConstSMul` `NormedSpace.toIsBoundedSMul` `Finset.aestronglyMeasurable_sum` `IsTopologicalAddGroup.toContinuousAdd` `SeminormedAddCommGroup.toIsTopologicalAddGroup` `Finset.smul_sum` `Finset.indicator_sum` `le_trans` `eLpNorm_sum_le` `AEStronglyMeasurable.indicator` `Set.indicator_const_smul` `eq_or_ne` `Finset.sum_congr` `CharP.cast_eq_zero` `CharP.ofCharZero` `RCLike.charZero_rclike` `inv_zero` `zero_smul` `Set.indicator_zero'` `eLpNorm_zero` `Finset.sum_const_zero` `ENNReal.instCanonicallyOrderedAdd` `eLpNorm_const_smul` `NormedSpace.toNormSMulClass` `enorm_inv` `ne_of_gt` `Nat.cast_pos'` `IsOrderedAddMonoid.toAddLeftMono` `Real.instIsOrderedAddMonoid` `Real.instZeroLEOneClass` `NeZero.charZero_one` `lt_of_le_of_ne'` `zero_le` `Nat.instCanonicallyOrderedAdd` `Real.enorm_natCast` `ENNReal.div_eq_inv_mul` `ENNReal.div_le_of_le_mul'` `Finset.card_range` `nsmul_eq_mul` `Finset.sum_le_card_nsmul` `ENNReal.instIsOrderedAddMonoid` `LE.le.trans`
`uniformIntegrable_average_real` 📖	mathematical	`ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `UniformIntegrable` `Real` `Real.normedAddCommGroup`	`Pi.instDiv` `DivInvMonoid.toDiv` `Real.instDivInvMonoid` `Finset.sum` `Pi.addCommMonoid` `Real.instAddCommMonoid` `Finset.range` `Pi.instNatCast` `Real.instNatCast`	—	`Finset.sum_apply` `div_eq_inv_mul` `uniformIntegrable_average`
`uniformIntegrable_congr_ae` 📖	mathematical	`Filter.EventuallyEq` `ae` `Measure` `Measure.instFunLike` `Measure.instOuterMeasureClass`	`UniformIntegrable`	—	`Measure.instOuterMeasureClass` `UniformIntegrable.ae_eq` `Filter.EventuallyEq.symm`
`uniformIntegrable_const` 📖	mathematical	`ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `MemLp` `ContinuousENorm.toENorm` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `NormedAddCommGroup.toSeminormedAddCommGroup` `SeminormedAddGroup.toContinuousENorm` `SeminormedAddCommGroup.toPseudoMetricSpace`	`UniformIntegrable`	—	`unifIntegrable_const` `le_of_eq` `ENNReal.coe_toNNReal` `LT.lt.ne`
`uniformIntegrable_finite` 📖	mathematical	`ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `MemLp` `ContinuousENorm.toENorm` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `NormedAddCommGroup.toSeminormedAddCommGroup` `SeminormedAddGroup.toContinuousENorm` `SeminormedAddCommGroup.toPseudoMetricSpace`	`UniformIntegrable`	—	`nonempty_fintype` `unifIntegrable_finite` `Finset.mem_image` `Finset.mem_univ` `ENNReal.coe_toNNReal` `ne_of_lt` `Finset.max'_lt_iff` `Finset.le_max'`
`uniformIntegrable_iff` 📖	mathematical	`ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal`	`UniformIntegrable` `AEStronglyMeasurable` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddCommGroup.toPseudoMetricSpace` `NormedAddCommGroup.toSeminormedAddCommGroup` `eLpNorm` `ContinuousENorm.toENorm` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `SeminormedAddGroup.toContinuousENorm` `Set.indicator` `NegZeroClass.toZero` `SubNegZeroMonoid.toNegZeroClass` `SubtractionMonoid.toSubNegZeroMonoid` `SubtractionCommMonoid.toSubtractionMonoid` `AddCommGroup.toDivisionAddCommMonoid` `NormedAddCommGroup.toAddCommGroup` `setOf` `NNReal` `instPartialOrderNNReal` `NNNorm.nnnorm` `SeminormedAddGroup.toNNNorm` `ENNReal.ofReal`	—	`UniformIntegrable.spec` `LT.lt.ne` `lt_of_lt_of_le` `zero_lt_one` `IsOrderedRing.toZeroLEOneClass` `ENNReal.instIsOrderedRing` `NeZero.charZero_one` `ENNReal.instCharZero` `uniformIntegrable_of`
`uniformIntegrable_of` 📖	mathematical	`ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `AEStronglyMeasurable` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddCommGroup.toPseudoMetricSpace` `NormedAddCommGroup.toSeminormedAddCommGroup` `eLpNorm` `ContinuousENorm.toENorm` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `SeminormedAddGroup.toContinuousENorm` `Set.indicator` `NegZeroClass.toZero` `SubNegZeroMonoid.toNegZeroClass` `SubtractionMonoid.toSubNegZeroMonoid` `SubtractionCommMonoid.toSubtractionMonoid` `AddCommGroup.toDivisionAddCommMonoid` `NormedAddCommGroup.toAddCommGroup` `setOf` `NNReal` `instPartialOrderNNReal` `NNNorm.nnnorm` `SeminormedAddGroup.toNNNorm` `ENNReal.ofReal`	`UniformIntegrable`	—	`Measure.instOuterMeasureClass` `Filter.EventuallyEq.symm` `UniformIntegrable.ae_eq` `uniformIntegrable_of'` `le_trans` `le_of_eq` `eLpNorm_congr_ae` `Filter.mp_mem` `Filter.univ_mem'` `Set.indicator_of_mem` `Set.mem_setOf` `Set.indicator_of_notMem`
`uniformIntegrable_of'` 📖	mathematical	`ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `StronglyMeasurable` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddCommGroup.toPseudoMetricSpace` `NormedAddCommGroup.toSeminormedAddCommGroup` `eLpNorm` `ContinuousENorm.toENorm` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `SeminormedAddGroup.toContinuousENorm` `Set.indicator` `NegZeroClass.toZero` `SubNegZeroMonoid.toNegZeroClass` `SubtractionMonoid.toSubNegZeroMonoid` `SubtractionCommMonoid.toSubtractionMonoid` `AddCommGroup.toDivisionAddCommMonoid` `NormedAddCommGroup.toAddCommGroup` `setOf` `NNReal` `instPartialOrderNNReal` `NNNorm.nnnorm` `SeminormedAddGroup.toNNNorm` `ENNReal.ofReal`	`UniformIntegrable`	—	`StronglyMeasurable.aestronglyMeasurable` `unifIntegrable_of` `one_pos` `Real.instZeroLEOneClass` `NeZero.charZero_one` `RCLike.charZero_rclike` `le_trans` `eLpNorm_mono_enorm` `Pi.add_apply` `Set.indicator_apply` `Set.indicator_of_notMem` `add_zero` `le_refl` `Set.indicator_of_mem` `zero_add` `eLpNorm_add_le` `StronglyMeasurable.indicator` `StronglyMeasurable.measurableSet_lt` `OrderTopology.to_orderClosedTopology` `NNReal.instOrderTopology` `PseudoEMetricSpace.pseudoMetrizableSpace` `StronglyMeasurable.nnnorm` `stronglyMeasurable_const` `StronglyMeasurable.measurableSet_le` `Measure.instOuterMeasureClass` `Filter.univ_mem'` `nnnorm_indicator_eq_indicator_nnnorm` `Set.indicator_le` `NNReal.instCanonicallyOrderedAdd` `LT.lt.le` `add_le_add` `IsOrderedAddMonoid.toAddLeftMono` `ENNReal.instIsOrderedAddMonoid` `covariant_swap_add_of_covariant_add` `eLpNorm_le_of_ae_bound` `ENNReal.ofReal_coe_nnreal` `mul_comm` `le_rfl` `ENNReal.ofReal_one` `ENNReal.coe_toNNReal` `ENNReal.Finiteness.add_ne_top` `ENNReal.mul_ne_top` `ENNReal.coe_ne_top` `ENNReal.rpow_ne_top_of_nonneg` `inv_nonneg_of_nonneg` `PosMulReflectLE.toPosMulReflectLT` `PosMulStrictMono.toPosMulReflectLE` `IsStrictOrderedRing.toPosMulStrictMono` `Real.instIsStrictOrderedRing` `ENNReal.toReal_nonneg` `measure_ne_top` `ENNReal.one_ne_top`
`uniformIntegrable_subsingleton` 📖	mathematical	`ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `MemLp` `ContinuousENorm.toENorm` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `NormedAddCommGroup.toSeminormedAddCommGroup` `SeminormedAddGroup.toContinuousENorm` `SeminormedAddCommGroup.toPseudoMetricSpace`	`UniformIntegrable`	—	`uniformIntegrable_finite` `Finite.of_subsingleton`
`uniformIntegrable_zero_meas` 📖	mathematical	—	`UniformIntegrable` `Measure` `Measure.instZero`	—	`aestronglyMeasurable_zero_measure` `unifIntegrable_zero_meas` `Eq.le` `eLpNorm_measure_zero`

MeasureTheory.MemLp

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`eLpNormEssSup_indicator_norm_ge_eq_zero` 📖	mathematical	`MeasureTheory.MemLp` `ContinuousENorm.toENorm` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `NormedAddCommGroup.toSeminormedAddCommGroup` `SeminormedAddGroup.toContinuousENorm` `SeminormedAddCommGroup.toPseudoMetricSpace` `Top.top` `ENNReal` `instTopENNReal` `MeasureTheory.StronglyMeasurable`	`MeasureTheory.eLpNormEssSup` `Set.indicator` `NegZeroClass.toZero` `SubNegZeroMonoid.toNegZeroClass` `SubtractionMonoid.toSubNegZeroMonoid` `SubtractionCommMonoid.toSubtractionMonoid` `AddCommGroup.toDivisionAddCommMonoid` `NormedAddCommGroup.toAddCommGroup` `setOf` `Real` `Real.instLE` `NNReal.toReal` `NNNorm.nnnorm` `SeminormedAddGroup.toNNNorm` `instZeroENNReal`	—	`eLpNorm_lt_top` `MeasureTheory.eLpNormEssSup_indicator_eq_eLpNormEssSup_restrict` `measurableSet_le` `BorelSpace.opensMeasurable` `Real.borelSpace` `HasSolidNorm.orderClosedTopology` `instHasSolidNormReal` `Real.instIsOrderedAddMonoid` `instSecondCountableTopologyReal` `measurable_const` `Measurable.subtype_coe` `MeasureTheory.StronglyMeasurable.measurable` `PseudoEMetricSpace.pseudoMetrizableSpace` `NNReal.borelSpace` `MeasureTheory.StronglyMeasurable.nnnorm` `MeasureTheory.eLpNorm_exponent_top` `Set.mem_setOf_eq` `ENNReal.toReal_lt_toReal` `LT.lt.ne` `ENNReal.coe_lt_top` `ENNReal.coe_toReal` `coe_nnnorm` `lt_of_lt_of_le` `ENNReal.Finiteness.add_ne_top` `ne_top_of_lt` `ENNReal.one_ne_top` `ENNReal.lt_add_right` `one_ne_zero` `NeZero.charZero_one` `ENNReal.instCharZero` `nonpos_iff_eq_zero` `ENNReal.instCanonicallyOrderedAdd` `LE.le.trans` `MeasureTheory.measure_mono` `MeasureTheory.Measure.instOuterMeasureClass` `MeasureTheory.enorm_ae_le_eLpNormEssSup` `MeasureTheory.eLpNormEssSup_measure_zero`
`eLpNorm_indicator_le` 📖	mathematical	`ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `MeasureTheory.MemLp` `ContinuousENorm.toENorm` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `NormedAddCommGroup.toSeminormedAddCommGroup` `SeminormedAddGroup.toContinuousENorm` `SeminormedAddCommGroup.toPseudoMetricSpace` `Real` `Real.instLT` `Real.instZero`	`MeasureTheory.eLpNorm` `Set.indicator` `NegZeroClass.toZero` `SubNegZeroMonoid.toNegZeroClass` `SubtractionMonoid.toSubNegZeroMonoid` `SubtractionCommMonoid.toSubtractionMonoid` `AddCommGroup.toDivisionAddCommMonoid` `NormedAddCommGroup.toAddCommGroup` `ENNReal.ofReal`	—	`MeasureTheory.Measure.instOuterMeasureClass` `eLpNorm_indicator_le_of_meas` `ae_eq` `MeasureTheory.eLpNorm_indicator_eq_eLpNorm_restrict` `MeasureTheory.eLpNorm_congr_ae` `Filter.EventuallyEq.restrict`
`eLpNorm_indicator_le'` 📖	mathematical	`ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `MeasureTheory.MemLp` `ContinuousENorm.toENorm` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `NormedAddCommGroup.toSeminormedAddCommGroup` `SeminormedAddGroup.toContinuousENorm` `SeminormedAddCommGroup.toPseudoMetricSpace` `MeasureTheory.StronglyMeasurable` `Real` `Real.instLT` `Real.instZero`	`MeasureTheory.eLpNorm` `Set.indicator` `NegZeroClass.toZero` `SubNegZeroMonoid.toNegZeroClass` `SubtractionMonoid.toSubNegZeroMonoid` `SubtractionCommMonoid.toSubtractionMonoid` `AddCommGroup.toDivisionAddCommMonoid` `NormedAddCommGroup.toAddCommGroup` `Distrib.toMul` `NonUnitalNonAssocSemiring.toDistrib` `NonAssocSemiring.toNonUnitalNonAssocSemiring` `Semiring.toNonAssocSemiring` `CommSemiring.toSemiring` `ENNReal.instCommSemiring` `instOfNatAtLeastTwo` `AddMonoidWithOne.toNatCast` `Nat.instAtLeastTwoHAddOfNat` `ENNReal.ofReal`	—	`Nat.instAtLeastTwoHAddOfNat` `eLpNorm_indicator_norm_ge_pos_le` `MeasureTheory.eLpNorm_indicator_le_of_bound` `norm_indicator_eq_indicator_norm` `Set.indicator_apply` `Pi.add_apply` `Set.indicator_of_mem` `Set.indicator_of_notMem` `add_zero` `zero_add` `MeasureTheory.eLpNorm_indicator_eq_eLpNorm_restrict` `le_trans` `MeasureTheory.eLpNorm_add_le` `MeasureTheory.StronglyMeasurable.aestronglyMeasurable` `MeasureTheory.StronglyMeasurable.indicator` `measurableSet_le` `BorelSpace.opensMeasurable` `Real.borelSpace` `HasSolidNorm.orderClosedTopology` `instHasSolidNormReal` `Real.instIsOrderedAddMonoid` `instSecondCountableTopologyReal` `measurable_const` `Measurable.subtype_coe` `MeasureTheory.StronglyMeasurable.measurable` `PseudoEMetricSpace.pseudoMetrizableSpace` `NNReal.borelSpace` `MeasureTheory.StronglyMeasurable.nnnorm` `measurableSet_lt` `two_mul` `add_le_add` `IsOrderedAddMonoid.toAddLeftMono` `ENNReal.instIsOrderedAddMonoid` `covariant_swap_add_of_covariant_add` `MeasureTheory.eLpNorm_mono_measure` `MeasureTheory.Measure.restrict_le_self`
`eLpNorm_indicator_le_of_meas` 📖	mathematical	`ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `MeasureTheory.MemLp` `ContinuousENorm.toENorm` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `NormedAddCommGroup.toSeminormedAddCommGroup` `SeminormedAddGroup.toContinuousENorm` `SeminormedAddCommGroup.toPseudoMetricSpace` `MeasureTheory.StronglyMeasurable` `Real` `Real.instLT` `Real.instZero`	`MeasureTheory.eLpNorm` `Set.indicator` `NegZeroClass.toZero` `SubNegZeroMonoid.toNegZeroClass` `SubtractionMonoid.toSubNegZeroMonoid` `SubtractionCommMonoid.toSubtractionMonoid` `AddCommGroup.toDivisionAddCommMonoid` `NormedAddCommGroup.toAddCommGroup` `ENNReal.ofReal`	—	`Nat.instAtLeastTwoHAddOfNat` `eLpNorm_indicator_le'` `half_pos` `PosMulReflectLE.toPosMulReflectLT` `PosMulStrictMono.toPosMulReflectLE` `IsStrictOrderedRing.toPosMulStrictMono` `Real.instIsStrictOrderedRing` `le_trans` `ENNReal.ofReal_div_of_pos` `zero_lt_two` `Real.instZeroLEOneClass` `NeZero.charZero_one` `RCLike.charZero_rclike` `IsOrderedAddMonoid.toAddLeftMono` `Real.instIsOrderedAddMonoid` `ENNReal.ofReal_ofNat` `ENNReal.mul_div_cancel` `Mathlib.Meta.NormNum.isNat_eq_false` `ENNReal.instCharZero` `Mathlib.Meta.NormNum.isNat_ofNat` `Mathlib.Meta.NormNum.instAtLeastTwo` `Nat.cast_zero` `le_refl`
`eLpNorm_indicator_norm_ge_le` 📖	mathematical	`MeasureTheory.MemLp` `ContinuousENorm.toENorm` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `NormedAddCommGroup.toSeminormedAddCommGroup` `SeminormedAddGroup.toContinuousENorm` `SeminormedAddCommGroup.toPseudoMetricSpace` `MeasureTheory.StronglyMeasurable` `Real` `Real.instLT` `Real.instZero`	`ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `MeasureTheory.eLpNorm` `Set.indicator` `NegZeroClass.toZero` `SubNegZeroMonoid.toNegZeroClass` `SubtractionMonoid.toSubNegZeroMonoid` `SubtractionCommMonoid.toSubtractionMonoid` `AddCommGroup.toDivisionAddCommMonoid` `NormedAddCommGroup.toAddCommGroup` `setOf` `Real.instLE` `NNReal.toReal` `NNNorm.nnnorm` `SeminormedAddGroup.toNNNorm` `ENNReal.ofReal`	—	`MeasureTheory.eLpNorm_exponent_zero` `ENNReal.instCanonicallyOrderedAdd` `eLpNormEssSup_indicator_norm_ge_eq_zero` `MeasureTheory.eLpNorm_exponent_top` `integral_indicator_norm_ge_nonneg_le` `norm_rpow` `Real.rpow_pos_of_pos` `MeasureTheory.eLpNorm_eq_lintegral_rpow_enorm_toReal` `ENNReal.rpow_one` `mul_one_div_cancel` `LT.lt.ne` `ENNReal.toReal_pos` `ENNReal.rpow_mul` `ENNReal.rpow_le_rpow_iff` `one_div_pos` `PosMulReflectLE.toPosMulReflectLT` `PosMulStrictMono.toPosMulReflectLE` `IsStrictOrderedRing.toPosMulStrictMono` `Real.instIsStrictOrderedRing` `ENNReal.ofReal_rpow_of_pos` `enorm_indicator_eq_indicator_enorm` `coe_nnnorm` `Real.norm_rpow_of_nonneg` `norm_nonneg` `norm_norm` `Real.rpow_le_rpow_iff` `Real.rpow_nonneg` `Real.rpow_mul` `Real.rpow_one` `Set.indicator_of_mem` `Set.mem_setOf_eq` `Real.enorm_of_nonneg` `ENNReal.ofReal_rpow_of_nonneg` `ENNReal.toReal_nonneg` `ofReal_norm` `Set.indicator_of_notMem` `ENNReal.zero_rpow_of_pos`
`eLpNorm_indicator_norm_ge_pos_le` 📖	mathematical	`MeasureTheory.MemLp` `ContinuousENorm.toENorm` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `NormedAddCommGroup.toSeminormedAddCommGroup` `SeminormedAddGroup.toContinuousENorm` `SeminormedAddCommGroup.toPseudoMetricSpace` `MeasureTheory.StronglyMeasurable` `Real` `Real.instLT` `Real.instZero`	`ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `MeasureTheory.eLpNorm` `Set.indicator` `NegZeroClass.toZero` `SubNegZeroMonoid.toNegZeroClass` `SubtractionMonoid.toSubNegZeroMonoid` `SubtractionCommMonoid.toSubtractionMonoid` `AddCommGroup.toDivisionAddCommMonoid` `NormedAddCommGroup.toAddCommGroup` `setOf` `Real.instLE` `NNReal.toReal` `NNNorm.nnnorm` `SeminormedAddGroup.toNNNorm` `ENNReal.ofReal`	—	`eLpNorm_indicator_norm_ge_le` `lt_of_lt_of_le` `zero_lt_one` `Real.instZeroLEOneClass` `NeZero.charZero_one` `RCLike.charZero_rclike` `le_max_right` `le_trans` `MeasureTheory.eLpNorm_mono` `norm_indicator_eq_indicator_norm` `le_imp_le_of_le_of_le` `Set.indicator_le_indicator_apply_of_subset` `Set.setOf_subset_setOf_of_imp` `le_max_left` `le_refl` `norm_nonneg`
`integral_indicator_norm_ge_le` 📖	mathematical	`MeasureTheory.MemLp` `ContinuousENorm.toENorm` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `NormedAddCommGroup.toSeminormedAddCommGroup` `SeminormedAddGroup.toContinuousENorm` `SeminormedAddCommGroup.toPseudoMetricSpace` `ENNReal` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `MeasureTheory.StronglyMeasurable` `Real` `Real.instLT` `Real.instZero`	`Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `MeasureTheory.lintegral` `ENNReal.ofNNReal` `NNNorm.nnnorm` `SeminormedAddGroup.toNNNorm` `Set.indicator` `NegZeroClass.toZero` `SubNegZeroMonoid.toNegZeroClass` `SubtractionMonoid.toSubNegZeroMonoid` `SubtractionCommMonoid.toSubtractionMonoid` `AddCommGroup.toDivisionAddCommMonoid` `NormedAddCommGroup.toAddCommGroup` `setOf` `Real.instLE` `NNReal.toReal` `ENNReal.ofReal`	—	`MeasureTheory.Measure.instOuterMeasureClass` `Filter.univ_mem'` `MeasureTheory.tendsto_indicator_ge` `MeasureTheory.AEStronglyMeasurable.indicator` `MeasureTheory.StronglyMeasurable.measurableSet_le` `HasSolidNorm.orderClosedTopology` `instHasSolidNormReal` `Real.instIsOrderedAddMonoid` `PseudoEMetricSpace.pseudoMetrizableSpace` `MeasureTheory.stronglyMeasurable_const` `Measurable.stronglyMeasurable` `instSecondCountableTopologyReal` `BorelSpace.opensMeasurable` `Real.borelSpace` `Measurable.coe_nnreal_real` `MeasureTheory.StronglyMeasurable.measurable` `NNReal.borelSpace` `MeasureTheory.StronglyMeasurable.nnnorm` `MeasureTheory.Integrable.norm` `MeasureTheory.memLp_one_iff_integrable` `MeasureTheory.tendsto_lintegral_norm_of_dominated_convergence` `Set.indicator_of_mem` `le_refl` `Set.indicator_of_notMem` `norm_zero` `norm_nonneg` `ENNReal.tendsto_atTop_zero` `ENNReal.ofReal_pos` `ENNReal.ofReal_eq_coe_nnreal` `sub_zero` `le_rfl`
`integral_indicator_norm_ge_nonneg_le` 📖	mathematical	`MeasureTheory.MemLp` `ContinuousENorm.toENorm` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `NormedAddCommGroup.toSeminormedAddCommGroup` `SeminormedAddGroup.toContinuousENorm` `SeminormedAddCommGroup.toPseudoMetricSpace` `ENNReal` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `Real` `Real.instLT` `Real.instZero`	`Real.instLE` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `MeasureTheory.lintegral` `ENorm.enorm` `Set.indicator` `NegZeroClass.toZero` `SubNegZeroMonoid.toNegZeroClass` `SubtractionMonoid.toSubNegZeroMonoid` `SubtractionCommMonoid.toSubtractionMonoid` `AddCommGroup.toDivisionAddCommMonoid` `NormedAddCommGroup.toAddCommGroup` `setOf` `NNReal.toReal` `NNNorm.nnnorm` `SeminormedAddGroup.toNNNorm` `ENNReal.ofReal`	—	`MeasureTheory.memLp_congr_ae` `integral_indicator_norm_ge_nonneg_le_of_meas` `LE.le.trans` `le_of_eq` `MeasureTheory.lintegral_congr_ae` `Filter.mp_mem` `MeasureTheory.Measure.instOuterMeasureClass` `Filter.univ_mem'` `Set.indicator_apply`
`integral_indicator_norm_ge_nonneg_le_of_meas` 📖	mathematical	`MeasureTheory.MemLp` `ContinuousENorm.toENorm` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `NormedAddCommGroup.toSeminormedAddCommGroup` `SeminormedAddGroup.toContinuousENorm` `SeminormedAddCommGroup.toPseudoMetricSpace` `ENNReal` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `MeasureTheory.StronglyMeasurable` `Real` `Real.instLT` `Real.instZero`	`Real.instLE` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `MeasureTheory.lintegral` `ENorm.enorm` `Set.indicator` `NegZeroClass.toZero` `SubNegZeroMonoid.toNegZeroClass` `SubtractionMonoid.toSubNegZeroMonoid` `SubtractionCommMonoid.toSubtractionMonoid` `AddCommGroup.toDivisionAddCommMonoid` `NormedAddCommGroup.toAddCommGroup` `setOf` `NNReal.toReal` `NNNorm.nnnorm` `SeminormedAddGroup.toNNNorm` `ENNReal.ofReal`	—	`integral_indicator_norm_ge_le` `le_max_right`

MeasureTheory.UnifIntegrable

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`add` 📖	mathematical	`MeasureTheory.UnifIntegrable` `ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `MeasureTheory.AEStronglyMeasurable` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddCommGroup.toPseudoMetricSpace` `NormedAddCommGroup.toSeminormedAddCommGroup`	`Pi.instAdd` `AddCommMagma.toAdd` `AddCommSemigroup.toAddCommMagma` `AddCommMonoid.toAddCommSemigroup` `ESeminormedAddCommMonoid.toAddCommMonoid` `ENormedAddCommMonoid.toESeminormedAddCommMonoid` `NormedAddCommGroup.toENormedAddCommMonoid`	—	`Nat.instAtLeastTwoHAddOfNat` `half_pos` `PosMulReflectLE.toPosMulReflectLT` `PosMulStrictMono.toPosMulReflectLE` `IsStrictOrderedRing.toPosMulStrictMono` `Real.instIsStrictOrderedRing` `lt_min` `Set.indicator_add'` `LE.le.trans` `MeasureTheory.eLpNorm_add_le` `MeasureTheory.AEStronglyMeasurable.indicator` `ENNReal.ofReal_add` `LT.lt.le` `add_halves` `ZeroLEOneClass.neZero.two` `Real.instZeroLEOneClass` `NeZero.charZero_one` `RCLike.charZero_rclike` `IsOrderedAddMonoid.toAddLeftMono` `Real.instIsOrderedAddMonoid` `add_le_add` `ENNReal.instIsOrderedAddMonoid` `covariant_swap_add_of_covariant_add` `ENNReal.ofReal_le_ofReal` `min_le_left` `min_le_right`
`ae_eq` 📖	—	`MeasureTheory.UnifIntegrable` `Filter.EventuallyEq` `MeasureTheory.ae` `MeasureTheory.Measure` `MeasureTheory.Measure.instFunLike` `MeasureTheory.Measure.instOuterMeasureClass`	—	—	`MeasureTheory.Measure.instOuterMeasureClass` `LE.le.trans` `le_of_eq` `MeasureTheory.eLpNorm_congr_ae` `Filter.mp_mem` `Filter.univ_mem'` `Set.indicator_apply`
`indicator` 📖	mathematical	`MeasureTheory.UnifIntegrable`	`Set.indicator` `NegZeroClass.toZero` `SubNegZeroMonoid.toNegZeroClass` `SubtractionMonoid.toSubNegZeroMonoid` `SubtractionCommMonoid.toSubtractionMonoid` `AddCommGroup.toDivisionAddCommMonoid` `NormedAddCommGroup.toAddCommGroup`	—	`Set.indicator_indicator` `Set.inter_comm` `MeasureTheory.eLpNorm_indicator_le`
`neg` 📖	mathematical	`MeasureTheory.UnifIntegrable`	`Pi.instNeg` `NegZeroClass.toNeg` `SubNegZeroMonoid.toNegZeroClass` `SubtractionMonoid.toSubNegZeroMonoid` `SubtractionCommMonoid.toSubtractionMonoid` `AddCommGroup.toDivisionAddCommMonoid` `NormedAddCommGroup.toAddCommGroup`	—	`Set.indicator_neg'` `MeasureTheory.eLpNorm_neg`
`restrict` 📖	mathematical	`MeasureTheory.UnifIntegrable`	`MeasureTheory.Measure.restrict`	—	`MeasureTheory.eLpNorm_indicator_eq_eLpNorm_restrict` `MeasureTheory.Measure.restrict_restrict` `MeasureTheory.eLpNorm_mono_measure` `MeasureTheory.Measure.restrict_mono` `MeasureTheory.subset_toMeasurable` `le_rfl` `MeasureTheory.measurableSet_toMeasurable` `MeasureTheory.measure_toMeasurable` `MeasureTheory.Measure.restrict_apply`
`sub` 📖	mathematical	`MeasureTheory.UnifIntegrable` `ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `MeasureTheory.AEStronglyMeasurable` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddCommGroup.toPseudoMetricSpace` `NormedAddCommGroup.toSeminormedAddCommGroup`	`Pi.instSub` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `NormedAddGroup.toAddGroup` `NormedAddCommGroup.toNormedAddGroup`	—	`sub_eq_add_neg` `add` `neg` `MeasureTheory.AEStronglyMeasurable.neg` `IsTopologicalAddGroup.toContinuousNeg` `SeminormedAddCommGroup.toIsTopologicalAddGroup`
`unifIntegrable_of_ae_tendsto` 📖	mathematical	`MeasureTheory.UnifIntegrable` `MeasureTheory.AEStronglyMeasurable` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddCommGroup.toPseudoMetricSpace` `NormedAddCommGroup.toSeminormedAddCommGroup`	`Set.Elem` `setOf` `Filter.Eventually` `Filter.Tendsto` `nhds` `MeasureTheory.ae` `MeasureTheory.Measure` `MeasureTheory.Measure.instFunLike` `MeasureTheory.Measure.instOuterMeasureClass` `Set` `Set.instMembership`	—	`MeasureTheory.Measure.instOuterMeasureClass` `MeasureTheory.Lp.eLpNorm_le_of_ae_tendsto` `Filter.Eventually.of_forall` `MeasureTheory.AEStronglyMeasurable.indicator` `Filter.mp_mem` `Filter.univ_mem'` `Set.indicator_of_mem` `Set.indicator_of_notMem` `T5Space.toT1Space` `T6Space.toT5Space` `instT6SpaceOfMetrizableSpace` `EMetricSpace.metrizableSpace`
`unifIntegrable_of_tendstoInMeasure` 📖	mathematical	`MeasureTheory.UnifIntegrable` `MeasureTheory.AEStronglyMeasurable` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddCommGroup.toPseudoMetricSpace` `NormedAddCommGroup.toSeminormedAddCommGroup`	`Set.Elem` `setOf` `MeasureTheory.TendstoInMeasure` `PseudoEMetricSpace.toEDist` `EMetricSpace.toPseudoEMetricSpace` `MetricSpace.toEMetricSpace` `NormedAddCommGroup.toMetricSpace` `Set` `Set.instMembership`	—	`MeasureTheory.eLpNorm_le_of_tendstoInMeasure` `Filter.Eventually.of_forall` `MeasureTheory.TendstoInMeasure.indicator` `MeasureTheory.AEStronglyMeasurable.indicator`

MeasureTheory.UniformIntegrable

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`ae_eq` 📖	—	`MeasureTheory.UniformIntegrable` `Filter.EventuallyEq` `MeasureTheory.ae` `MeasureTheory.Measure` `MeasureTheory.Measure.instFunLike` `MeasureTheory.Measure.instOuterMeasureClass`	—	—	`MeasureTheory.Measure.instOuterMeasureClass` `MeasureTheory.AEStronglyMeasurable.congr` `MeasureTheory.unifIntegrable_congr_ae` `MeasureTheory.eLpNorm_congr_ae`
`aestronglyMeasurable` 📖	mathematical	`MeasureTheory.UniformIntegrable`	`MeasureTheory.AEStronglyMeasurable` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddCommGroup.toPseudoMetricSpace` `NormedAddCommGroup.toSeminormedAddCommGroup`	—	—
`integrable_of_ae_tendsto` 📖	mathematical	`MeasureTheory.UniformIntegrable` `ENNReal` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `Filter.Eventually` `Filter.Tendsto` `nhds` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddCommGroup.toPseudoMetricSpace` `NormedAddCommGroup.toSeminormedAddCommGroup` `MeasureTheory.ae` `MeasureTheory.Measure` `MeasureTheory.Measure.instFunLike` `MeasureTheory.Measure.instOuterMeasureClass`	`MeasureTheory.Integrable` `SeminormedAddGroup.toContinuousENorm` `SeminormedAddCommGroup.toSeminormedAddGroup`	—	`MeasureTheory.Measure.instOuterMeasureClass` `MeasureTheory.memLp_one_iff_integrable` `memLp_of_ae_tendsto`
`integrable_of_tendstoInMeasure` 📖	mathematical	`MeasureTheory.UniformIntegrable` `ENNReal` `AddMonoidWithOne.toOne` `AddCommMonoidWithOne.toAddMonoidWithOne` `instAddCommMonoidWithOneENNReal` `MeasureTheory.TendstoInMeasure` `PseudoEMetricSpace.toEDist` `EMetricSpace.toPseudoEMetricSpace` `MetricSpace.toEMetricSpace` `NormedAddCommGroup.toMetricSpace`	`MeasureTheory.Integrable` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddCommGroup.toPseudoMetricSpace` `NormedAddCommGroup.toSeminormedAddCommGroup` `SeminormedAddGroup.toContinuousENorm` `SeminormedAddCommGroup.toSeminormedAddGroup`	—	`MeasureTheory.memLp_one_iff_integrable` `memLp_of_tendstoInMeasure`
`memLp` 📖	mathematical	`MeasureTheory.UniformIntegrable`	`MeasureTheory.MemLp` `ContinuousENorm.toENorm` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `NormedAddCommGroup.toSeminormedAddCommGroup` `SeminormedAddGroup.toContinuousENorm` `SeminormedAddCommGroup.toPseudoMetricSpace`	—	`lt_of_le_of_lt` `ENNReal.coe_lt_top`
`memLp_of_ae_tendsto` 📖	mathematical	`MeasureTheory.UniformIntegrable` `Filter.Eventually` `Filter.Tendsto` `nhds` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddCommGroup.toPseudoMetricSpace` `NormedAddCommGroup.toSeminormedAddCommGroup` `MeasureTheory.ae` `MeasureTheory.Measure` `MeasureTheory.Measure.instFunLike` `MeasureTheory.Measure.instOuterMeasureClass`	`MeasureTheory.MemLp` `ContinuousENorm.toENorm` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `SeminormedAddGroup.toContinuousENorm`	—	`MeasureTheory.Measure.instOuterMeasureClass` `memLp` `uniformIntegrable_of_ae_tendsto`
`memLp_of_tendstoInMeasure` 📖	mathematical	`MeasureTheory.UniformIntegrable` `MeasureTheory.TendstoInMeasure` `PseudoEMetricSpace.toEDist` `EMetricSpace.toPseudoEMetricSpace` `MetricSpace.toEMetricSpace` `NormedAddCommGroup.toMetricSpace`	`MeasureTheory.MemLp` `ContinuousENorm.toENorm` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `NormedAddCommGroup.toSeminormedAddCommGroup` `SeminormedAddGroup.toContinuousENorm` `SeminormedAddCommGroup.toPseudoMetricSpace`	—	`memLp` `uniformIntegrable_of_tendstoInMeasure`
`spec` 📖	mathematical	`MeasureTheory.UniformIntegrable` `Real` `Real.instLT` `Real.instZero`	`ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `MeasureTheory.eLpNorm` `ContinuousENorm.toENorm` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `NormedAddCommGroup.toSeminormedAddCommGroup` `SeminormedAddGroup.toContinuousENorm` `Set.indicator` `NegZeroClass.toZero` `SubNegZeroMonoid.toNegZeroClass` `SubtractionMonoid.toSubNegZeroMonoid` `SubtractionCommMonoid.toSubtractionMonoid` `AddCommGroup.toDivisionAddCommMonoid` `NormedAddCommGroup.toAddCommGroup` `setOf` `NNReal` `instPartialOrderNNReal` `NNNorm.nnnorm` `SeminormedAddGroup.toNNNorm` `ENNReal.ofReal`	—	`MeasureTheory.Measure.instOuterMeasureClass` `ae_eq` `spec'` `le_trans` `le_of_eq` `MeasureTheory.eLpNorm_congr_ae` `Filter.mp_mem` `Filter.univ_mem'` `Set.indicator_of_mem` `Set.mem_setOf` `Set.indicator_of_notMem`
`spec'` 📖	mathematical	`MeasureTheory.StronglyMeasurable` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddCommGroup.toPseudoMetricSpace` `NormedAddCommGroup.toSeminormedAddCommGroup` `MeasureTheory.UniformIntegrable` `Real` `Real.instLT` `Real.instZero`	`ENNReal` `Preorder.toLE` `PartialOrder.toPreorder` `ENNReal.instPartialOrder` `MeasureTheory.eLpNorm` `ContinuousENorm.toENorm` `SeminormedAddGroup.toPseudoMetricSpace` `SeminormedAddCommGroup.toSeminormedAddGroup` `SeminormedAddGroup.toContinuousENorm` `Set.indicator` `NegZeroClass.toZero` `SubNegZeroMonoid.toNegZeroClass` `SubtractionMonoid.toSubNegZeroMonoid` `SubtractionCommMonoid.toSubtractionMonoid` `AddCommGroup.toDivisionAddCommMonoid` `NormedAddCommGroup.toAddCommGroup` `setOf` `NNReal` `instPartialOrderNNReal` `NNNorm.nnnorm` `SeminormedAddGroup.toNNNorm` `ENNReal.ofReal`	—	`CanLift.prf` `NNReal.canLift` `LT.lt.le` `ENNReal.smul_def` `smul_eq_mul` `mul_le_mul'` `IsOrderedMonoid.toMulLeftMono` `ENNReal.instIsOrderedMonoid` `covariant_swap_mul_of_covariant_mul` `le_rfl` `ENNReal.rpow_le_rpow` `ENNReal.ofReal_coe_nnreal` `one_div_nonneg` `PosMulReflectLE.toPosMulReflectLT` `PosMulStrictMono.toPosMulReflectLE` `IsStrictOrderedRing.toPosMulStrictMono` `Real.instIsStrictOrderedRing` `ENNReal.toReal_nonneg` `MeasureTheory.le_eLpNorm_of_bddBelow` `measurableSet_le` `BorelSpace.opensMeasurable` `NNReal.borelSpace` `OrderTopology.to_orderClosedTopology` `NNReal.instOrderTopology` `NNReal.instSecondCountableTopology` `measurable_const` `MeasureTheory.StronglyMeasurable.measurable` `PseudoEMetricSpace.pseudoMetrizableSpace` `MeasureTheory.StronglyMeasurable.nnnorm` `Filter.Eventually.of_forall` `MeasureTheory.Measure.instOuterMeasureClass` `nnnorm_indicator_eq_indicator_nnnorm` `Set.indicator_of_mem` `MeasureTheory.eLpNorm_indicator_le` `LT.lt.ne` `Nat.instAtLeastTwoHAddOfNat` `lt_of_le_of_lt` `le_trans` `le_max_left` `lt_of_lt_of_le` `ENNReal.coe_one` `ENNReal.coe_max` `ENNReal.coe_mul` `ENNReal.coe_lt_coe` `lt_two_mul_self` `LinearOrderedCommMonoidWithZero.toZeroLeOneClass` `IsStrictOrderedRing.toMulPosStrictMono` `NNReal.instIsStrictOrderedRing` `NeZero.charZero_one` `IsStrictOrderedRing.toCharZero` `IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono` `AddLeftCancelSemigroup.toIsLeftCancelAdd` `NNReal.addLeftMono` `lt_max_of_lt_right` `one_pos` `NNReal.inv_rpow` `mul_one` `inv_mul_cancel₀` `NNReal.rpow_pos` `NNReal.coe_pos` `mul_assoc` `ENNReal.coe_smul` `IsScalarTower.right` `IsScalarTower.left` `ENNReal.coe_rpow_of_nonneg` `Mathlib.Tactic.Push.not_forall_eq`
`unifIntegrable` 📖	mathematical	`MeasureTheory.UniformIntegrable`	`MeasureTheory.UnifIntegrable`	—	—
`uniformIntegrable_of_ae_tendsto` 📖	mathematical	`MeasureTheory.UniformIntegrable`	`Set.Elem` `setOf` `Filter.Eventually` `Filter.Tendsto` `nhds` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddCommGroup.toPseudoMetricSpace` `NormedAddCommGroup.toSeminormedAddCommGroup` `MeasureTheory.ae` `MeasureTheory.Measure` `MeasureTheory.Measure.instFunLike` `MeasureTheory.Measure.instOuterMeasureClass` `Set` `Set.instMembership`	—	`MeasureTheory.Measure.instOuterMeasureClass` `aestronglyMeasurable_of_tendsto_ae` `PseudoEMetricSpace.pseudoMetrizableSpace` `MeasureTheory.UnifIntegrable.unifIntegrable_of_ae_tendsto` `MeasureTheory.Lp.eLpNorm_le_of_ae_tendsto` `Filter.Eventually.of_forall`
`uniformIntegrable_of_tendstoInMeasure` 📖	mathematical	`MeasureTheory.UniformIntegrable`	`Set.Elem` `setOf` `MeasureTheory.TendstoInMeasure` `PseudoEMetricSpace.toEDist` `EMetricSpace.toPseudoEMetricSpace` `MetricSpace.toEMetricSpace` `NormedAddCommGroup.toMetricSpace` `Set` `Set.instMembership`	—	`MeasureTheory.TendstoInMeasure.aestronglyMeasurable` `MeasureTheory.UnifIntegrable.unifIntegrable_of_tendstoInMeasure` `MeasureTheory.eLpNorm_le_of_tendstoInMeasure` `Filter.Eventually.of_forall`

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