Documentation Verification Report

Basic

📁 Source: Mathlib/Probability/Independence/Basic.lean

Statistics

MetricCount
DefinitionsIndepSet, IndepSets, iIndep, iIndepFun, iIndepSet, iIndepSets, «term_⟂ᵢ[_]_», «term_⟂ᵢ_»
8
TheoremsindepSet_of_measurableSet, indepSets, comp, comp₀, map_add_eq_map_conv_map, map_add_eq_map_conv_map', map_add_eq_map_conv_map₀, map_add_eq_map_conv_map₀', map_mul_eq_map_mconv_map, map_mul_eq_map_mconv_map', map_mul_eq_map_mconv_map₀, map_mul_eq_map_mconv_map₀', meas_inter, measure_inter_preimage_eq_mul, neg_left, neg_right, IndepFun_iff, IndepFun_iff_Indep, measure_inter_eq_mul, IndepSet_iff, IndepSet_iff_Indep, bInter, bUnion, biUnion, iInter, iUnion, indep, indep', indepSet_of_mem, inter, union, union_iff, IndepSets_iff, Indep_iff, Indep_iff_IndepSets, cond_iInter, iIndepSets, iIndepSets', indep, isProbabilityMeasure, meas_biInter, meas_iInter, of_precomp, of_subsingleton, precomp, comp, comp₀, cond, iIndep, indepFun, indepFun_add_add, indepFun_add_left, indepFun_add_left₀, indepFun_add_mul₀, indepFun_add_right, indepFun_add_right₀, indepFun_div_div, indepFun_div_div₀, indepFun_div_left, indepFun_div_left₀, indepFun_div_right, indepFun_div_right₀, indepFun_finset, indepFun_finset_prod_of_notMem, indepFun_finset_prod_of_notMem₀, indepFun_finset_sum_of_notMem, indepFun_finset_sum_of_notMem₀, indepFun_finset₀, indepFun_mul_left, indepFun_mul_left₀, indepFun_mul_mul, indepFun_mul_mul₀, indepFun_mul_right, indepFun_mul_right₀, indepFun_prodMk, indepFun_prodMk_prodMk, indepFun_prodMk_prodMk₀, indepFun_prodMk₀, indepFun_prod_range_succ, indepFun_prod_range_succ₀, indepFun_sub_div₀, indepFun_sub_left, indepFun_sub_left₀, indepFun_sub_right, indepFun_sub_right₀, indepFun_sub_sub, indepFun_sum_range_succ, indepFun_sum_range_succ₀, isProbabilityMeasure, meas_biInter, meas_iInter, measure_inter_preimage_eq_mul, of_precomp, of_subsingleton, precomp, iIndepFun_congr, iIndepFun_iff, iIndepFun_iff_finset, iIndepFun_iff_iIndep, iIndepFun_iff_map_fun_eq_pi_map, iIndepFun_iff_measure_inter_preimage_eq_mul, iIndepFun_pi, iIndepFun_precomp_of_bijective, iIndepFun_indicator, iIndep_comap_mem, indep_generateFrom_le, indep_generateFrom_le_nat, indep_generateFrom_lt, indep_generateFrom_of_disjoint, isProbabilityMeasure, meas_biInter, of_precomp, precomp, iIndepSet_iff, iIndepSet_iff_iIndep, iIndepSet_iff_iIndepSets_singleton, iIndepSet_iff_meas_biInter, iIndepSet_precomp_of_bijective, iIndep, iIndepSet_of_mem, indepSets, isProbabilityMeasure, meas_biInter, meas_iInter, of_precomp, of_subsingleton, piiUnionInter_of_notMem, precomp, iIndepSets_iff, iIndepSets_precomp_of_bijective, iIndepSets_singleton_iff, iIndep_comap_mem_iff, iIndep_iff, iIndep_iff_iIndepSets, iIndep_of_iIndep_of_le, iIndep_precomp_of_bijective, indepFun_const_left, indepFun_const_right, indepFun_iff_indepSet_preimage, indepFun_iff_map_prod_eq_prod_map_map, indepFun_iff_map_prod_eq_prod_map_map', indepFun_iff_measure_inter_preimage_eq_mul, indepFun_prod, indepFun_prod₀, indepSet_empty_left, indepSet_empty_right, indepSet_iff_indepSets_singleton, indepSet_iff_measure_inter_eq_mul, indepSets_iff_singleton_indepSets, indepSets_of_indepSets_of_le_left, indepSets_of_indepSets_of_le_right, indepSets_piiUnionInter_of_disjoint, indepSets_singleton_iff, indep_bot_left, indep_bot_right, indep_iSup_of_antitone, indep_iSup_of_directed_le, indep_iSup_of_disjoint, indep_iSup_of_monotone, indep_iff_forall_indepSet, indep_of_indep_of_le_left, indep_of_indep_of_le_right
162
Total170

ProbabilityTheory

Definitions

NameCategoryTheorems
IndepSet 📖MathDef
10 mathmath: IndepSet_iff_Indep, IndepSet_iff, indep_iff_forall_indepSet, indepFun_iff_indepSet_preimage, indepSet_iff_measure_inter_eq_mul, IndepSets.indepSet_of_mem, indepSet_iff_indepSets_singleton, indepSet_empty_left, Indep.indepSet_of_measurableSet, indepSet_empty_right
IndepSets 📖MathDef
16 mathmath: iIndepSets.piiUnionInter_of_notMem, indepSets_piiUnionInter_of_disjoint, Indep.indepSets, indepSets_singleton_iff, indepSets_comap_of_bcf, iIndepSets.indepSets, IndepSets.union_iff, indepSets_comap_pi_of_bcf, indepSets_comap_pi_of_prod_bcf, indepSets_comap_process_of_bcf, Indep_iff_IndepSets, indepSets_iff_singleton_indepSets, indepSet_iff_indepSets_singleton, indepSets_comap_process_of_prod_bcf, IndepSets_iff, IndepFun.singleton_indepSets_of_indicator
iIndep 📖MathDef
10 mathmath: iIndep_iff_iIndepSets, iIndep.of_subsingleton, iIndep_precomp_of_bijective, iIndep_comap_mem_iff, iIndepFun_iff_iIndep, iIndepFun.iIndep, iIndepSet.iIndep_comap_mem, iIndep_iff, iIndepSet_iff_iIndep, iIndepSets.iIndep
iIndepFun 📖MathDef
25 mathmath: iIndepFun_iff, iIndepFun_iff_map_fun_eq_infinitePi_map₀, iIndepFun_iff_measure_inter_preimage_eq_mul, iIndepFun_infinitePi, exists_hasLaw_indepFun, HasGaussianLaw.iIndepFun_of_covariance_strongDual, iIndepFun_pi, exists_iid, iIndepFun_iff_iIndep, iIndepFun_iff_finset, HasGaussianLaw.iIndepFun_of_covariance_eq_zero, iIndepFun_iff_charFunDual_pi, iIndepFun_precomp_of_bijective, iIndepFun_congr, iIndepFun_iff_map_fun_eq_infinitePi_map₀', iIndepFun_uncurry_infinitePi, iIndepFun_iff_charFunDual_pi', iIndepFun_iff_charFun_pi, iIndepFun_uncurry_infinitePi', iIndepSet.iIndepFun_indicator, iIndepFun_iff_map_fun_eq_infinitePi_map, HasGaussianLaw.iIndepFun_of_covariance_eval, HasGaussianLaw.iIndepFun_of_covariance_inner, iIndepFun_iff_map_fun_eq_pi_map, iIndepFun.of_subsingleton
iIndepSet 📖MathDef
7 mathmath: iIndepSets.iIndepSet_of_mem, iIndep_comap_mem_iff, iIndepSet_precomp_of_bijective, iIndepSet_iff, iIndepSet_iff_iIndep, iIndepSet_iff_meas_biInter, iIndepSet_iff_iIndepSets_singleton
iIndepSets 📖MathDef
8 mathmath: iIndep_iff_iIndepSets, iIndep.iIndepSets, iIndepSets_iff, iIndep.iIndepSets', iIndepSets.of_subsingleton, iIndepSet_iff_iIndepSets_singleton, iIndepSets_precomp_of_bijective, iIndepSets_singleton_iff
«term_⟂ᵢ[_]_» 📖CompOp
«term_⟂ᵢ_» 📖CompOp

Theorems

NameKindAssumesProvesValidatesDepends On
IndepFun_iff 📖mathematicalIndepFun
DFunLike.coe
MeasureTheory.Measure
Set
ENNReal
MeasureTheory.Measure.instFunLike
Set.instInter
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
ENNReal.instCommSemiring
IndepFun_iff_Indep
Indep_iff
IndepFun_iff_Indep 📖mathematicalIndepFun
Indep
MeasurableSpace.comap
IndepSet_iff 📖mathematicalIndepSet
DFunLike.coe
MeasureTheory.Measure
Set
ENNReal
MeasureTheory.Measure.instFunLike
Set.instInter
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
ENNReal.instCommSemiring
IndepSet_iff_Indep 📖mathematicalIndepSet
Indep
MeasurableSpace.generateFrom
Set
Set.instSingletonSet
IndepSets_iff 📖mathematicalIndepSets
DFunLike.coe
MeasureTheory.Measure
Set
ENNReal
MeasureTheory.Measure.instFunLike
Set.instInter
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
ENNReal.instCommSemiring
MeasureTheory.Measure.instOuterMeasureClass
MeasureTheory.ae_dirac_eq
instMeasurableSingletonClassOfMeasurableEq
instMeasurableEqOfSecondCountableTopologyOfT2Space
BorelSpace.opensMeasurable
Unit.borelSpace
TopologicalSpace.instSecondCountableTopologyOfLindelofSpaceOfPseudoMetrizableSpace
Countable.LindelofSpace
instCountablePUnit
PseudoEMetricSpace.pseudoMetrizableSpace
TopologicalSpace.t2Space_of_metrizableSpace
EMetricSpace.metrizableSpace
Indep_iff 📖mathematicalIndep
DFunLike.coe
MeasureTheory.Measure
Set
ENNReal
MeasureTheory.Measure.instFunLike
Set.instInter
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
ENNReal.instCommSemiring
Indep_iff_IndepSets
IndepSets_iff
Indep_iff_IndepSets 📖mathematicalIndep
IndepSets
setOf
Set
MeasurableSet
cond_iInter 📖mathematicalMeasurable
iIndepFun
Prod.instMeasurableSpace
MeasurableSet
MeasurableSpace.comap
DFunLike.coe
MeasureTheory.Measure
Set
ENNReal
MeasureTheory.Measure.instFunLike
cond
Set.iInter
Set.preimage
Finset
Finset.instMembership
Finset.prod
CommSemiring.toCommMonoid
ENNReal.instCommSemiring
iIndepFun.isProbabilityMeasure
nonempty_fintype
cond_apply
MeasurableSet.iInter
Finite.to_countable
Set.iInter_ite
Set.iInter_univ
Set.inter_univ
Set.iInter_inter_distrib
Set.iInter_congr
iIndepFun.meas_iInter
MeasurableSet.prod
MeasurableSet.univ
Set.ext
MeasurableSet.inter
Finset.prod_mul_distrib
ENNReal.prod_inv_distrib
MeasureTheory.measure_ne_top
MeasureTheory.IsZeroOrProbabilityMeasure.toIsFiniteMeasure
MeasureTheory.instIsZeroOrProbabilityMeasureOfIsProbabilityMeasure
Finset.prod_congr
ENNReal.inv_mul_cancel
Finset.prod_ite_mem
Finset.univ_inter
iIndepFun_congr 📖mathematicalFilter.EventuallyEq
MeasureTheory.ae
MeasureTheory.Measure
MeasureTheory.Measure.instFunLike
MeasureTheory.Measure.instOuterMeasureClass
iIndepFunMeasureTheory.Measure.instOuterMeasureClass
Kernel.iIndepFun_congr'
MeasureTheory.ae_dirac_eq
instMeasurableSingletonClassOfMeasurableEq
instMeasurableEqOfSecondCountableTopologyOfT2Space
BorelSpace.opensMeasurable
Unit.borelSpace
TopologicalSpace.instSecondCountableTopologyOfLindelofSpaceOfPseudoMetrizableSpace
Countable.LindelofSpace
instCountablePUnit
PseudoEMetricSpace.pseudoMetrizableSpace
TopologicalSpace.t2Space_of_metrizableSpace
EMetricSpace.metrizableSpace
iIndepFun_iff 📖mathematicaliIndepFun
DFunLike.coe
MeasureTheory.Measure
Set
ENNReal
MeasureTheory.Measure.instFunLike
Set.iInter
Finset
Finset.instMembership
Finset.prod
CommSemiring.toCommMonoid
ENNReal.instCommSemiring
iIndepFun_iff_finset 📖mathematicaliIndepFun
Finset
SetLike.instMembership
Finset.instSetLike
Finset.restrict
iIndepFun.precomp
Subtype.val_injective
iIndepFun_iff
Set.ext
Finset.prod_coe_sort
iIndepFun.meas_iInter
iIndepFun_iff_iIndep 📖mathematicaliIndepFun
iIndep
MeasurableSpace.comap
iIndepFun_iff_map_fun_eq_pi_map 📖mathematicalAEMeasurableiIndepFun
MeasureTheory.Measure.map
MeasurableSpace.pi
MeasureTheory.Measure.pi
iIndepFun_iff_measure_inter_preimage_eq_mul
MeasureTheory.Measure.map_apply_of_aemeasurable
aemeasurable_pi_lambda
Finite.to_countable
Finite.of_fintype
MeasurableSet.univ_pi
Set.ext
MeasureTheory.Measure.pi_eq
MeasureTheory.IsFiniteMeasure.toSigmaFinite
MeasureTheory.Measure.isFiniteMeasure_map
MeasureTheory.IsZeroOrProbabilityMeasure.toIsFiniteMeasure
MeasureTheory.instIsZeroOrProbabilityMeasureOfIsProbabilityMeasure
Set.iInter_congr_Prop
Set.iInter_true
Finset.prod_congr
MeasureTheory.IsProbabilityMeasure.measure_univ
Finset.prod_ite_mem
Finset.univ_inter
Finset.prod_ite
Finset.filter_univ_mem
Set.iInter_ite
Set.iInter_univ
Set.inter_univ
MeasureTheory.Measure.pi_pi
iIndepFun_iff_measure_inter_preimage_eq_mul 📖mathematicaliIndepFun
DFunLike.coe
MeasureTheory.Measure
Set
ENNReal
MeasureTheory.Measure.instFunLike
Set.iInter
Finset
Finset.instMembership
Set.preimage
Finset.prod
CommSemiring.toCommMonoid
ENNReal.instCommSemiring
MeasureTheory.Measure.instOuterMeasureClass
Finset.prod_congr
MeasureTheory.ae_dirac_eq
instMeasurableSingletonClassOfMeasurableEq
instMeasurableEqOfSecondCountableTopologyOfT2Space
BorelSpace.opensMeasurable
Unit.borelSpace
TopologicalSpace.instSecondCountableTopologyOfLindelofSpaceOfPseudoMetrizableSpace
Countable.LindelofSpace
instCountablePUnit
PseudoEMetricSpace.pseudoMetrizableSpace
TopologicalSpace.t2Space_of_metrizableSpace
EMetricSpace.metrizableSpace
iIndepFun_pi 📖mathematicalMeasureTheory.IsProbabilityMeasure
AEMeasurable
iIndepFun
MeasurableSpace.pi
MeasureTheory.Measure.pi
iIndepFun_iff_map_fun_eq_pi_map
MeasureTheory.Measure.pi.instIsProbabilityMeasure
AEMeasurable.comp_quasiMeasurePreserving
MeasureTheory.Measure.quasiMeasurePreserving_eval
MeasureTheory.IsFiniteMeasure.toSigmaFinite
MeasureTheory.IsZeroOrProbabilityMeasure.toIsFiniteMeasure
MeasureTheory.instIsZeroOrProbabilityMeasureOfIsProbabilityMeasure
MeasureTheory.Measure.pi_map_pi
MeasureTheory.Measure.isFiniteMeasure_map
MeasureTheory.MeasurePreserving.map_eq
MeasureTheory.measurePreserving_eval
AEMeasurable.map_map_of_aemeasurable
Measurable.aemeasurable
measurable_pi_apply
iIndepFun_precomp_of_bijective 📖mathematicalFunction.BijectiveiIndepFunKernel.iIndepFun_precomp_of_bijective
iIndepSet_iff 📖mathematicaliIndepSet
DFunLike.coe
MeasureTheory.Measure
Set
ENNReal
MeasureTheory.Measure.instFunLike
Set.iInter
Finset
Finset.instMembership
Finset.prod
CommSemiring.toCommMonoid
ENNReal.instCommSemiring
iIndepSet_iff_iIndep 📖mathematicaliIndepSet
iIndep
MeasurableSpace.generateFrom
Set
Set.instSingletonSet
iIndepSet_iff_iIndepSets_singleton 📖mathematicalMeasurableSetiIndepSet
iIndepSets
Set
Set.instSingletonSet
Kernel.iIndepSet_iff_iIndepSets_singleton
iIndepSet_iff_meas_biInter 📖mathematicalMeasurableSetiIndepSet
DFunLike.coe
MeasureTheory.Measure
Set
ENNReal
MeasureTheory.Measure.instFunLike
Set.iInter
Finset
Finset.instMembership
Finset.prod
CommSemiring.toCommMonoid
ENNReal.instCommSemiring
MeasureTheory.Measure.instOuterMeasureClass
Kernel.iIndepSet_iff_meas_biInter
MeasureTheory.ae_dirac_eq
instMeasurableSingletonClassOfMeasurableEq
instMeasurableEqOfSecondCountableTopologyOfT2Space
BorelSpace.opensMeasurable
Unit.borelSpace
TopologicalSpace.instSecondCountableTopologyOfLindelofSpaceOfPseudoMetrizableSpace
Countable.LindelofSpace
instCountablePUnit
PseudoEMetricSpace.pseudoMetrizableSpace
TopologicalSpace.t2Space_of_metrizableSpace
EMetricSpace.metrizableSpace
Finset.prod_congr
iIndepSet_precomp_of_bijective 📖mathematicalFunction.BijectiveiIndepSet
Set
Kernel.iIndepSet_precomp_of_bijective
iIndepSets_iff 📖mathematicaliIndepSets
DFunLike.coe
MeasureTheory.Measure
Set
ENNReal
MeasureTheory.Measure.instFunLike
Set.iInter
Finset
Finset.instMembership
Finset.prod
CommSemiring.toCommMonoid
ENNReal.instCommSemiring
MeasureTheory.Measure.instOuterMeasureClass
Finset.prod_congr
MeasureTheory.ae_dirac_eq
instMeasurableSingletonClassOfMeasurableEq
instMeasurableEqOfSecondCountableTopologyOfT2Space
BorelSpace.opensMeasurable
Unit.borelSpace
TopologicalSpace.instSecondCountableTopologyOfLindelofSpaceOfPseudoMetrizableSpace
Countable.LindelofSpace
instCountablePUnit
PseudoEMetricSpace.pseudoMetrizableSpace
TopologicalSpace.t2Space_of_metrizableSpace
EMetricSpace.metrizableSpace
iIndepSets_precomp_of_bijective 📖mathematicalFunction.BijectiveiIndepSets
Set
Kernel.iIndepSets_precomp_of_bijective
iIndepSets_singleton_iff 📖mathematicaliIndepSets
Set
Set.instSingletonSet
DFunLike.coe
MeasureTheory.Measure
ENNReal
MeasureTheory.Measure.instFunLike
Set.iInter
Finset
Finset.instMembership
Finset.prod
CommSemiring.toCommMonoid
ENNReal.instCommSemiring
MeasureTheory.Measure.instOuterMeasureClass
MeasureTheory.ae_dirac_eq
instMeasurableSingletonClassOfMeasurableEq
instMeasurableEqOfSecondCountableTopologyOfT2Space
BorelSpace.opensMeasurable
Unit.borelSpace
TopologicalSpace.instSecondCountableTopologyOfLindelofSpaceOfPseudoMetrizableSpace
Countable.LindelofSpace
instCountablePUnit
PseudoEMetricSpace.pseudoMetrizableSpace
TopologicalSpace.t2Space_of_metrizableSpace
EMetricSpace.metrizableSpace
Finset.prod_congr
iIndep_comap_mem_iff 📖mathematicaliIndep
MeasurableSpace.comap
Set
Set.instMembership
Top.top
MeasurableSpace
OrderTop.toTop
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
CompleteLattice.toLattice
MeasurableSpace.instCompleteLattice
BoundedOrder.toOrderTop
CompleteLattice.toBoundedOrder
iIndepSet
Kernel.iIndep_comap_mem_iff
iIndep_iff 📖mathematicaliIndep
DFunLike.coe
MeasureTheory.Measure
Set
ENNReal
MeasureTheory.Measure.instFunLike
Set.iInter
Finset
Finset.instMembership
Finset.prod
CommSemiring.toCommMonoid
ENNReal.instCommSemiring
iIndep_iff_iIndepSets 📖mathematicaliIndep
iIndepSets
setOf
Set
MeasurableSet
iIndep_of_iIndep_of_le 📖iIndep
MeasurableSpace
MeasurableSpace.instLE
Kernel.iIndep_of_iIndep_of_le
iIndep_precomp_of_bijective 📖mathematicalFunction.BijectiveiIndep
MeasurableSpace
Kernel.iIndep_precomp_of_bijective
indepFun_const_left 📖mathematicalIndepFunKernel.indepFun_const_left
Kernel.const.instIsZeroOrMarkovKernel
indepFun_const_right 📖mathematicalIndepFunKernel.indepFun_const_right
Kernel.const.instIsZeroOrMarkovKernel
indepFun_iff_indepSet_preimage 📖mathematicalMeasurableIndepFun
IndepSet
Set.preimage
Kernel.indepFun_iff_indepSet_preimage
Kernel.const.instIsZeroOrMarkovKernel
indepFun_iff_map_prod_eq_prod_map_map 📖mathematicalAEMeasurableIndepFun
MeasureTheory.Measure.map
Prod.instMeasurableSpace
MeasureTheory.Measure.prod
indepFun_iff_map_prod_eq_prod_map_map'
MeasureTheory.IsFiniteMeasure.toSigmaFinite
MeasureTheory.Measure.isFiniteMeasure_map
indepFun_iff_map_prod_eq_prod_map_map' 📖mathematicalAEMeasurableIndepFun
MeasureTheory.Measure.map
Prod.instMeasurableSpace
MeasureTheory.Measure.prod
indepFun_iff_measure_inter_preimage_eq_mul
MeasureTheory.Measure.map_apply_of_aemeasurable
AEMeasurable.prodMk
MeasurableSet.prod
MeasureTheory.Measure.prod_eq
MeasureTheory.Measure.prod_prod
MeasureTheory.instSFiniteOfSigmaFinite
indepFun_iff_measure_inter_preimage_eq_mul 📖mathematicalIndepFun
DFunLike.coe
MeasureTheory.Measure
Set
ENNReal
MeasureTheory.Measure.instFunLike
Set.instInter
Set.preimage
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
ENNReal.instCommSemiring
MeasureTheory.Measure.instOuterMeasureClass
MeasureTheory.ae_dirac_eq
instMeasurableSingletonClassOfMeasurableEq
instMeasurableEqOfSecondCountableTopologyOfT2Space
BorelSpace.opensMeasurable
Unit.borelSpace
TopologicalSpace.instSecondCountableTopologyOfLindelofSpaceOfPseudoMetrizableSpace
Countable.LindelofSpace
instCountablePUnit
PseudoEMetricSpace.pseudoMetrizableSpace
TopologicalSpace.t2Space_of_metrizableSpace
EMetricSpace.metrizableSpace
indepFun_prod 📖mathematicalMeasurableIndepFun
Prod.instMeasurableSpace
MeasureTheory.Measure.prod
indepFun_iff_map_prod_eq_prod_map_map
MeasureTheory.Measure.prod.instIsFiniteMeasure
MeasureTheory.IsZeroOrProbabilityMeasure.toIsFiniteMeasure
MeasureTheory.instIsZeroOrProbabilityMeasureOfIsProbabilityMeasure
Measurable.comp_aemeasurable'
AEMeasurable.fst
aemeasurable_id
AEMeasurable.snd
MeasureTheory.Measure.map_map
measurable_fst
IsScalarTower.right
MeasureTheory.Measure.map_fst_prod
MeasureTheory.instSFiniteOfSigmaFinite
MeasureTheory.IsFiniteMeasure.toSigmaFinite
MeasureTheory.IsProbabilityMeasure.measure_univ
one_smul
measurable_snd
MeasureTheory.Measure.map_snd_prod
MeasureTheory.Measure.map_prod_map
indepFun_prod₀ 📖mathematicalAEMeasurableIndepFun
Prod.instMeasurableSpace
MeasureTheory.Measure.prod
indepFun_prod
IndepFun.congr
MeasureTheory.Measure.instOuterMeasureClass
MeasureTheory.ae_eq_comp
Measurable.aemeasurable
measurable_fst
MeasureTheory.MeasurePreserving.map_eq
MeasureTheory.measurePreserving_fst
MeasureTheory.instSFiniteOfSigmaFinite
MeasureTheory.IsFiniteMeasure.toSigmaFinite
MeasureTheory.IsZeroOrProbabilityMeasure.toIsFiniteMeasure
MeasureTheory.instIsZeroOrProbabilityMeasureOfIsProbabilityMeasure
Filter.EventuallyEq.symm
measurable_snd
MeasureTheory.measurePreserving_snd
indepSet_empty_left 📖mathematicalIndepSet
Set
Set.instEmptyCollection
Kernel.indepSet_empty_left
Kernel.const.instIsZeroOrMarkovKernel
indepSet_empty_right 📖mathematicalIndepSet
Set
Set.instEmptyCollection
Kernel.indepSet_empty_right
Kernel.const.instIsZeroOrMarkovKernel
indepSet_iff_indepSets_singleton 📖mathematicalMeasurableSetIndepSet
IndepSets
Set
Set.instSingletonSet
Kernel.indepSet_iff_indepSets_singleton
Kernel.const.instIsZeroOrMarkovKernel
indepSet_iff_measure_inter_eq_mul 📖mathematicalMeasurableSetIndepSet
DFunLike.coe
MeasureTheory.Measure
Set
ENNReal
MeasureTheory.Measure.instFunLike
Set.instInter
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
ENNReal.instCommSemiring
indepSet_iff_indepSets_singleton
indepSets_singleton_iff
indepSets_iff_singleton_indepSets 📖mathematicalIndepSets
Set
Set.instSingletonSet
indepSets_of_indepSets_of_le_left
Set.singleton_subset_iff
Set.biUnion_of_singleton
IndepSets.biUnion
indepSets_of_indepSets_of_le_left 📖IndepSets
Set
Set.instHasSubset
Kernel.indepSets_of_indepSets_of_le_left
indepSets_of_indepSets_of_le_right 📖IndepSets
Set
Set.instHasSubset
Kernel.indepSets_of_indepSets_of_le_right
indepSets_piiUnionInter_of_disjoint 📖mathematicaliIndepSets
Disjoint
Set
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
IndepSets
piiUnionInter
Kernel.indepSets_piiUnionInter_of_disjoint
indepSets_singleton_iff 📖mathematicalIndepSets
Set
Set.instSingletonSet
DFunLike.coe
MeasureTheory.Measure
ENNReal
MeasureTheory.Measure.instFunLike
Set.instInter
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
ENNReal.instCommSemiring
MeasureTheory.Measure.instOuterMeasureClass
MeasureTheory.ae_dirac_eq
instMeasurableSingletonClassOfMeasurableEq
instMeasurableEqOfSecondCountableTopologyOfT2Space
BorelSpace.opensMeasurable
Unit.borelSpace
TopologicalSpace.instSecondCountableTopologyOfLindelofSpaceOfPseudoMetrizableSpace
Countable.LindelofSpace
instCountablePUnit
PseudoEMetricSpace.pseudoMetrizableSpace
TopologicalSpace.t2Space_of_metrizableSpace
EMetricSpace.metrizableSpace
indep_bot_left 📖mathematicalIndep
Bot.bot
MeasurableSpace
OrderBot.toBot
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
CompleteLattice.toLattice
MeasurableSpace.instCompleteLattice
BoundedOrder.toOrderBot
CompleteLattice.toBoundedOrder
Indep.symm
indep_bot_right
indep_bot_right 📖mathematicalIndep
Bot.bot
MeasurableSpace
OrderBot.toBot
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
CompleteLattice.toLattice
MeasurableSpace.instCompleteLattice
BoundedOrder.toOrderBot
CompleteLattice.toBoundedOrder
Kernel.indep_bot_right
Kernel.const.instIsZeroOrMarkovKernel
indep_iSup_of_antitone 📖mathematicalIndep
MeasurableSpace
MeasurableSpace.instLE
Antitone
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
MeasurableSpace.instPartialOrder
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
MeasurableSpace.instCompleteLattice
Kernel.indep_iSup_of_antitone
Kernel.const.instIsZeroOrMarkovKernel
indep_iSup_of_directed_le 📖mathematicalIndep
MeasurableSpace
MeasurableSpace.instLE
Directed
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
MeasurableSpace.instCompleteLattice
Kernel.indep_iSup_of_directed_le
Kernel.const.instIsZeroOrMarkovKernel
indep_iSup_of_disjoint 📖mathematicalMeasurableSpace
MeasurableSpace.instLE
iIndep
Disjoint
Set
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
Indep
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
MeasurableSpace.instCompleteLattice
Set.instMembership
Kernel.indep_iSup_of_disjoint
indep_iSup_of_monotone 📖mathematicalIndep
MeasurableSpace
MeasurableSpace.instLE
Monotone
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
MeasurableSpace.instPartialOrder
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
MeasurableSpace.instCompleteLattice
Kernel.indep_iSup_of_monotone
Kernel.const.instIsZeroOrMarkovKernel
indep_iff_forall_indepSet 📖mathematicalIndep
IndepSet
Kernel.indep_iff_forall_indepSet
indep_of_indep_of_le_left 📖Indep
MeasurableSpace
MeasurableSpace.instLE
Kernel.indep_of_indep_of_le_left
indep_of_indep_of_le_right 📖Indep
MeasurableSpace
MeasurableSpace.instLE
Kernel.indep_of_indep_of_le_right

ProbabilityTheory.Indep

Theorems

NameKindAssumesProvesValidatesDepends On
indepSet_of_measurableSet 📖mathematicalProbabilityTheory.Indep
MeasurableSet
ProbabilityTheory.IndepSetProbabilityTheory.Kernel.Indep.indepSet_of_measurableSet
indepSets 📖mathematicalProbabilityTheory.Indep
MeasurableSpace.generateFrom
ProbabilityTheory.IndepSetsProbabilityTheory.Kernel.Indep.indepSets

ProbabilityTheory.IndepFun

Theorems

NameKindAssumesProvesValidatesDepends On
comp 📖ProbabilityTheory.IndepFun
Measurable
ProbabilityTheory.Kernel.IndepFun.comp
comp₀ 📖ProbabilityTheory.IndepFun
AEMeasurable
MeasureTheory.Measure.map
ProbabilityTheory.Kernel.IndepFun.comp₀
IsScalarTower.right
MeasureTheory.Measure.const_comp
MeasureTheory.IsProbabilityMeasure.measure_univ
MeasureTheory.Measure.dirac.isProbabilityMeasure
one_smul
map_add_eq_map_conv_map 📖mathematicalMeasurable
ProbabilityTheory.IndepFun
MeasureTheory.Measure.map
Pi.instAdd
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
MeasureTheory.Measure.conv
map_add_eq_map_conv_map₀
Measurable.aemeasurable
map_add_eq_map_conv_map' 📖mathematicalMeasurable
ProbabilityTheory.IndepFun
MeasureTheory.Measure.map
Pi.instAdd
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
MeasureTheory.Measure.conv
map_add_eq_map_conv_map₀'
Measurable.aemeasurable
map_add_eq_map_conv_map₀ 📖mathematicalAEMeasurable
ProbabilityTheory.IndepFun
MeasureTheory.Measure.map
Pi.instAdd
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
MeasureTheory.Measure.conv
map_add_eq_map_conv_map₀'
MeasureTheory.IsFiniteMeasure.toSigmaFinite
MeasureTheory.Measure.isFiniteMeasure_map
map_add_eq_map_conv_map₀' 📖mathematicalAEMeasurable
ProbabilityTheory.IndepFun
MeasureTheory.Measure.map
Pi.instAdd
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
MeasureTheory.Measure.conv
AEMeasurable.map_map_of_aemeasurable
Measurable.aemeasurable
MeasurableAdd₂.measurable_add
AEMeasurable.prodMk
ProbabilityTheory.indepFun_iff_map_prod_eq_prod_map_map'
MeasureTheory.Measure.conv.eq_1
map_mul_eq_map_mconv_map 📖mathematicalMeasurable
ProbabilityTheory.IndepFun
MeasureTheory.Measure.map
Pi.instMul
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
MeasureTheory.Measure.mconv
map_mul_eq_map_mconv_map₀
Measurable.aemeasurable
map_mul_eq_map_mconv_map' 📖mathematicalMeasurable
ProbabilityTheory.IndepFun
MeasureTheory.Measure.map
Pi.instMul
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
MeasureTheory.Measure.mconv
map_mul_eq_map_mconv_map₀'
Measurable.aemeasurable
map_mul_eq_map_mconv_map₀ 📖mathematicalAEMeasurable
ProbabilityTheory.IndepFun
MeasureTheory.Measure.map
Pi.instMul
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
MeasureTheory.Measure.mconv
map_mul_eq_map_mconv_map₀'
MeasureTheory.IsFiniteMeasure.toSigmaFinite
MeasureTheory.Measure.isFiniteMeasure_map
map_mul_eq_map_mconv_map₀' 📖mathematicalAEMeasurable
ProbabilityTheory.IndepFun
MeasureTheory.Measure.map
Pi.instMul
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
MeasureTheory.Measure.mconv
AEMeasurable.map_map_of_aemeasurable
Measurable.aemeasurable
MeasurableMul₂.measurable_mul
AEMeasurable.prodMk
ProbabilityTheory.indepFun_iff_map_prod_eq_prod_map_map'
MeasureTheory.Measure.mconv.eq_1
meas_inter 📖mathematicalProbabilityTheory.IndepFun
MeasurableSet
MeasurableSpace.comap
DFunLike.coe
MeasureTheory.Measure
Set
ENNReal
MeasureTheory.Measure.instFunLike
Set.instInter
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
ENNReal.instCommSemiring
ProbabilityTheory.IndepFun_iff
measure_inter_preimage_eq_mul 📖mathematicalProbabilityTheory.IndepFun
MeasurableSet
DFunLike.coe
MeasureTheory.Measure
Set
ENNReal
MeasureTheory.Measure.instFunLike
Set.instInter
Set.preimage
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
ENNReal.instCommSemiring
ProbabilityTheory.indepFun_iff_measure_inter_preimage_eq_mul
neg_left 📖mathematicalProbabilityTheory.IndepFunPi.instNegcomp
MeasurableNeg.measurable_neg
measurable_id
neg_right 📖mathematicalProbabilityTheory.IndepFunPi.instNegcomp
measurable_id
MeasurableNeg.measurable_neg

ProbabilityTheory.IndepSet

Theorems

NameKindAssumesProvesValidatesDepends On
measure_inter_eq_mul 📖mathematicalProbabilityTheory.IndepSetDFunLike.coe
MeasureTheory.Measure
Set
ENNReal
MeasureTheory.Measure.instFunLike
Set.instInter
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
ENNReal.instCommSemiring
MeasureTheory.Measure.instOuterMeasureClass
MeasureTheory.ae_dirac_eq
instMeasurableSingletonClassOfMeasurableEq
instMeasurableEqOfSecondCountableTopologyOfT2Space
BorelSpace.opensMeasurable
Unit.borelSpace
TopologicalSpace.instSecondCountableTopologyOfLindelofSpaceOfPseudoMetrizableSpace
Countable.LindelofSpace
instCountablePUnit
PseudoEMetricSpace.pseudoMetrizableSpace
TopologicalSpace.t2Space_of_metrizableSpace
EMetricSpace.metrizableSpace
ProbabilityTheory.Kernel.IndepSet.measure_inter_eq_mul

ProbabilityTheory.IndepSets

Theorems

NameKindAssumesProvesValidatesDepends On
bInter 📖mathematicalSet
Set.instMembership
ProbabilityTheory.IndepSets
Set.iInterProbabilityTheory.Kernel.IndepSets.bInter
bUnion 📖mathematicalProbabilityTheory.IndepSetsSet.iUnion
Set
Set.instMembership
biUnion
biUnion 📖mathematicalProbabilityTheory.IndepSetsSet.iUnion
Set
Set.instMembership
ProbabilityTheory.Kernel.IndepSets.biUnion
iInter 📖mathematicalProbabilityTheory.IndepSetsSet.iInter
Set
ProbabilityTheory.Kernel.IndepSets.iInter
iUnion 📖mathematicalProbabilityTheory.IndepSetsSet.iUnion
Set
ProbabilityTheory.Kernel.IndepSets.iUnion
indep 📖mathematicalMeasurableSpace
MeasurableSpace.instLE
IsPiSystem
MeasurableSpace.generateFrom
ProbabilityTheory.IndepSets
ProbabilityTheory.IndepProbabilityTheory.Kernel.IndepSets.indep
ProbabilityTheory.Kernel.const.instIsZeroOrMarkovKernel
indep' 📖mathematicalMeasurableSet
IsPiSystem
ProbabilityTheory.IndepSets
ProbabilityTheory.Indep
MeasurableSpace.generateFrom
ProbabilityTheory.Kernel.IndepSets.indep'
ProbabilityTheory.Kernel.const.instIsZeroOrMarkovKernel
indepSet_of_mem 📖mathematicalSet
Set.instMembership
MeasurableSet
ProbabilityTheory.IndepSets
ProbabilityTheory.IndepSetProbabilityTheory.Kernel.IndepSets.indepSet_of_mem
ProbabilityTheory.Kernel.const.instIsZeroOrMarkovKernel
inter 📖mathematicalProbabilityTheory.IndepSetsSet
Set.instInter
ProbabilityTheory.Kernel.IndepSets.inter
union 📖mathematicalProbabilityTheory.IndepSetsSet
Set.instUnion
ProbabilityTheory.Kernel.IndepSets.union
union_iff 📖mathematicalProbabilityTheory.IndepSets
Set
Set.instUnion
ProbabilityTheory.Kernel.IndepSets.union_iff

ProbabilityTheory.iIndep

Theorems

NameKindAssumesProvesValidatesDepends On
iIndepSets 📖mathematicalMeasurableSpace.generateFrom
ProbabilityTheory.iIndep
ProbabilityTheory.iIndepSetsProbabilityTheory.Kernel.iIndep.iIndepSets
iIndepSets' 📖mathematicalProbabilityTheory.iIndepProbabilityTheory.iIndepSets
setOf
Set
MeasurableSet
ProbabilityTheory.iIndep_iff_iIndepSets
indep 📖mathematicalProbabilityTheory.iIndepProbabilityTheory.IndepProbabilityTheory.Kernel.iIndep.indep
isProbabilityMeasure 📖mathematicalProbabilityTheory.iIndepMeasureTheory.IsProbabilityMeasureProbabilityTheory.iIndepSets.isProbabilityMeasure
iIndepSets'
meas_biInter 📖mathematicalProbabilityTheory.iIndep
MeasurableSet
DFunLike.coe
MeasureTheory.Measure
Set
ENNReal
MeasureTheory.Measure.instFunLike
Set.iInter
Finset
Finset.instMembership
Finset.prod
CommSemiring.toCommMonoid
ENNReal.instCommSemiring
ProbabilityTheory.iIndep_iff
meas_iInter 📖mathematicalProbabilityTheory.iIndep
MeasurableSet
DFunLike.coe
MeasureTheory.Measure
Set
ENNReal
MeasureTheory.Measure.instFunLike
Set.iInter
Finset.prod
CommSemiring.toCommMonoid
ENNReal.instCommSemiring
Finset.univ
meas_biInter
Set.iInter_congr_Prop
Set.iInter_true
of_precomp 📖ProbabilityTheory.iIndep
MeasurableSpace
ProbabilityTheory.Kernel.iIndep.of_precomp
of_subsingleton 📖mathematicalProbabilityTheory.iIndepProbabilityTheory.Kernel.iIndep.of_subsingleton
ProbabilityTheory.Kernel.const.instIsMarkovKernel
precomp 📖mathematicalProbabilityTheory.iIndepMeasurableSpaceProbabilityTheory.Kernel.iIndep.precomp

ProbabilityTheory.iIndepFun

Theorems

NameKindAssumesProvesValidatesDepends On
comp 📖ProbabilityTheory.iIndepFun
Measurable
ProbabilityTheory.Kernel.iIndepFun.comp
comp₀ 📖ProbabilityTheory.iIndepFun
AEMeasurable
MeasureTheory.Measure.map
ProbabilityTheory.Kernel.iIndepFun.comp₀
IsScalarTower.right
MeasureTheory.Measure.const_comp
MeasureTheory.IsProbabilityMeasure.measure_univ
MeasureTheory.Measure.dirac.isProbabilityMeasure
one_smul
cond 📖mathematicalMeasurable
ProbabilityTheory.iIndepFun
Prod.instMeasurableSpace
MeasurableSet
ProbabilityTheory.cond
Set.iInter
Set.preimage
ProbabilityTheory.iIndepFun_iff
Finset.prod_congr
Set.iInter_congr_Prop
Set.iInter_iInter_eq_left
Finset.prod_singleton
ProbabilityTheory.cond_iInter
Finset.mem_singleton
iIndep 📖mathematicalProbabilityTheory.iIndepFunProbabilityTheory.iIndep
MeasurableSpace.comap
indepFun 📖mathematicalProbabilityTheory.iIndepFunProbabilityTheory.IndepFunProbabilityTheory.Kernel.iIndepFun.indepFun
indepFun_add_add 📖mathematicalProbabilityTheory.iIndepFun
Measurable
ProbabilityTheory.IndepFun
Pi.instAdd
ProbabilityTheory.Kernel.iIndepFun.indepFun_add_add
indepFun_add_left 📖mathematicalProbabilityTheory.iIndepFun
Measurable
ProbabilityTheory.IndepFun
Pi.instAdd
ProbabilityTheory.Kernel.iIndepFun.indepFun_add_left
indepFun_add_left₀ 📖mathematicalProbabilityTheory.iIndepFun
AEMeasurable
ProbabilityTheory.IndepFun
Pi.instAdd
ProbabilityTheory.Kernel.iIndepFun.indepFun_add_left₀
IsScalarTower.right
MeasureTheory.Measure.const_comp
MeasureTheory.IsProbabilityMeasure.measure_univ
MeasureTheory.Measure.dirac.isProbabilityMeasure
one_smul
indepFun_add_mul₀ 📖mathematicalProbabilityTheory.iIndepFun
AEMeasurable
ProbabilityTheory.IndepFun
Pi.instAdd
ProbabilityTheory.Kernel.iIndepFun.indepFun_add_mul₀
IsScalarTower.right
MeasureTheory.Measure.const_comp
MeasureTheory.IsProbabilityMeasure.measure_univ
MeasureTheory.Measure.dirac.isProbabilityMeasure
one_smul
indepFun_add_right 📖mathematicalProbabilityTheory.iIndepFun
Measurable
ProbabilityTheory.IndepFun
Pi.instAdd
ProbabilityTheory.Kernel.iIndepFun.indepFun_add_right
indepFun_add_right₀ 📖mathematicalProbabilityTheory.iIndepFun
AEMeasurable
ProbabilityTheory.IndepFun
Pi.instAdd
ProbabilityTheory.Kernel.iIndepFun.indepFun_add_right₀
IsScalarTower.right
MeasureTheory.Measure.const_comp
MeasureTheory.IsProbabilityMeasure.measure_univ
MeasureTheory.Measure.dirac.isProbabilityMeasure
one_smul
indepFun_div_div 📖mathematicalProbabilityTheory.iIndepFun
Measurable
ProbabilityTheory.IndepFun
Pi.instDiv
ProbabilityTheory.Kernel.iIndepFun.indepFun_div_div
indepFun_div_div₀ 📖mathematicalProbabilityTheory.iIndepFun
AEMeasurable
ProbabilityTheory.IndepFun
Pi.instDiv
ProbabilityTheory.Kernel.iIndepFun.indepFun_div_div₀
IsScalarTower.right
MeasureTheory.Measure.const_comp
MeasureTheory.IsProbabilityMeasure.measure_univ
MeasureTheory.Measure.dirac.isProbabilityMeasure
one_smul
indepFun_div_left 📖mathematicalProbabilityTheory.iIndepFun
Measurable
ProbabilityTheory.IndepFun
Pi.instDiv
ProbabilityTheory.Kernel.iIndepFun.indepFun_div_left
indepFun_div_left₀ 📖mathematicalProbabilityTheory.iIndepFun
AEMeasurable
ProbabilityTheory.IndepFun
Pi.instDiv
ProbabilityTheory.Kernel.iIndepFun.indepFun_div_left₀
IsScalarTower.right
MeasureTheory.Measure.const_comp
MeasureTheory.IsProbabilityMeasure.measure_univ
MeasureTheory.Measure.dirac.isProbabilityMeasure
one_smul
indepFun_div_right 📖mathematicalProbabilityTheory.iIndepFun
Measurable
ProbabilityTheory.IndepFun
Pi.instDiv
ProbabilityTheory.Kernel.iIndepFun.indepFun_div_right
indepFun_div_right₀ 📖mathematicalProbabilityTheory.iIndepFun
AEMeasurable
ProbabilityTheory.IndepFun
Pi.instDiv
ProbabilityTheory.Kernel.iIndepFun.indepFun_div_right₀
IsScalarTower.right
MeasureTheory.Measure.const_comp
MeasureTheory.IsProbabilityMeasure.measure_univ
MeasureTheory.Measure.dirac.isProbabilityMeasure
one_smul
indepFun_finset 📖mathematicalDisjoint
Finset
Finset.partialOrder
Finset.instOrderBot
ProbabilityTheory.iIndepFun
Measurable
ProbabilityTheory.IndepFun
MeasurableSpace.pi
SetLike.instMembership
Finset.instSetLike
ProbabilityTheory.Kernel.iIndepFun.indepFun_finset
indepFun_finset_prod_of_notMem 📖mathematicalProbabilityTheory.iIndepFun
Measurable
Finset
Finset.instMembership
ProbabilityTheory.IndepFun
Finset.prod
Pi.commMonoid
ProbabilityTheory.Kernel.iIndepFun.indepFun_finset_prod_of_notMem
indepFun_finset_prod_of_notMem₀ 📖mathematicalProbabilityTheory.iIndepFun
AEMeasurable
Finset
Finset.instMembership
ProbabilityTheory.IndepFun
Finset.prod
Pi.commMonoid
ProbabilityTheory.Kernel.iIndepFun.indepFun_finset_prod_of_notMem₀
IsScalarTower.right
MeasureTheory.Measure.const_comp
MeasureTheory.IsProbabilityMeasure.measure_univ
MeasureTheory.Measure.dirac.isProbabilityMeasure
one_smul
indepFun_finset_sum_of_notMem 📖mathematicalProbabilityTheory.iIndepFun
Measurable
Finset
Finset.instMembership
ProbabilityTheory.IndepFun
Finset.sum
Pi.addCommMonoid
ProbabilityTheory.Kernel.iIndepFun.indepFun_finset_sum_of_notMem
indepFun_finset_sum_of_notMem₀ 📖mathematicalProbabilityTheory.iIndepFun
AEMeasurable
Finset
Finset.instMembership
ProbabilityTheory.IndepFun
Finset.sum
Pi.addCommMonoid
ProbabilityTheory.Kernel.iIndepFun.indepFun_finset_sum_of_notMem₀
IsScalarTower.right
MeasureTheory.Measure.const_comp
MeasureTheory.IsProbabilityMeasure.measure_univ
MeasureTheory.Measure.dirac.isProbabilityMeasure
one_smul
indepFun_finset₀ 📖mathematicalDisjoint
Finset
Finset.partialOrder
Finset.instOrderBot
ProbabilityTheory.iIndepFun
AEMeasurable
ProbabilityTheory.IndepFun
MeasurableSpace.pi
SetLike.instMembership
Finset.instSetLike
ProbabilityTheory.Kernel.iIndepFun.indepFun_finset₀
IsScalarTower.right
MeasureTheory.Measure.const_comp
MeasureTheory.IsProbabilityMeasure.measure_univ
MeasureTheory.Measure.dirac.isProbabilityMeasure
one_smul
indepFun_mul_left 📖mathematicalProbabilityTheory.iIndepFun
Measurable
ProbabilityTheory.IndepFun
Pi.instMul
ProbabilityTheory.Kernel.iIndepFun.indepFun_mul_left
indepFun_mul_left₀ 📖mathematicalProbabilityTheory.iIndepFun
AEMeasurable
ProbabilityTheory.IndepFun
Pi.instMul
ProbabilityTheory.Kernel.iIndepFun.indepFun_mul_left₀
IsScalarTower.right
MeasureTheory.Measure.const_comp
MeasureTheory.IsProbabilityMeasure.measure_univ
MeasureTheory.Measure.dirac.isProbabilityMeasure
one_smul
indepFun_mul_mul 📖mathematicalProbabilityTheory.iIndepFun
Measurable
ProbabilityTheory.IndepFun
Pi.instMul
ProbabilityTheory.Kernel.iIndepFun.indepFun_mul_mul
indepFun_mul_mul₀ 📖mathematicalProbabilityTheory.iIndepFun
AEMeasurable
ProbabilityTheory.IndepFun
Pi.instMul
ProbabilityTheory.Kernel.iIndepFun.indepFun_mul_mul₀
IsScalarTower.right
MeasureTheory.Measure.const_comp
MeasureTheory.IsProbabilityMeasure.measure_univ
MeasureTheory.Measure.dirac.isProbabilityMeasure
one_smul
indepFun_mul_right 📖mathematicalProbabilityTheory.iIndepFun
Measurable
ProbabilityTheory.IndepFun
Pi.instMul
ProbabilityTheory.Kernel.iIndepFun.indepFun_mul_right
indepFun_mul_right₀ 📖mathematicalProbabilityTheory.iIndepFun
AEMeasurable
ProbabilityTheory.IndepFun
Pi.instMul
ProbabilityTheory.Kernel.iIndepFun.indepFun_mul_right₀
IsScalarTower.right
MeasureTheory.Measure.const_comp
MeasureTheory.IsProbabilityMeasure.measure_univ
MeasureTheory.Measure.dirac.isProbabilityMeasure
one_smul
indepFun_prodMk 📖mathematicalProbabilityTheory.iIndepFun
Measurable
ProbabilityTheory.IndepFun
Prod.instMeasurableSpace
ProbabilityTheory.Kernel.iIndepFun.indepFun_prodMk
indepFun_prodMk_prodMk 📖mathematicalProbabilityTheory.iIndepFun
Measurable
ProbabilityTheory.IndepFun
Prod.instMeasurableSpace
ProbabilityTheory.Kernel.iIndepFun.indepFun_prodMk_prodMk
indepFun_prodMk_prodMk₀ 📖mathematicalProbabilityTheory.iIndepFun
AEMeasurable
ProbabilityTheory.IndepFun
Prod.instMeasurableSpace
ProbabilityTheory.Kernel.iIndepFun.indepFun_prodMk_prodMk₀
IsScalarTower.right
MeasureTheory.Measure.const_comp
MeasureTheory.IsProbabilityMeasure.measure_univ
MeasureTheory.Measure.dirac.isProbabilityMeasure
one_smul
indepFun_prodMk₀ 📖mathematicalProbabilityTheory.iIndepFun
AEMeasurable
ProbabilityTheory.IndepFun
Prod.instMeasurableSpace
ProbabilityTheory.Kernel.iIndepFun.indepFun_prodMk₀
IsScalarTower.right
MeasureTheory.Measure.const_comp
MeasureTheory.IsProbabilityMeasure.measure_univ
MeasureTheory.Measure.dirac.isProbabilityMeasure
one_smul
indepFun_prod_range_succ 📖mathematicalProbabilityTheory.iIndepFun
Measurable
ProbabilityTheory.IndepFun
Finset.prod
Pi.commMonoid
Finset.range
ProbabilityTheory.Kernel.iIndepFun.indepFun_prod_range_succ
indepFun_prod_range_succ₀ 📖mathematicalProbabilityTheory.iIndepFun
AEMeasurable
ProbabilityTheory.IndepFun
Finset.prod
Pi.commMonoid
Finset.range
indepFun_finset_prod_of_notMem₀
indepFun_sub_div₀ 📖mathematicalProbabilityTheory.iIndepFun
AEMeasurable
ProbabilityTheory.IndepFun
Pi.instSub
ProbabilityTheory.Kernel.iIndepFun.indepFun_sub_div₀
IsScalarTower.right
MeasureTheory.Measure.const_comp
MeasureTheory.IsProbabilityMeasure.measure_univ
MeasureTheory.Measure.dirac.isProbabilityMeasure
one_smul
indepFun_sub_left 📖mathematicalProbabilityTheory.iIndepFun
Measurable
ProbabilityTheory.IndepFun
Pi.instSub
ProbabilityTheory.Kernel.iIndepFun.indepFun_sub_left
indepFun_sub_left₀ 📖mathematicalProbabilityTheory.iIndepFun
AEMeasurable
ProbabilityTheory.IndepFun
Pi.instSub
ProbabilityTheory.Kernel.iIndepFun.indepFun_sub_left₀
IsScalarTower.right
MeasureTheory.Measure.const_comp
MeasureTheory.IsProbabilityMeasure.measure_univ
MeasureTheory.Measure.dirac.isProbabilityMeasure
one_smul
indepFun_sub_right 📖mathematicalProbabilityTheory.iIndepFun
Measurable
ProbabilityTheory.IndepFun
Pi.instSub
ProbabilityTheory.Kernel.iIndepFun.indepFun_sub_right
indepFun_sub_right₀ 📖mathematicalProbabilityTheory.iIndepFun
AEMeasurable
ProbabilityTheory.IndepFun
Pi.instSub
ProbabilityTheory.Kernel.iIndepFun.indepFun_sub_right₀
IsScalarTower.right
MeasureTheory.Measure.const_comp
MeasureTheory.IsProbabilityMeasure.measure_univ
MeasureTheory.Measure.dirac.isProbabilityMeasure
one_smul
indepFun_sub_sub 📖mathematicalProbabilityTheory.iIndepFun
Measurable
ProbabilityTheory.IndepFun
Pi.instSub
ProbabilityTheory.Kernel.iIndepFun.indepFun_sub_sub
indepFun_sum_range_succ 📖mathematicalProbabilityTheory.iIndepFun
Measurable
ProbabilityTheory.IndepFun
Finset.sum
Pi.addCommMonoid
Finset.range
ProbabilityTheory.Kernel.iIndepFun.indepFun_sum_range_succ
indepFun_sum_range_succ₀ 📖mathematicalProbabilityTheory.iIndepFun
AEMeasurable
ProbabilityTheory.IndepFun
Finset.sum
Pi.addCommMonoid
Finset.range
indepFun_finset_sum_of_notMem₀
Finset.mem_range
lt_self_iff_false
isProbabilityMeasure 📖mathematicalProbabilityTheory.iIndepFunMeasureTheory.IsProbabilityMeasureSet.iInter_congr_Prop
Set.iInter_of_empty
instIsEmptyFalse
Set.iInter_univ
Finset.prod_const
pow_zero
meas_biInter
meas_biInter 📖mathematicalProbabilityTheory.iIndepFun
MeasurableSet
MeasurableSpace.comap
DFunLike.coe
MeasureTheory.Measure
Set
ENNReal
MeasureTheory.Measure.instFunLike
Set.iInter
Finset
Finset.instMembership
Finset.prod
CommSemiring.toCommMonoid
ENNReal.instCommSemiring
ProbabilityTheory.iIndep.meas_biInter
iIndep
meas_iInter 📖mathematicalProbabilityTheory.iIndepFun
MeasurableSet
MeasurableSpace.comap
DFunLike.coe
MeasureTheory.Measure
Set
ENNReal
MeasureTheory.Measure.instFunLike
Set.iInter
Finset.prod
CommSemiring.toCommMonoid
ENNReal.instCommSemiring
Finset.univ
ProbabilityTheory.iIndep.meas_iInter
iIndep
measure_inter_preimage_eq_mul 📖mathematicalProbabilityTheory.iIndepFun
MeasurableSet
DFunLike.coe
MeasureTheory.Measure
Set
ENNReal
MeasureTheory.Measure.instFunLike
Set.iInter
Finset
Finset.instMembership
Set.preimage
Finset.prod
CommSemiring.toCommMonoid
ENNReal.instCommSemiring
ProbabilityTheory.iIndepFun_iff_measure_inter_preimage_eq_mul
of_precomp 📖ProbabilityTheory.iIndepFunProbabilityTheory.Kernel.iIndepFun.of_precomp
of_subsingleton 📖mathematicalProbabilityTheory.iIndepFunProbabilityTheory.Kernel.iIndepFun.of_subsingleton
ProbabilityTheory.Kernel.const.instIsMarkovKernel
precomp 📖ProbabilityTheory.iIndepFunProbabilityTheory.Kernel.iIndepFun.precomp

ProbabilityTheory.iIndepSet

Theorems

NameKindAssumesProvesValidatesDepends On
iIndepFun_indicator 📖mathematicalProbabilityTheory.iIndepSetProbabilityTheory.iIndepFun
Set.indicator
ProbabilityTheory.Kernel.iIndepSet.iIndepFun_indicator
iIndep_comap_mem 📖mathematicalProbabilityTheory.iIndepSetProbabilityTheory.iIndep
MeasurableSpace.comap
Set
Set.instMembership
Top.top
MeasurableSpace
OrderTop.toTop
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
CompleteLattice.toLattice
MeasurableSpace.instCompleteLattice
BoundedOrder.toOrderTop
CompleteLattice.toBoundedOrder
ProbabilityTheory.iIndep_comap_mem_iff
indep_generateFrom_le 📖mathematicalMeasurableSet
ProbabilityTheory.iIndepSet
Preorder.toLT
ProbabilityTheory.Indep
MeasurableSpace.generateFrom
Set
Set.instSingletonSet
setOf
Preorder.toLE
ProbabilityTheory.Kernel.iIndepSet.indep_generateFrom_le
indep_generateFrom_le_nat 📖mathematicalMeasurableSet
ProbabilityTheory.iIndepSet
ProbabilityTheory.Indep
MeasurableSpace.generateFrom
Set
Set.instSingletonSet
setOf
ProbabilityTheory.Kernel.iIndepSet.indep_generateFrom_le_nat
indep_generateFrom_lt 📖mathematicalMeasurableSet
ProbabilityTheory.iIndepSet
ProbabilityTheory.Indep
MeasurableSpace.generateFrom
Set
Set.instSingletonSet
setOf
Preorder.toLT
ProbabilityTheory.Kernel.iIndepSet.indep_generateFrom_lt
indep_generateFrom_of_disjoint 📖mathematicalMeasurableSet
ProbabilityTheory.iIndepSet
Disjoint
Set
OmegaCompletePartialOrder.toPartialOrder
CompleteLattice.instOmegaCompletePartialOrder
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
ProbabilityTheory.Indep
MeasurableSpace.generateFrom
setOf
Set.instMembership
ProbabilityTheory.Kernel.iIndepSet.indep_generateFrom_of_disjoint
isProbabilityMeasure 📖mathematicalProbabilityTheory.iIndepSetMeasureTheory.IsProbabilityMeasureProbabilityTheory.iIndep.isProbabilityMeasure
ProbabilityTheory.iIndepSet_iff_iIndep
meas_biInter 📖mathematicalProbabilityTheory.iIndepSetDFunLike.coe
MeasureTheory.Measure
Set
ENNReal
MeasureTheory.Measure.instFunLike
Set.iInter
Finset
Finset.instMembership
Finset.prod
CommSemiring.toCommMonoid
ENNReal.instCommSemiring
MeasureTheory.Measure.instOuterMeasureClass
Finset.prod_congr
MeasureTheory.ae_dirac_eq
instMeasurableSingletonClassOfMeasurableEq
instMeasurableEqOfSecondCountableTopologyOfT2Space
BorelSpace.opensMeasurable
Unit.borelSpace
TopologicalSpace.instSecondCountableTopologyOfLindelofSpaceOfPseudoMetrizableSpace
Countable.LindelofSpace
instCountablePUnit
PseudoEMetricSpace.pseudoMetrizableSpace
TopologicalSpace.t2Space_of_metrizableSpace
EMetricSpace.metrizableSpace
ProbabilityTheory.Kernel.iIndepSet.meas_biInter
of_precomp 📖ProbabilityTheory.iIndepSet
Set
ProbabilityTheory.Kernel.iIndepSet.of_precomp
precomp 📖mathematicalProbabilityTheory.iIndepSetSetProbabilityTheory.Kernel.iIndepSet.precomp

ProbabilityTheory.iIndepSets

Theorems

NameKindAssumesProvesValidatesDepends On
iIndep 📖mathematicalMeasurableSpace
MeasurableSpace.instLE
IsPiSystem
MeasurableSpace.generateFrom
ProbabilityTheory.iIndepSets
ProbabilityTheory.iIndepProbabilityTheory.Kernel.iIndepSets.iIndep
iIndepSet_of_mem 📖mathematicalSet
Set.instMembership
MeasurableSet
ProbabilityTheory.iIndepSets
ProbabilityTheory.iIndepSetProbabilityTheory.Kernel.iIndepSets.iIndepSet_of_mem
indepSets 📖mathematicalProbabilityTheory.iIndepSetsProbabilityTheory.IndepSetsProbabilityTheory.Kernel.iIndepSets.indepSets
isProbabilityMeasure 📖mathematicalProbabilityTheory.iIndepSetsMeasureTheory.IsProbabilityMeasureMeasureTheory.Measure.instOuterMeasureClass
instIsEmptyFalse
Set.iInter_congr_Prop
Set.iInter_of_empty
Set.iInter_univ
Finset.prod_congr
Finset.prod_const
pow_zero
MeasureTheory.ae_dirac_eq
instMeasurableSingletonClassOfMeasurableEq
instMeasurableEqOfSecondCountableTopologyOfT2Space
BorelSpace.opensMeasurable
Unit.borelSpace
TopologicalSpace.instSecondCountableTopologyOfLindelofSpaceOfPseudoMetrizableSpace
Countable.LindelofSpace
instCountablePUnit
PseudoEMetricSpace.pseudoMetrizableSpace
TopologicalSpace.t2Space_of_metrizableSpace
EMetricSpace.metrizableSpace
meas_biInter 📖mathematicalProbabilityTheory.iIndepSets
Set
Set.instMembership
DFunLike.coe
MeasureTheory.Measure
ENNReal
MeasureTheory.Measure.instFunLike
Set.iInter
Finset
Finset.instMembership
Finset.prod
CommSemiring.toCommMonoid
ENNReal.instCommSemiring
ProbabilityTheory.iIndepSets_iff
meas_iInter 📖mathematicalProbabilityTheory.iIndepSets
Set
Set.instMembership
DFunLike.coe
MeasureTheory.Measure
ENNReal
MeasureTheory.Measure.instFunLike
Set.iInter
Finset.prod
CommSemiring.toCommMonoid
ENNReal.instCommSemiring
Finset.univ
meas_biInter
Set.iInter_congr_Prop
Set.iInter_true
of_precomp 📖ProbabilityTheory.iIndepSets
Set
ProbabilityTheory.Kernel.iIndepSets.of_precomp
of_subsingleton 📖mathematicalProbabilityTheory.iIndepSetsProbabilityTheory.Kernel.iIndepSets.of_subsingleton
ProbabilityTheory.Kernel.const.instIsMarkovKernel
piiUnionInter_of_notMem 📖mathematicalProbabilityTheory.iIndepSets
Finset
Finset.instMembership
ProbabilityTheory.IndepSets
piiUnionInter
SetLike.coe
Finset.instSetLike
ProbabilityTheory.Kernel.iIndepSets.piiUnionInter_of_notMem
precomp 📖mathematicalProbabilityTheory.iIndepSetsSetProbabilityTheory.Kernel.iIndepSets.precomp

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