Documentation Verification Report

Real

📁 Source: Mathlib/Topology/Order/Real.lean

Statistics

MetricCount
DefinitionsinstTopologicalSpace, instTopologicalSpace
2
TheoremsinstOrderTopology, instT2Space, instT4Space, instT5Space, denseRange_ratCast, instOrderTopology, instT2Space, instT5Space
8
Total10

ENNReal

Definitions

NameCategoryTheorems
instTopologicalSpace 📖CompOp
516 mathmath: MeasureTheory.hasFiniteIntegral_count_iff_enorm, ContinuousAt.enorm, nhds_zero, PMF.tsum_coe, MeasureTheory.tendsto_Lp_of_tendsto_ae, MeasureTheory.Lp.tendsto_Lp_iff_tendsto_eLpNorm'', continuous_nnreal_sub, ContinuousWithinAt.ereal_toENNReal, MeasureTheory.ae_tendsto_enorm, MeasureTheory.measure_of_cont_bdd_of_tendsto_indicator, MeasureTheory.IsAddFundamentalDomain.lintegral_eq_tsum'', instT4Space, aefinStronglyMeasurable_of_aemeasurable, MeasureTheory.VectorMeasure.equivMeasure_apply, tsum_set_const, MeasureTheory.Lp.cauchySeq_Lp_iff_cauchySeq_eLpNorm, MeasureTheory.Lp.cauchy_tendsto_of_tendsto, thickenedIndicatorAux_tendsto_indicator_closure, PMF.toMeasure_bind_apply, PMF.toMeasure_ofFinset_apply, MeasureTheory.lintegral_iUnion₀, MeasureTheory.SimpleFunc.lintegral_sum, MeasureTheory.measure_biUnion, MeasureTheory.tendsto_Lp_of_tendstoInMeasure, MeasureTheory.isTightMeasureSet_iff_inner_tendsto, MeasureTheory.piContent_tendsto_zero, tsum_geometric_encode_lt_top, tendsto_nat_tsum, MeasureTheory.tendsto_measure_iUnion_atBot, Asymptotics.IsLittleOTVS.tendsto_div, MeasureTheory.measure_preimage_fst_singleton_eq_tsum, tsum_le_tsum, tsum_iUnion_le_tsum, MeasureTheory.measure_biUnion_le, exists_pos_sum_of_countable', tsum_mul_left, Besicovitch.ae_tendsto_measure_inter_div_of_measurableSet, tendsto_coe_sub, isOpenEmbedding_coe, tsum_geometric_lt_top, tendsto_toNNReal_iff', MeasureTheory.SimpleFunc.tendsto_approxOn_range_L1_enorm, tsum_sub, HasFPowerSeriesWithinAt.eventually, tsum_eq_top_of_eq_top, tsum_lt_tsum, tendsto_pow_atTop_nhds_zero_iff, MeasureTheory.Measure.le_sum_apply, Besicovitch.exists_closedBall_covering_tsum_measure_le, MeasureTheory.lintegral_biUnion, tendsto_rpow_atTop_of_one_lt_base, PMF.toMeasure_ofFintype_apply, MeasureTheory.lintegral_countable, MeasureTheory.Measure.mkMetric_apply, tendsto_const_mul_rpow_nhds_zero_of_pos, MeasureTheory.Measure.sum_apply_of_countable, IntervalIntegrable.enorm, PMF.toMeasure_apply, PMF.toMeasure_ofMultiset_apply, nhds_top', Metric.PiNatEmbed.edist_def, tendsto_nhds_top_iff_nnreal, tsum_toNNReal_eq, PMF.toOuterMeasure_ofFinset_apply, MeasureTheory.tendsto_measure_biInter_gt, MeasureTheory.ProbabilityMeasure.continuous_lintegral_boundedContinuousFunction, MeasureTheory.Measure.toENNRealVectorMeasure_apply, tsum_add, MeasureTheory.tendsto_measure_iUnion_atTop, tendsto_atTop_zero_iff_lt_of_antitone, tsum_biUnion_le, hasBasis_nhds_of_ne_top, continuous_log, EReal.tendsto_exp_nhds_zero_nhds_one, MeasureTheory.MemLp.integrable_enorm_pow', continuousOn_sub, tendsto_toReal, MeasureTheory.OuterMeasure.iInf_apply', EReal.isClosedEmbedding_coe_ennreal, VitaliFamily.FineSubfamilyOn.measure_le_tsum, finStronglyMeasurable_of_measurable, MeasureTheory.SignedMeasure.mutuallySingular_ennreal_iff, PiCountable.edist_eq_tsum, MeasureTheory.AEEqFun.lintegral_coeFn, tsum_biUnion, IsAddFoelner.tendsto_meas_vadd_symmDiff_vadd, MeasureTheory.tendstoInMeasure_iff_tendsto_Lp, tsum_const_eq_top_of_ne_zero, MeasureTheory.SignedMeasure.mutuallySingular_singularPart, MeasureTheory.AEEqFun.lintegral_add, MeasureTheory.tendsto_measure_of_tendsto_indicator, MeasureTheory.SimpleFunc.tendsto_eapprox, MeasureTheory.tendstoInMeasure_iff_norm, AbsolutelyContinuousOnInterval.tendsto_volume_totalLengthFilter_nhds_zero, ContinuousOn.ereal_toENNReal, tendsto_cofinite_zero_of_tsum_ne_top, MeasureTheory.StronglyMeasurable.edist, MeasureTheory.MemLp.enorm, MeasureTheory.Integrable.tendsto_eLpNorm_condExp, continuous_sub_right, MeasureTheory.Integrable.of_mem_Icc_enorm, multipliable_of_le_one, IsFoelner.tendsto_meas_smul_symmDiff, edist_le_tsum_of_edist_le_of_tendsto₀, MeasureTheory.hasSum_lintegral_measure, ProbabilityTheory.Kernel.compProd_eq_tsum_compProd, VitaliFamily.ae_tendsto_lintegral_div, VitaliFamily.ae_tendsto_measure_inter_div_of_measurableSet, MeasureTheory.FiniteMeasure.tendsto_iff_forall_lintegral_tendsto, lowerSemicontinuousWithinAt_tsum, tendsto_tsum_compl_atTop_zero, MeasureTheory.Content.innerContent_iSup_nat, Filter.Tendsto.enorm, MeasureTheory.AEEqFun.lintegral_eq_zero_iff, MeasureTheory.SignedMeasure.absolutelyContinuous_iff_withDensityᵥ_rnDeriv_eq, edist_le_tsum_of_edist_le_of_tendsto, ProbabilityTheory.Kernel.trajContent_tendsto_zero, ProbabilityTheory.stronglyMeasurable_condExpKernel, continuous_const_mul, MeasureTheory.lintegral_iUnion_le, continuousAt_mul_const, MeasureTheory.IsFundamentalDomain.lintegral_eq_tsum_of_ac, Besicovitch.ae_tendsto_rnDeriv, MeasureTheory.IsFundamentalDomain.lintegral_eq_tsum', MeasureTheory.MemLp.integrable_enorm_rpow', coe_tsum, tsum_set_one, MeasureTheory.tendsto_measure_of_ae_tendsto_indicator_of_isFiniteMeasure, MeasureTheory.IsFundamentalDomain.setLIntegral_eq_tsum, instContinuousInv, MeasureTheory.SignedMeasure.absolutelyContinuous_ennreal_iff, MeasureTheory.tendstoInMeasure_iff_dist, continuous_of_le_add_edist, instMetrizableSpace, tendsto_atTop_zero_of_tsum_ne_top, tendsto_coe_toNNReal, spectrum.gelfand_formula, tsum_schlomilch_le, ProbabilityTheory.tsum_prob_mem_Ioi_lt_top, tendsto_measure_cthickening, VitaliFamily.ae_tendsto_limRatio, tendsto_measure_Icc, isOpen_ne_top, tendsto_atTop, instContinuousAdd, continuous_edist, ContinuousENorm.continuous_enorm, MeasureTheory.OuterMeasure.mkMetric'.tendsto_pre, tsum_iSup_eq, tendsto_pow_atTop_nhds_zero_of_lt_one, AddCircle.volume_of_add_preimage_eq, MeasureTheory.Measure.sum_apply, tsum_top, continuous_coe_iff, StieltjesFunction.length_subadditive_Icc_Ioo, tendsto_rpow_at_top, tsum_union_le, tsum_le_of_sum_range_le, summable, tendsto_pow_atTop_nhds_top_iff, tsum_geometric_add_one, tsum_eq_iSup_nat', tsum_sigma', EReal.tendsto_coe_ennreal, EReal.expHomeomorph_symm, Metric.eventually_nhds_zero_forall_closedEBall_subset, MeasureTheory.OuterMeasure.iInf_apply, MeasureTheory.withDensity_tsum, MeasureTheory.AEEqFun.lintegral_zero, MeasureTheory.tendsto_measure_vadd_diff_isCompact_isClosed, hasBasis_nhds_of_ne_top', MeasureTheory.tendsto_measure_Iic_atTop, tendsto_toNNReal, MeasureTheory.FiniteMeasure.continuous_iff_forall_continuous_lintegral, ofReal_tsum_of_nonneg, MeasureTheory.tsum_measure_le_measure_univ, MeasureTheory.tendsto_measure_of_tendsto_indicator_of_isFiniteMeasure, tsum_eq_iSup_sum, spectrum.pow_nnnorm_pow_one_div_tendsto_nhds_spectralRadius, MeasureTheory.measure_of_cont_bdd_of_tendsto_filter_indicator, continuousAt_const_mul, MeasureTheory.Measure.restrict_iUnion_apply, instT5Space, MeasureTheory.OuterMeasure.biInf_apply, MeasureTheory.VectorMeasure.ennrealToMeasure_apply, coe_range_mem_nhds, tendsto_atTop_zero_iff_le_of_antitone, MeasureTheory.OuterMeasure.mkMetric'.tendsto_pre_nat, MeasureTheory.Measure.withDensityᵥ_absolutelyContinuous, MeasureTheory.integrable_enorm_iff, ProbabilityTheory.lintegral_exp_mul_sq_norm_le_of_map_rotation_eq_self, MeasureTheory.IsAddFundamentalDomain.setLIntegral_eq_tsum', MeasureTheory.tendsto_measure_iInter_le, tendsto_measure_thickening, MeasureTheory.FiniteMeasure.continuous_iff_forall_continuousMap_continuous_lintegral, tendsto_mul, continuousOn_sub_left, MeasureTheory.tendsto_measure_norm_gt_of_isTightMeasureSet, nhds_coe, continuous_ofReal, PMF.toOuterMeasure_ofMultiset_apply, continuous_pow, lowerSemicontinuousOn_tsum, MeasureTheory.Measure.sum_apply₀, MeasureTheory.MemLp.integrable_enorm_pow, MeasureTheory.measure_iUnion₀, MeasureTheory.Measure.m_iUnion, tendsto_nat_nhds_top, continuous_zpow, tsum_eq_iSup_sum', tsum_prod', MeasureTheory.IsAddFundamentalDomain.lintegral_eq_tsum, MeasureTheory.lintegral_count, nhdsGT_zero_neBot, MeasureTheory.VectorMeasure.equivMeasure_symm_apply, Asymptotics.isLittleOTVS_iff_tendsto_div, MeasureTheory.OuterMeasure.ofFunction_eq_iInf_mem, tsum_two_zpow_neg_add_one, MeasureTheory.VectorMeasure.AbsolutelyContinuous.ennrealToMeasure, MeasureTheory.eLpNormEssSup_smul_measure, ProbabilityTheory.Kernel.withDensity_tsum, tendsto_measure_Icc_nhdsWithin_right, MeasureTheory.lintegral_countable', continuous_coe_ennreal_ereal, Probability.stronglyMeasurable_cauchyPDF, continuous_enorm, PMF.seq_apply, MeasureTheory.tendsto_measure_Ioc_atBot, MeasureTheory.preVariation.iUnion, tsum_mul_right, ProbabilityTheory.IsKolmogorovProcess.stronglyMeasurable_edist, nhds_top_basis, PMF.toOuterMeasure_ofFintype_apply, EReal.tendsto_exp_nhds_top_nhds_top, MeasureTheory.FiniteMeasure.tendsto_lintegral_nn_of_le_const, MeasureTheory.IntegrableAtFilter.enorm, isFoelner_iff, nhds_of_ne_top, continuousAt_toReal, nhdsGT_coe_neBot, IsFoelner.tendsto_meas_smul_symmDiff_smul, MeasureTheory.IsFundamentalDomain.setLIntegral_eq_tsum', AbsolutelyContinuousOnInterval.tendsto_volume_restrict_totalLengthFilter_disjWithin_nhds_zero, MeasureTheory.integrable_enorm_rpow_iff, tsum_geometric_two, MeasureTheory.tsum_meas_le_meas_iUnion_of_disjoint, tsum_le_tsum_comp_of_surjective, MeasureTheory.all_ae_tendsto_ofReal_norm, ContinuousAt.ereal_toENNReal, MeasureTheory.ProbabilityMeasure.continuous_lintegral_continuousMap, MeasureTheory.measure_iUnion, TopologicalSpace.Closeds.continuous_infEDist, HasFPowerSeriesAt.eventually, VitaliFamily.ae_tendsto_div, VitaliFamily.ae_tendsto_measure_inter_div, MeasureTheory.tendsto_measure_symmDiff_preimage_nhds_zero, MeasureTheory.IsFundamentalDomain.lintegral_eq_tsum, tsum_const_smul, MeasureTheory.OuterMeasure.iUnion_nat, MeasureTheory.Measure.toENNRealVectorMeasure_add, tsum_sigma, tsum_eq_limsup_sum_nat, VitaliFamily.ae_eventually_measure_zero_of_singular, Besicovitch.ae_tendsto_measure_inter_div, PMF.toOuterMeasure_bindOnSupport_apply, continuous_mul_const, tendsto_toNNReal_iff, le_tsum, exists_pos_sum_of_countable, le_tsum_of_forall_lt_exists_sum, VitaliFamily.ae_tendsto_lintegral_div', MeasureTheory.MemLp.enorm_rpow, MeasureTheory.IsFundamentalDomain.lintegral_eq_tsum'', ProbabilityTheory.Kernel.tendsto_eLpNorm_one_densityProcess_limitProcess, tsum_condensed_le, MeasureTheory.IsFundamentalDomain.measure_eq_tsum_of_ac, EReal.tendsto_exp_nhds_bot_nhds_zero, tendsto_coe, MeasureTheory.tendsto_measure_sigmaFiniteSetGE, MeasureTheory.ProbabilityMeasure.continuous_iff_forall_continuous_lintegral, tendsto_ofReal_atTop, EMetric.cauchySeq_iff_le_tendsto_0, eHolderNorm_nsmul, MeasureTheory.tendsto_lintegral_norm_of_dominated_convergence, MeasureTheory.Integrable.withDensityᵥ_trim_absolutelyContinuous, MeasureTheory.IsAddFundamentalDomain.lintegral_eq_tsum_of_ac, exists_pos_tsum_mul_lt_of_countable, HasOuterApproxClosed.tendsto_lintegral_apprSeq, MeasureTheory.IsAddFundamentalDomain.measure_eq_card_smul_of_vadd_ae_eq_self, Metric.exists_continuous_ennreal_forall_closedEBall_subset, MeasureTheory.IsAddFundamentalDomain.measure_eq_tsum, continuous_sub_left, nhds_zero_basis, MeasureTheory.measure_iUnion_le, tendsto_inv_nat_nhds_zero, NNReal.tsum_eq_toNNReal_tsum, MeasureTheory.tendsto_eLpNorm_condExp, tsum_coe_eq, continuous_exp, tendsto_toReal_zero_iff, MeasureTheory.IsFundamentalDomain.measure_eq_tsum', lowerSemicontinuousAt_tsum, MeasureTheory.OuterMeasure.sInf_apply, continuousOn_toNNReal, hasSum, nhds_zero_basis_Iic, tsum_comp_le_tsum_of_injective, sum_le_tsum, MeasureTheory.tendsto_measure_iInter_atTop, MeasureTheory.tendsto_measure_Ici_atBot, tendsto_rpow_atTop_of_base_lt_one, measure_eq_measure_preimage_add_measure_tsum_Ico_zpow, MeasureTheory.Content.innerContent_iUnion_nat, MeasureTheory.OuterMeasure.boundedBy_apply, MeasureTheory.IsAddFundamentalDomain.setLIntegral_eq_tsum, MeasureTheory.tendsto_lintegral_thickenedIndicator_of_isClosed, tendsto_const_sub_nhds_zero_iff, le_tsum_condensed, continuousAt_coe_iff, EReal.continuous_toENNReal, Measurable.ennreal_tsum, MeasureTheory.Submartingale.tendsto_eLpNorm_one_limitProcess, hasSum_iff_tendsto_nat, ProbabilityTheory.strong_law_Lp, MeasureTheory.memLp_enorm_iff, instOrderTopology, tendsto_nhds, tendsto_sub, sum_add_tsum_compl, MeasureTheory.lintegral_count', MeasureTheory.AECover.lintegral_tendsto_of_countably_generated, MeasureTheory.lintegral_iUnion, tsum_biUnion', MeasureTheory.Integrable.enorm, tendsto_sum_nat_add, VitaliFamily.ae_tendsto_rnDeriv_of_absolutelyContinuous, MeasureTheory.addContent_iUnion_eq_tsum_of_disjoint_of_IsSigmaSubadditive, tendsto_measure_cthickening_of_isClosed, nhdsLT_neBot, MeasureTheory.tendsto_measure_iInter_atBot, tsum_prod, EReal.isEmbedding_coe_ennreal, MeasureTheory.tsum_measure_preimage_singleton, spectrum.pow_norm_pow_one_div_tendsto_nhds_spectralRadius, VitaliFamily.ae_tendsto_rnDeriv, continuous_truncateToReal, MeasureTheory.IsAddFundamentalDomain.measure_eq_tsum', tendsto_rpow_atBot_of_one_lt_base, enorm_tsum_le_tsum_enorm, MeasureTheory.MeasuredSets.continuous_measure, logHomeomorph_symm, tsum_coe_eq_top_iff_not_summable_coe, MeasureTheory.Measure.le_count_apply, tsum_eq_iSup_nat, MeasureTheory.FiniteMeasure.continuous_lintegral_continuousMap, MeasureTheory.Measure.restrict_iUnion_apply_ae, MeasureTheory.tendsto_measure_of_ae_tendsto_indicator, MeasureTheory.Measure.tendsto_IicSnd_atTop, borelSpace, tendsto_iff_edist_tendsto_0, MeasureTheory.measure_sUnion₀, tendsto_nhds_coe_iff, nhdsGT_ofNat_neBot, MeasureTheory.IsFundamentalDomain.measure_eq_card_smul_of_smul_ae_eq_self, tsum_eq_liminf_sum_nat, MeasureTheory.Measure.hausdorffMeasure_apply, Metric.continuous_infEDist, tendsto_measure_Icc_nhdsWithin_right', tendsto_atTop_zero, VitaliFamily.FineSubfamilyOn.measure_le_tsum_of_absolutelyContinuous, AEMeasurable.ennreal_tsum, PMF.hasSum_coe_one, MeasureTheory.OuterMeasure.sInf_apply', IsAddFoelner.tendsto_nhds_mean, tendsto_ofReal, nhdsGT_nat_neBot, MeasureTheory.Egorov.measure_notConvergentSeq_tendsto_zero, tsum_geometric_two_encode_le_two, PMF.toMeasure_bindOnSupport_apply, tsum_comm, MeasureTheory.Lp.tendsto_Lp_iff_tendsto_eLpNorm', MeasureTheory.tendsto_measure_iUnion_accumulate, MeasureTheory.ComplexMeasure.absolutelyContinuous_ennreal_iff, MeasureTheory.OuterMeasure.sum_apply, PMF.toOuterMeasure_apply, tendsto_measure_cthickening_of_isCompact, tendsto_nhds_zero, MeasureTheory.Measure.infinitePi_singleton, ProbabilityTheory.Fernique.lintegral_exp_mul_sq_norm_le_mul, ContinuousOn.enorm, continuous_thickenedIndicatorAux, logHomeomorph_apply, MeasureTheory.preVariation.sum_le_preVariationFun_iUnion, MeasureTheory.tendsto_measure_compl_closedBall_of_isTightMeasureSet, IsAddFoelner.tendsto_meas_vadd_symmDiff, MeasureTheory.tendstoInMeasure_iff_enorm, MeasureTheory.ae_tendsto_ofReal_norm, isOpen_Ico_zero, MeasureTheory.Measure.tsum_indicator_apply_singleton, MeasureTheory.LocallyIntegrableOn.enorm, Continuous.ereal_toENNReal, MeasureTheory.memLp_enorm_rpow_iff, tsum_fiberwise, MeasureTheory.tendsto_measure_of_le_liminf_measure_of_limsup_measure_le, MeasureTheory.tendsto_Lp_finite_of_tendstoInMeasure, MeasureTheory.tendsto_measure_smul_diff_isCompact_isClosed, continuousOn_toReal, continuous_coe, tsum_apply, MeasureTheory.SimpleFunc.tsum_eapproxDiff, continuous_rpow_const, MeasureTheory.isTightMeasureSet_iff_tendsto_measure_norm_gt, EMetric.exists_continuous_eNNReal_forall_closedBall_subset, MeasureTheory.Measure.tendsto_IicSnd_atBot, MeasureTheory.IsAddFundamentalDomain.lintegral_eq_tsum', tendsto_nhds_top_iff_nat, hasProd_iInf_prod, nhds_top, instSecondCountableTopology, EReal.continuous_coe_ennreal_iff, MeasureTheory.measure_sUnion, PMF.filter_apply, MeasureTheory.ProbabilityMeasure.tendsto_measure_of_null_frontier_of_tendsto', tsum_mono_subtype, MeasureTheory.IsAddFundamentalDomain.measure_eq_tsum_of_ac, biInf_le_nhds, MeasureTheory.ProbabilityMeasure.continuous_iff_forall_continuousMap_continuous_lintegral, hasSum_coe, ProbabilityTheory.Kernel.tendsto_eLpNorm_one_restrict_densityProcess_limitProcess, MeasureTheory.OuterMeasureClass.measure_iUnion_nat_le, ProbabilityTheory.Kernel.sum_apply', MeasureTheory.OuterMeasure.biInf_apply', MeasureTheory.MemLp.integrable_enorm_rpow, tsum_geometric, MeasureTheory.MemLp.enorm_rpow_div, tsum_const, EMetric.eventually_nhds_zero_forall_closedBall_subset, MeasureTheory.Measure.toENNRealVectorMeasure_apply_measurable, tendsto_toReal_iff, PMF.bind_apply, MeasureTheory.SimpleFunc.tendsto_approxOn_L1_enorm, VitaliFamily.ae_tendsto_limRatioMeas, MeasureTheory.OuterMeasure.f_iUnion, MeasureTheory.lintegral_tsum, tendsto_ofReal_nhds_top, MeasureTheory.tendsto_lintegral_nn_filter_of_le_const, lowerSemicontinuous_tsum, MeasureTheory.OuterMeasure.iUnion_eq_of_caratheodory, MeasureTheory.lintegral_biUnion₀, MeasureTheory.tendsto_measure_Ico_atTop, ProbabilityTheory.IsAEKolmogorovProcess.aestronglyMeasurable_edist, MeasureTheory.isTightMeasureSet_iff_tendsto_measure_compl_closedBall, PMF.map_apply, tendsto_rpow_atBot_of_base_lt_one, MeasureTheory.tendsto_measure_of_null_frontier, Continuous.enorm, MeasureTheory.measure_preimage_snd_singleton_eq_tsum, tendsto_nhds_top, tendsto_measure_thickening_of_isClosed, tendsto_coe_nhds_top, MeasureTheory.lintegral_sum_measure, Filter.Tendsto.enorm', PMF.toOuterMeasure_bind_apply, MeasureTheory.tendsto_measure_biUnion_Ici_zero_of_pairwise_disjoint, Filter.Tendsto.edist, tsum_iUnion_le, EReal.expHomeomorph_apply, EMetric.continuous_infEdist, tprod_eq_iInf_prod, IntervalIntegrable.intervalIntegrable_enorm_iff, continuous_div_const, VitaliFamily.ae_tendsto_lintegral_enorm_sub_div_of_integrable, MeasureTheory.OuterMeasure.ofFunction_apply, tsum_one, VitaliFamily.ae_tendsto_lintegral_enorm_sub_div, PMF.bindOnSupport_apply, Icc_mem_nhds, VitaliFamily.ae_tendsto_lintegral_enorm_sub_div'_of_integrable, IsFoelner.tendsto_nhds_mean, MeasureTheory.tendstoInMeasure_iff_tendsto_Lp_finite, isEmbedding_coe, isAddFoelner_iff, tsum_biUnion_le_tsum, MeasureTheory.Measure.toENNRealVectorMeasure_zero, Measurable.ennreal_tsum', MeasureTheory.Lp.tendsto_Lp_iff_tendsto_eLpNorm, MeasureTheory.tendsto_Lp_finite_of_tendsto_ae_of_meas, MeasureTheory.SimpleFunc.tendsto_approxOn_Lp_eLpNorm, tsum_eq_zero, MeasureTheory.Measure.infinitePi_pi_univ, tsum_eq_add_tsum_ite, Summable.tsum_ofReal_lt_top, MeasureTheory.IsFundamentalDomain.measure_eq_tsum, MeasureTheory.FiniteMeasure.continuous_lintegral_boundedContinuousFunction, HasFiniteFPowerSeriesAt.eventually, tsum_toReal_eq, MeasureTheory.Measure.infinitePi_pi_of_countable, nhdsGT_one_neBot, exists_countable_dense_no_zero_top, MeasureTheory.tsum_meas_le_meas_iUnion_of_disjoint₀, EMetric.continuous_infEdist_hausdorffEdist, MeasureTheory.AECover.lintegral_tendsto_of_nat, le_tsum_schlomilch, MeasureTheory.Lp.cauchy_complete_eLpNorm, instT2Space, nhds_coe_coe, MeasureTheory.SimpleFunc.tendsto_approxOn_range_Lp_eLpNorm, tendsto_nhds_of_Icc, MeasureTheory.tendsto_Lp_finite_of_tendsto_ae, ContinuousWithinAt.enorm, PolishSpace.instENNReal, Continuous.edist, MeasureTheory.ProbabilityMeasure.tendsto_iff_forall_lintegral_tendsto, MeasureTheory.measure_biUnion₀, PMF.toMeasure_apply_eq_tsum, PMF.normalize_apply

Theorems

NameKindAssumesProvesValidatesDepends On
instOrderTopology 📖mathematicalOrderTopology
ENNReal
instTopologicalSpace
PartialOrder.toPreorder
instPartialOrder
instT2Space 📖mathematicalT2Space
ENNReal
instTopologicalSpace
T25Space.t2Space
T3Space.t25Space
T4Space.t3Space
T5Space.toT4Space
OrderTopology.t5Space
instOrderTopology
instT4Space 📖mathematicalT4Space
ENNReal
instTopologicalSpace
T5Space.toT4Space
instT5Space
instT5Space 📖mathematicalT5Space
ENNReal
instTopologicalSpace
OrderTopology.t5Space
instOrderTopology

EReal

Definitions

NameCategoryTheorems
instTopologicalSpace 📖CompOp
63 mathmath: nhds_top_basis, continuousAt_add_coe_bot, continuousAt_add_coe_top, continuousAt_add_coe_coe, instT2Space, nhds_coe, continuousAt_add_bot_bot, isOpenEmbedding_coe, ENNReal.continuous_log, tendsto_exp_nhds_zero_nhds_one, tendsto_coe_atTop, isClosedEmbedding_coe_ennreal, nhds_bot_basis, nhds_top', tendsto_toReal_atTop, instT5Space, tendsto_toReal_atBot, mem_nhds_bot_iff, tendsto_coe_ennreal, expHomeomorph_symm, instSecondCountableTopology, tendsto_const_div_atTop_nhds_zero_nat, continuousAt_add_bot_coe, continuousOn_toReal, continuous_coe_ennreal_ereal, LinearGrowth.tendsto_atTop_of_linearGrowthInf_pos, tendsto_exp_nhds_top_nhds_top, continuous_coe_real_ereal, tendsto_coe, tendsto_exp_nhds_bot_nhds_zero, ENNReal.continuous_exp, tendsto_coe_nhds_bot_iff, tendsto_coe_atBot, instPolishSpaceEReal, continuousAt_add_top_coe, borelSpace, continuous_coe_iff, continuous_toENNReal, tendsto_coe_nhds_top_iff, isEmbedding_coe_ennreal, nhds_bot', ENNReal.logHomeomorph_symm, tendsto_nhds_top_iff_real, nhds_bot, lowerSemicontinuous_add, continuousAt_add, tendsto_toReal, continuousAt_add_top_top, ENNReal.logHomeomorph_apply, nhds_top, tendsto_mul, isEmbedding_coe, denseRange_ratCast, continuous_coe_ennreal_iff, nhds_coe_coe, continuousAt_mul, instContinuousNeg, nhdsWithin_top, expHomeomorph_apply, nhdsWithin_bot, mem_nhds_top_iff, instOrderTopology, tendsto_nhds_bot_iff_real

Theorems

NameKindAssumesProvesValidatesDepends On
denseRange_ratCast 📖mathematicalDenseRange
EReal
instTopologicalSpace
Real.toEReal
Real
Real.instRatCast
dense_of_exists_between
instOrderTopology
Set.exists_range_iff
exists_rat_btwn_of_lt
instOrderTopology 📖mathematicalOrderTopology
EReal
instTopologicalSpace
PartialOrder.toPreorder
instPartialOrderEReal
instT2Space 📖mathematicalT2Space
EReal
instTopologicalSpace
T25Space.t2Space
T3Space.t25Space
T4Space.t3Space
T5Space.toT4Space
instT5Space
instT5Space 📖mathematicalT5Space
EReal
instTopologicalSpace
OrderTopology.t5Space
instOrderTopology

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