Documentation Verification Report

Basic

๐Ÿ“ Source: Mathlib/Probability/Moments/Basic.lean

Statistics

MetricCount
DefinitionscentralMoment, cgf, mgf, moment
4
Theoremsintegral_comp_id_comm, integral_comp_id_comm', integral_id_map, cgf_add, exp_mul, integrable_exp_mul_add, mgf_add, mgf_add', aemeasurable_exp_mul, aestronglyMeasurable_exp_mul_add, aestronglyMeasurable_exp_mul_sum, centralMoment_congr_ae, centralMoment_one, centralMoment_one', centralMoment_two_eq_variance, centralMoment_zero, centralMoment_zero_measure, cgf_const, cgf_const', cgf_neg, cgf_undef, cgf_zero, cgf_zero', cgf_zero_fun, cgf_zero_measure, exp_cgf, cgf_sum, cgf_sumโ‚€, integrable_exp_mul_sum, mgf_sum, mgf_sumโ‚€, integrable_exp_mul_of_le, integrable_exp_mul_of_mem_Icc, measure_ge_le_exp_cgf, measure_ge_le_exp_mul_mgf, measure_le_le_exp_cgf, measure_le_le_exp_mul_mgf, mgf_add_const, mgf_add_measure, mgf_anti_of_nonpos, mgf_congr, mgf_congr_identDistrib, mgf_congr_of_identDistrib, mgf_const, mgf_const', mgf_const_add, mgf_const_mul, mgf_dirac, mgf_dirac', mgf_id_map, mgf_map, mgf_mono_of_nonneg, mgf_neg, mgf_nonneg, mgf_pos, mgf_pos', mgf_pos_iff, mgf_smul_left, mgf_smul_measure, mgf_sum_measure, mgf_sum_of_identDistrib, mgf_sum_of_identDistribโ‚€, mgf_undef, mgf_zero, mgf_zero', mgf_zero_fun, mgf_zero_measure, moment_def, moment_one, moment_zero, moment_zero_measure
71
Total75

ContinuousLinearMap

Theorems

NameKindAssumesProvesValidatesDepends On
integral_comp_id_comm ๐Ÿ“–mathematicalMeasureTheory.Integrable
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
NormedAddCommGroup.toSeminormedAddCommGroup
SeminormedAddGroup.toContinuousENorm
SeminormedAddCommGroup.toSeminormedAddGroup
MeasureTheory.integral
DFunLike.coe
ContinuousLinearMap
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
DenselyNormedField.toNormedField
RCLike.toDenselyNormedField
RingHom.id
Semiring.toNonAssocSemiring
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
NormedAddCommGroup.toSeminormedAddCommGroup
ESeminormedAddCommMonoid.toAddCommMonoid
ENormedAddCommMonoid.toESeminormedAddCommMonoid
NormedAddCommGroup.toENormedAddCommMonoid
NormedSpace.toModule
funLike
โ€”integral_comp_id_comm'
integral_comp_id_comm' ๐Ÿ“–mathematicalMeasureTheory.Integrable
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
NormedAddCommGroup.toSeminormedAddCommGroup
SeminormedAddGroup.toContinuousENorm
SeminormedAddCommGroup.toSeminormedAddGroup
MeasureTheory.integral
DFunLike.coe
ContinuousLinearMap
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
DenselyNormedField.toNormedField
RCLike.toDenselyNormedField
RingHom.id
Semiring.toNonAssocSemiring
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
NormedAddCommGroup.toSeminormedAddCommGroup
ESeminormedAddCommMonoid.toAddCommMonoid
ENormedAddCommMonoid.toESeminormedAddCommMonoid
NormedAddCommGroup.toENormedAddCommMonoid
NormedSpace.toModule
funLike
โ€”integral_comp_comm
integral_id_map ๐Ÿ“–mathematicalMeasureTheory.Integrable
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
NormedAddCommGroup.toSeminormedAddCommGroup
SeminormedAddGroup.toContinuousENorm
SeminormedAddCommGroup.toSeminormedAddGroup
MeasureTheory.integral
MeasureTheory.Measure.map
DFunLike.coe
ContinuousLinearMap
DivisionSemiring.toSemiring
Semifield.toDivisionSemiring
Field.toSemifield
NormedField.toField
DenselyNormedField.toNormedField
RCLike.toDenselyNormedField
RingHom.id
Semiring.toNonAssocSemiring
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
SeminormedAddCommGroup.toPseudoMetricSpace
NormedAddCommGroup.toSeminormedAddCommGroup
ESeminormedAddCommMonoid.toAddCommMonoid
ENormedAddCommMonoid.toESeminormedAddCommMonoid
NormedAddCommGroup.toENormedAddCommMonoid
NormedSpace.toModule
funLike
โ€”MeasureTheory.integral_map
Continuous.aemeasurable
Continuous.clm_apply
continuous_const
SeminormedAddCommGroup.toIsTopologicalAddGroup
continuous_id'
aestronglyMeasurable_id
PseudoEMetricSpace.pseudoMetrizableSpace
BorelSpace.opensMeasurable
integral_comp_id_comm

ProbabilityTheory

Definitions

NameCategoryTheorems
centralMoment ๐Ÿ“–CompOp
6 mathmath: centralMoment_two_eq_variance, centralMoment_one, centralMoment_one', centralMoment_zero, centralMoment_zero_measure, centralMoment_congr_ae
cgf ๐Ÿ“–CompOp
33 mathmath: measure_ge_le_exp_cgf, cgf_gaussianReal, tilted_mul_apply_cgf, exp_cgf, cgf_zero_fun, cgf_zero, analyticOnNhd_cgf, exists_cgf_eq_iteratedDeriv_two_cgf_mul, tilted_mul_apply_eq_ofReal_integral_cgf', analyticOn_cgf, analyticAt_cgf, cgf_const, iIndepFun.cgf_sum, setIntegral_tilted_mul_eq_cgf', IndepFun.cgf_add, cgf_neg, cgf_zero', HasSubgaussianMGF.cgf_le, cgf_const', iteratedDeriv_two_cgf, deriv_cgf, iteratedDeriv_two_cgf_eq_integral, tilted_mul_apply_eq_ofReal_integral_cgf, setIntegral_tilted_mul_eq_cgf, deriv_cgf_zero, variance_tilted_mul, iIndepFun.cgf_sumโ‚€, cgf_zero_measure, measure_le_le_exp_cgf, cgf_undef, integral_tilted_mul_self, tilted_mul_apply_cgf', integral_tilted_mul_eq_cgf
mgf ๐Ÿ“–CompOp
74 mathmath: mgf_anti_of_nonpos, analyticOn_mgf, mgf_const_add, mgf_nonneg, mgf_neg, mgf_congr_identDistrib, iIndepFun.mgf_sumโ‚€, mgf_dirac, mgf_zero_fun, mgf_mono_of_nonneg, HasSubgaussianMGF.mgf_le, mgf_le_of_mem_Icc_of_integral_eq_zero, exp_cgf, mgf_const, tilted_mul_apply_eq_ofReal_integral_mgf', analyticAt_mgf, differentiableAt_mgf, iteratedDeriv_mgf, tilted_mul_apply_eq_ofReal_integral_mgf, differentiableOn_mgf, mgf_sum_of_identDistribโ‚€, mgf_add_measure, setIntegral_tilted_mul_eq_mgf', mgf_zero, mgf_zero_measure, tilted_mul_apply_mgf', mgf_fun_id_gaussianReal, norm_complexMGF_le_mgf, hasFPowerSeriesAt_mgf, differentiableAt_iteratedDeriv_mgf, mgf_pos', mgf_add_const, iteratedDeriv_mgf_zero, deriv_mgf, integral_tilted_mul_eq_mgf, mgf_smul_left, hasDerivAt_iteratedDeriv_mgf, mgf_const', mgf_map, tilted_mul_apply_mgf, iteratedDeriv_two_cgf, deriv_mgf_zero, deriv_cgf, mgf_sum_of_identDistrib, analyticOnNhd_mgf, mgf_zero', iteratedDeriv_two_cgf_eq_integral, setIntegral_tilted_mul_eq_mgf, mgf_dirac', complexMGF_ofReal, mgf_gaussianReal, mgf_id_gaussianReal, mgf_id_map, measure_le_le_exp_mul_mgf, measure_ge_le_exp_mul_mgf, hasDerivAt_mgf, mgf_const_mul, IndepFun.mgf_add, re_complexMGF_ofReal', mgf_pos, continuous_mgf, mgf_congr, mgf_smul_measure, analyticAt_iteratedDeriv_mgf, analyticOn_iteratedDeriv_mgf, continuousOn_mgf, mgf_undef, mgf_sum_measure, mgf_pos_iff, iIndepFun.mgf_sum, mgf_congr_of_identDistrib, IndepFun.mgf_add', analyticOnNhd_iteratedDeriv_mgf, re_complexMGF_ofReal
moment ๐Ÿ“–CompOp
4 mathmath: moment_one, moment_zero, moment_zero_measure, moment_def

Theorems

NameKindAssumesProvesValidatesDepends On
aemeasurable_exp_mul ๐Ÿ“–mathematicalAEMeasurable
Real
Real.measurableSpace
MeasureTheory.AEStronglyMeasurable
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.exp
Real.instMul
โ€”AEMeasurable.aestronglyMeasurable
PseudoEMetricSpace.pseudoMetrizableSpace
BorelSpace.opensMeasurable
Real.borelSpace
instSecondCountableTopologyReal
Measurable.comp_aemeasurable
Real.measurable_exp
AEMeasurable.const_mul
ContinuousMul.measurableMul
IsSemitopologicalSemiring.toSeparatelyContinuousMul
IsSemitopologicalRing.toIsSemitopologicalSemiring
IsTopologicalRing.toIsSemitopologicalRing
instIsTopologicalRingReal
aestronglyMeasurable_exp_mul_add ๐Ÿ“–mathematicalMeasureTheory.AEStronglyMeasurable
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.exp
Real.instMul
MeasureTheory.AEStronglyMeasurable
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.exp
Real.instMul
Pi.instAdd
Real.instAdd
โ€”mul_add
Distrib.leftDistribClass
Real.exp_add
MeasureTheory.AEStronglyMeasurable.mul
IsTopologicalSemiring.toContinuousMul
IsTopologicalRing.toIsTopologicalSemiring
instIsTopologicalRingReal
aestronglyMeasurable_exp_mul_sum ๐Ÿ“–mathematicalMeasureTheory.AEStronglyMeasurable
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.exp
Real.instMul
MeasureTheory.AEStronglyMeasurable
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.exp
Real.instMul
Finset.sum
Pi.addCommMonoid
Real.instAddCommMonoid
โ€”Finset.induction_on
Finset.sum_apply
MulZeroClass.mul_zero
Real.exp_zero
MeasureTheory.aestronglyMeasurable_const
Finset.mem_insert_of_mem
Finset.sum_insert
aestronglyMeasurable_exp_mul_add
Finset.mem_insert_self
centralMoment_congr_ae ๐Ÿ“–mathematicalFilter.EventuallyEq
Real
MeasureTheory.ae
MeasureTheory.Measure
MeasureTheory.Measure.instFunLike
MeasureTheory.Measure.instOuterMeasureClass
centralMomentโ€”MeasureTheory.Measure.instOuterMeasureClass
MeasureTheory.integral_congr_ae
Filter.mp_mem
Filter.univ_mem'
centralMoment_one ๐Ÿ“–mathematicalโ€”centralMoment
Real
Real.instZero
โ€”MeasureTheory.eq_zero_or_isProbabilityMeasure
MeasureTheory.integral_zero_measure
sub_zero
pow_one
centralMoment_one'
MeasureTheory.IsZeroOrProbabilityMeasure.toIsFiniteMeasure
MeasureTheory.probReal_univ
sub_self
MulZeroClass.zero_mul
sub_add_cancel
MeasureTheory.Integrable.add
IsSemitopologicalSemiring.toContinuousAdd
IsSemitopologicalRing.toIsSemitopologicalSemiring
IsTopologicalRing.toIsSemitopologicalRing
instIsTopologicalRingReal
MeasureTheory.integrable_const
MeasureTheory.integral_undef
centralMoment_one' ๐Ÿ“–mathematicalMeasureTheory.Integrable
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
SeminormedAddGroup.toContinuousENorm
SeminormedAddCommGroup.toSeminormedAddGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
centralMoment
Real
Real.instMul
Real.instSub
Real.instOne
MeasureTheory.Measure.real
Set.univ
MeasureTheory.integral
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
โ€”pow_one
MeasureTheory.integral_sub
MeasureTheory.integrable_const
MeasureTheory.integral_const
Real.instCompleteSpace
sub_mul
one_mul
centralMoment_two_eq_variance ๐Ÿ“–mathematicalAEMeasurable
Real
Real.measurableSpace
centralMoment
variance
โ€”variance_eq_integral
centralMoment_zero ๐Ÿ“–mathematicalโ€”centralMoment
Pi.instZero
Real
Real.instZero
โ€”MeasureTheory.integral_const
Real.instCompleteSpace
MulZeroClass.mul_zero
zero_sub
neg_zero
zero_pow
centralMoment_zero_measure ๐Ÿ“–mathematicalโ€”centralMoment
MeasureTheory.Measure
MeasureTheory.Measure.instZero
Real
Real.instZero
โ€”MeasureTheory.integral_zero_measure
sub_zero
cgf_const ๐Ÿ“–mathematicalโ€”cgf
Real
Real.instMul
โ€”mgf_const
Real.log_exp
cgf_const' ๐Ÿ“–mathematicalโ€”cgf
Real
Real.instAdd
Real.log
MeasureTheory.Measure.real
Set.univ
Real.instMul
โ€”mgf_const'
Real.log_mul
MeasureTheory.measureReal_eq_zero_iff
MeasureTheory.measure_ne_top
MeasureTheory.Measure.measure_univ_eq_zero
LT.lt.ne'
Real.exp_pos
Real.log_exp
cgf_neg ๐Ÿ“–mathematicalโ€”cgf
Pi.instNeg
Real
Real.instNeg
โ€”mgf_neg
cgf_undef ๐Ÿ“–mathematicalMeasureTheory.Integrable
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
SeminormedAddGroup.toContinuousENorm
SeminormedAddCommGroup.toSeminormedAddGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
Real.exp
Real.instMul
cgf
Real
Real.instZero
โ€”mgf_undef
Real.log_zero
cgf_zero ๐Ÿ“–mathematicalโ€”cgf
Real
Real.instZero
โ€”MeasureTheory.eq_zero_or_isProbabilityMeasure
cgf_zero'
Real.log_zero
MeasureTheory.probReal_univ
Real.log_one
cgf_zero' ๐Ÿ“–mathematicalโ€”cgf
Real
Real.instZero
Real.log
MeasureTheory.Measure.real
Set.univ
โ€”mgf_zero'
cgf_zero_fun ๐Ÿ“–mathematicalโ€”cgf
Pi.instZero
Real
Real.instZero
Real.log
MeasureTheory.Measure.real
Set.univ
โ€”mgf_zero_fun
cgf_zero_measure ๐Ÿ“–mathematicalโ€”cgf
MeasureTheory.Measure
MeasureTheory.Measure.instZero
Pi.instZero
Real
Real.instZero
โ€”mgf_zero_measure
Real.log_zero
exp_cgf ๐Ÿ“–mathematicalMeasureTheory.Integrable
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
SeminormedAddGroup.toContinuousENorm
SeminormedAddCommGroup.toSeminormedAddGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
Real.exp
Real.instMul
Real.exp
cgf
mgf
โ€”cgf.eq_1
Real.exp_log
mgf_pos'
integrable_exp_mul_of_le ๐Ÿ“–mathematicalReal
Real.instLE
Real.instZero
AEMeasurable
Real.measurableSpace
Filter.Eventually
MeasureTheory.ae
MeasureTheory.Measure
MeasureTheory.Measure.instFunLike
MeasureTheory.Measure.instOuterMeasureClass
MeasureTheory.Integrable
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
SeminormedAddGroup.toContinuousENorm
SeminormedAddCommGroup.toSeminormedAddGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
Real.exp
Real.instMul
โ€”MeasureTheory.Measure.instOuterMeasureClass
MeasureTheory.Integrable.of_mem_Icc
Measurable.comp_aemeasurable
Real.measurable_exp
AEMeasurable.const_mul
ContinuousMul.measurableMul
Real.borelSpace
IsSemitopologicalSemiring.toSeparatelyContinuousMul
IsSemitopologicalRing.toIsSemitopologicalSemiring
IsTopologicalRing.toIsSemitopologicalRing
instIsTopologicalRingReal
Filter.mp_mem
Filter.univ_mem'
le_of_lt
Real.exp_pos
Real.exp_monotone
mul_le_mul_of_nonneg_left
IsOrderedRing.toPosMulMono
Real.instIsOrderedRing
integrable_exp_mul_of_mem_Icc ๐Ÿ“–mathematicalAEMeasurable
Real
Real.measurableSpace
Filter.Eventually
Set
Set.instMembership
Set.Icc
Real.instPreorder
MeasureTheory.ae
MeasureTheory.Measure
MeasureTheory.Measure.instFunLike
MeasureTheory.Measure.instOuterMeasureClass
MeasureTheory.Integrable
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
SeminormedAddGroup.toContinuousENorm
SeminormedAddCommGroup.toSeminormedAddGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
Real.exp
Real.instMul
โ€”MeasureTheory.Measure.instOuterMeasureClass
MeasureTheory.Integrable.of_mem_Icc
Measurable.comp_aemeasurable
Real.measurable_exp
AEMeasurable.const_mul
ContinuousMul.measurableMul
Real.borelSpace
IsSemitopologicalSemiring.toSeparatelyContinuousMul
IsSemitopologicalRing.toIsSemitopologicalSemiring
IsTopologicalRing.toIsSemitopologicalRing
instIsTopologicalRingReal
Filter.mp_mem
Filter.univ_mem'
le_of_not_gt
Mathlib.Tactic.Linarith.lt_irrefl
Mathlib.Tactic.Ring.of_eq
Mathlib.Tactic.Ring.add_congr
Mathlib.Tactic.Ring.sub_congr
Mathlib.Tactic.Ring.mul_congr
Mathlib.Tactic.Ring.atom_pf
Mathlib.Tactic.Ring.add_mul
Mathlib.Tactic.Ring.mul_add
Mathlib.Tactic.Ring.mul_pf_left
Mathlib.Tactic.Ring.mul_pf_right
Mathlib.Tactic.Ring.one_mul
Mathlib.Tactic.Ring.mul_zero
Mathlib.Tactic.Ring.add_pf_add_zero
Mathlib.Tactic.Ring.zero_mul
Mathlib.Tactic.Ring.sub_pf
Mathlib.Tactic.Ring.neg_add
Mathlib.Tactic.Ring.neg_mul
Mathlib.Tactic.Ring.neg_one_mul
Mathlib.Meta.NormNum.IsInt.to_raw_eq
Mathlib.Meta.NormNum.isInt_mul
Mathlib.Meta.NormNum.IsInt.of_raw
Mathlib.Meta.NormNum.IsNat.to_isInt
Mathlib.Meta.NormNum.IsNat.of_raw
Mathlib.Tactic.Ring.neg_zero
Mathlib.Tactic.Ring.add_pf_add_lt
Mathlib.Tactic.Ring.add_pf_zero_add
Mathlib.Tactic.Ring.cast_zero
Mathlib.Meta.NormNum.isNat_ofNat
Nat.cast_zero
Mathlib.Tactic.Ring.add_pf_add_gt
Mathlib.Meta.NormNum.IsNat.to_raw_eq
Mathlib.Meta.NormNum.IsInt.to_isNat
Mathlib.Tactic.Ring.add_pf_add_overlap_zero
Mathlib.Tactic.Ring.add_overlap_pf_zero
Mathlib.Meta.NormNum.isInt_add
Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
Real.instIsStrictOrderedRing
Mathlib.Tactic.Linarith.sub_neg_of_lt
Real.instIsOrderedRing
Mathlib.Tactic.Linarith.sub_nonpos_of_le
mul_nonneg_of_nonpos_of_nonpos
AddGroup.existsAddOfLE
IsOrderedRing.toMulPosMono
covariant_swap_add_of_covariant_add
IsOrderedAddMonoid.toAddLeftMono
Real.instIsOrderedAddMonoid
IsRightCancelAdd.addRightReflectLE_of_addRightReflectLT
instIsRightCancelAddOfAddRightReflectLE
contravariant_swap_add_of_contravariant_add
IsLeftCancelAdd.addLeftReflectLE_of_addLeftReflectLT
instIsLeftCancelAddOfAddLeftReflectLE
IsOrderedCancelAddMonoid.toAddLeftReflectLE
Real.instIsOrderedCancelAddMonoid
contravariant_lt_of_covariant_le
Mathlib.Tactic.Ring.mul_one
le_of_lt
lt_of_not_ge
measure_ge_le_exp_cgf ๐Ÿ“–mathematicalReal
Real.instLE
Real.instZero
MeasureTheory.Integrable
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
SeminormedAddGroup.toContinuousENorm
SeminormedAddCommGroup.toSeminormedAddGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
Real.exp
Real.instMul
Real
Real.instLE
MeasureTheory.Measure.real
setOf
Real.exp
Real.instAdd
Real.instMul
Real.instNeg
cgf
โ€”LE.le.trans
measure_ge_le_exp_mul_mgf
Real.exp_add
mul_le_mul
IsOrderedRing.toPosMulMono
Real.instIsOrderedRing
IsOrderedRing.toMulPosMono
le_rfl
Real.le_exp_log
mgf_nonneg
LT.lt.le
Real.exp_pos
measure_ge_le_exp_mul_mgf ๐Ÿ“–mathematicalReal
Real.instLE
Real.instZero
MeasureTheory.Integrable
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
SeminormedAddGroup.toContinuousENorm
SeminormedAddCommGroup.toSeminormedAddGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
Real.exp
Real.instMul
Real
Real.instLE
MeasureTheory.Measure.real
setOf
Real.instMul
Real.exp
Real.instNeg
mgf
โ€”LE.le.eq_or_lt
neg_zero
MulZeroClass.zero_mul
Real.exp_zero
mgf_zero'
one_mul
MeasureTheory.measureReal_mono
Set.subset_univ
MeasureTheory.measure_ne_top
Set.ext
mul_le_mul_of_nonneg_left
IsOrderedRing.toPosMulMono
Real.instIsOrderedRing
LT.lt.le
le_of_mul_le_mul_left
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
Real.instIsStrictOrderedRing
MeasureTheory.mul_meas_ge_le_integral_of_nonneg
MeasureTheory.ae_of_all
MeasureTheory.Measure.instOuterMeasureClass
Real.exp_pos
mul_comm
div_eq_mul_inv
le_div_iffโ‚€'
PosMulReflectLE.toPosMulReflectLT
neg_mul
Real.exp_neg
measure_le_le_exp_cgf ๐Ÿ“–mathematicalReal
Real.instLE
Real.instZero
MeasureTheory.Integrable
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
SeminormedAddGroup.toContinuousENorm
SeminormedAddCommGroup.toSeminormedAddGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
Real.exp
Real.instMul
Real
Real.instLE
MeasureTheory.Measure.real
setOf
Real.exp
Real.instAdd
Real.instMul
Real.instNeg
cgf
โ€”LE.le.trans
measure_le_le_exp_mul_mgf
Real.exp_add
mul_le_mul
IsOrderedRing.toPosMulMono
Real.instIsOrderedRing
IsOrderedRing.toMulPosMono
le_rfl
Real.le_exp_log
mgf_nonneg
LT.lt.le
Real.exp_pos
measure_le_le_exp_mul_mgf ๐Ÿ“–mathematicalReal
Real.instLE
Real.instZero
MeasureTheory.Integrable
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
SeminormedAddGroup.toContinuousENorm
SeminormedAddCommGroup.toSeminormedAddGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
Real.exp
Real.instMul
Real
Real.instLE
MeasureTheory.Measure.real
setOf
Real.instMul
Real.exp
Real.instNeg
mgf
โ€”neg_neg
mgf_neg
neg_mul_neg
Eq.trans_le
IsOrderedAddMonoid.toAddLeftMono
Real.instIsOrderedAddMonoid
covariant_swap_add_of_covariant_add
measure_ge_le_exp_mul_mgf
neg_nonneg
mgf_add_const ๐Ÿ“–mathematicalโ€”mgf
Real
Real.instAdd
Real.instMul
Real.exp
โ€”add_comm
mgf_const_add
mul_comm
mgf_add_measure ๐Ÿ“–mathematicalMeasureTheory.Integrable
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
SeminormedAddGroup.toContinuousENorm
SeminormedAddCommGroup.toSeminormedAddGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
Real.exp
Real.instMul
mgf
MeasureTheory.Measure
MeasureTheory.Measure.instAdd
Real
Real.instAdd
โ€”mgf.eq_1
MeasureTheory.integral_add_measure
mgf_anti_of_nonpos ๐Ÿ“–mathematicalFilter.EventuallyLE
Real
Real.instLE
MeasureTheory.ae
MeasureTheory.Measure
MeasureTheory.Measure.instFunLike
MeasureTheory.Measure.instOuterMeasureClass
Real.instZero
MeasureTheory.Integrable
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
SeminormedAddGroup.toContinuousENorm
SeminormedAddCommGroup.toSeminormedAddGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
Real.exp
Real.instMul
Real
Real.instLE
mgf
โ€”MeasureTheory.Measure.instOuterMeasureClass
MeasureTheory.integral_mono_ae
Real.instIsOrderedAddMonoid
IsStrictOrderedModule.toIsOrderedModule
IsStrictOrderedRing.toIsStrictOrderedModule
Real.instIsStrictOrderedRing
instClosedIciTopology
HasSolidNorm.orderClosedTopology
instHasSolidNormReal
Filter.mp_mem
Filter.univ_mem'
Real.exp_monotone
mul_le_mul_of_nonpos_left
AddGroup.existsAddOfLE
IsOrderedRing.toPosMulMono
Real.instIsOrderedRing
covariant_swap_add_of_covariant_add
IsOrderedAddMonoid.toAddLeftMono
IsRightCancelAdd.addRightReflectLE_of_addRightReflectLT
instIsRightCancelAddOfAddRightReflectLE
contravariant_swap_add_of_contravariant_add
IsLeftCancelAdd.addLeftReflectLE_of_addLeftReflectLT
instIsLeftCancelAddOfAddLeftReflectLE
IsOrderedCancelAddMonoid.toAddLeftReflectLE
Real.instIsOrderedCancelAddMonoid
contravariant_lt_of_covariant_le
mgf_undef
mgf_nonneg
mgf_congr ๐Ÿ“–mathematicalFilter.EventuallyEq
Real
MeasureTheory.ae
MeasureTheory.Measure
MeasureTheory.Measure.instFunLike
MeasureTheory.Measure.instOuterMeasureClass
mgfโ€”MeasureTheory.Measure.instOuterMeasureClass
MeasureTheory.integral_congr_ae
Filter.mp_mem
Filter.univ_mem'
mgf_congr_identDistrib ๐Ÿ“–mathematicalIdentDistrib
Real
Real.measurableSpace
mgfโ€”mgf_id_map
IdentDistrib.aemeasurable_fst
IdentDistrib.aemeasurable_snd
IdentDistrib.map_eq
mgf_congr_of_identDistrib ๐Ÿ“–mathematicalIdentDistrib
Real
Real.measurableSpace
mgfโ€”IdentDistrib.integral_eq
Real.borelSpace
IdentDistrib.comp
Measurable.exp
MeasurableMul.measurable_const_mul
ContinuousMul.measurableMul
IsSemitopologicalSemiring.toSeparatelyContinuousMul
IsSemitopologicalRing.toIsSemitopologicalSemiring
IsTopologicalRing.toIsSemitopologicalRing
instIsTopologicalRingReal
mgf_const ๐Ÿ“–mathematicalโ€”mgf
Real.exp
Real
Real.instMul
โ€”mgf_const'
MeasureTheory.probReal_univ
one_mul
mgf_const' ๐Ÿ“–mathematicalโ€”mgf
Real
Real.instMul
MeasureTheory.Measure.real
Set.univ
Real.exp
โ€”MeasureTheory.integral_const
Real.instCompleteSpace
mgf_const_add ๐Ÿ“–mathematicalโ€”mgf
Real
Real.instAdd
Real.instMul
Real.exp
โ€”mgf.eq_1
MeasureTheory.integral_const_mul
mul_add
Distrib.leftDistribClass
Real.exp_add
mgf_const_mul ๐Ÿ“–mathematicalโ€”mgf
Real
Real.instMul
โ€”mgf_smul_left
mgf_dirac ๐Ÿ“–mathematicalMeasureTheory.Measure.map
Real
Real.measurableSpace
MeasureTheory.Measure.dirac
mgf
Real.exp
Real
Real.instMul
โ€”MeasureTheory.Measure.dirac.isProbabilityMeasure
mgf_id_map
AEMeasurable.of_map_ne_zero
MeasureTheory.IsProbabilityMeasure.ne_zero
mgf.eq_1
MeasureTheory.integral_dirac
Real.instCompleteSpace
instMeasurableSingletonClassOfMeasurableEq
instMeasurableEqOfSecondCountableTopologyOfT2Space
BorelSpace.opensMeasurable
Real.borelSpace
instSecondCountableTopologyReal
TopologicalSpace.t2Space_of_metrizableSpace
EMetricSpace.metrizableSpace
mul_comm
mgf_dirac' ๐Ÿ“–mathematicalโ€”mgf
MeasureTheory.Measure.dirac
Real.exp
Real
Real.instMul
โ€”mgf.eq_1
MeasureTheory.integral_dirac
Real.instCompleteSpace
mgf_id_map ๐Ÿ“–mathematicalAEMeasurable
Real
Real.measurableSpace
mgf
Real
Real.measurableSpace
MeasureTheory.Measure.map
โ€”mgf_map
Measurable.aestronglyMeasurable
PseudoEMetricSpace.pseudoMetrizableSpace
instSecondCountableTopologyReal
BorelSpace.opensMeasurable
Real.borelSpace
Measurable.exp
MeasurableMul.measurable_const_mul
ContinuousMul.measurableMul
IsSemitopologicalSemiring.toSeparatelyContinuousMul
IsSemitopologicalRing.toIsSemitopologicalSemiring
IsTopologicalRing.toIsSemitopologicalRing
instIsTopologicalRingReal
mgf_map ๐Ÿ“–mathematicalAEMeasurable
MeasureTheory.AEStronglyMeasurable
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.exp
Real.instMul
MeasureTheory.Measure.map
mgf
MeasureTheory.Measure.map
Real
โ€”MeasureTheory.integral_map
mgf_mono_of_nonneg ๐Ÿ“–mathematicalFilter.EventuallyLE
Real
Real.instLE
MeasureTheory.ae
MeasureTheory.Measure
MeasureTheory.Measure.instFunLike
MeasureTheory.Measure.instOuterMeasureClass
Real.instZero
MeasureTheory.Integrable
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
SeminormedAddGroup.toContinuousENorm
SeminormedAddCommGroup.toSeminormedAddGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
Real.exp
Real.instMul
Real
Real.instLE
mgf
โ€”MeasureTheory.Measure.instOuterMeasureClass
MeasureTheory.integral_mono_ae
Real.instIsOrderedAddMonoid
IsStrictOrderedModule.toIsOrderedModule
IsStrictOrderedRing.toIsStrictOrderedModule
Real.instIsStrictOrderedRing
instClosedIciTopology
HasSolidNorm.orderClosedTopology
instHasSolidNormReal
Filter.mp_mem
Filter.univ_mem'
Real.exp_monotone
mul_le_mul_of_nonneg_left
IsOrderedRing.toPosMulMono
Real.instIsOrderedRing
mgf_undef
mgf_nonneg
mgf_neg ๐Ÿ“–mathematicalโ€”mgf
Pi.instNeg
Real
Real.instNeg
โ€”mul_neg
neg_mul
mgf_nonneg ๐Ÿ“–mathematicalโ€”Real
Real.instLE
Real.instZero
mgf
โ€”MeasureTheory.integral_nonneg
Real.instIsOrderedAddMonoid
IsStrictOrderedModule.toIsOrderedModule
IsStrictOrderedRing.toIsStrictOrderedModule
Real.instIsStrictOrderedRing
instClosedIciTopology
HasSolidNorm.orderClosedTopology
instHasSolidNormReal
le_of_lt
Real.exp_pos
mgf_pos ๐Ÿ“–mathematicalMeasureTheory.Integrable
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
SeminormedAddGroup.toContinuousENorm
SeminormedAddCommGroup.toSeminormedAddGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
Real.exp
Real.instMul
Real
Real.instLT
Real.instZero
mgf
โ€”mgf_pos'
MeasureTheory.IsProbabilityMeasure.ne_zero
mgf_pos' ๐Ÿ“–mathematicalMeasureTheory.Integrable
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
SeminormedAddGroup.toContinuousENorm
SeminormedAddCommGroup.toSeminormedAddGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
Real.exp
Real.instMul
Real
Real.instLT
Real.instZero
mgf
โ€”MeasureTheory.Measure.restrict_univ
MeasureTheory.setIntegral_pos_iff_support_of_nonneg_ae
Filter.univ_mem'
MeasureTheory.Measure.instOuterMeasureClass
Pi.zero_apply
LT.lt.le
Real.exp_pos
MeasureTheory.integrableOn_univ
Set.ext
LT.lt.ne'
Set.inter_univ
Ne.bot_lt
mgf_pos_iff ๐Ÿ“–mathematicalโ€”Real
Real.instLT
Real.instZero
mgf
MeasureTheory.Integrable
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
SeminormedAddGroup.toContinuousENorm
SeminormedAddCommGroup.toSeminormedAddGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
Real.exp
Real.instMul
โ€”Mathlib.Tactic.Contrapose.contraposeโ‚
mgf_undef
instReflLe
mgf_pos'
mgf_smul_left ๐Ÿ“–mathematicalโ€”mgf
Real
Function.hasSMul
Algebra.toSMul
Real.instCommSemiring
CommSemiring.toSemiring
Algebra.id
Real.instMul
โ€”mul_comm
mul_assoc
mgf_smul_measure ๐Ÿ“–mathematicalโ€”mgf
ENNReal
MeasureTheory.Measure
MeasureTheory.Measure.instSMul
Algebra.toSMul
ENNReal.instCommSemiring
CommSemiring.toSemiring
Algebra.id
IsScalarTower.right
Real
Real.instMul
ENNReal.toReal
โ€”IsScalarTower.right
mgf.eq_1
MeasureTheory.integral_smul_measure
smul_eq_mul
mgf_sum_measure ๐Ÿ“–mathematicalMeasureTheory.Integrable
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
SeminormedAddGroup.toContinuousENorm
SeminormedAddCommGroup.toSeminormedAddGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
Real.exp
Real.instMul
MeasureTheory.Measure.sum
mgf
MeasureTheory.Measure.sum
tsum
Real
Real.instAddCommMonoid
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
SummationFilter.unconditional
โ€”MeasureTheory.integral_sum_measure
mgf_sum_of_identDistrib ๐Ÿ“–mathematicalMeasurable
Real
Real.measurableSpace
iIndepFun
IdentDistrib
Finset
SetLike.instMembership
Finset.instSetLike
mgf
Finset.sum
Pi.addCommMonoid
Real
Real.instAddCommMonoid
Monoid.toPow
Real.instMonoid
Finset.card
โ€”mgf_sum_of_identDistribโ‚€
Measurable.aemeasurable
mgf_sum_of_identDistribโ‚€ ๐Ÿ“–mathematicalAEMeasurable
Real
Real.measurableSpace
iIndepFun
IdentDistrib
Finset
SetLike.instMembership
Finset.instSetLike
mgf
Finset.sum
Pi.addCommMonoid
Real
Real.instAddCommMonoid
Monoid.toPow
Real.instMonoid
Finset.card
โ€”iIndepFun.mgf_sumโ‚€
Finset.prod_eq_pow_card
mgf_congr_of_identDistrib
mgf_undef ๐Ÿ“–mathematicalMeasureTheory.Integrable
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
SeminormedAddGroup.toContinuousENorm
SeminormedAddCommGroup.toSeminormedAddGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
Real.exp
Real.instMul
mgf
Real
Real.instZero
โ€”MeasureTheory.integral_undef
mgf_zero ๐Ÿ“–mathematicalโ€”mgf
Real
Real.instZero
Real.instOne
โ€”mgf_zero'
MeasureTheory.probReal_univ
mgf_zero' ๐Ÿ“–mathematicalโ€”mgf
Real
Real.instZero
MeasureTheory.Measure.real
Set.univ
โ€”MulZeroClass.zero_mul
Real.exp_zero
MeasureTheory.integral_const
Real.instCompleteSpace
mul_one
mgf_zero_fun ๐Ÿ“–mathematicalโ€”mgf
Pi.instZero
Real
Real.instZero
MeasureTheory.Measure.real
Set.univ
โ€”MulZeroClass.mul_zero
Real.exp_zero
MeasureTheory.integral_const
Real.instCompleteSpace
mul_one
mgf_zero_measure ๐Ÿ“–mathematicalโ€”mgf
MeasureTheory.Measure
MeasureTheory.Measure.instZero
Pi.instZero
Real
Real.instZero
โ€”MeasureTheory.integral_zero_measure
moment_def ๐Ÿ“–mathematicalโ€”moment
MeasureTheory.integral
Real
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
Pi.instPow
Monoid.toPow
Real.instMonoid
โ€”โ€”
moment_one ๐Ÿ“–mathematicalโ€”moment
MeasureTheory.integral
Real
Real.normedAddCommGroup
InnerProductSpace.toNormedSpace
Real.instRCLike
NormedAddCommGroup.toSeminormedAddCommGroup
RCLike.toInnerProductSpaceReal
โ€”pow_one
moment_zero ๐Ÿ“–mathematicalโ€”moment
Pi.instZero
Real
Real.instZero
โ€”zero_pow
MeasureTheory.integral_const
Real.instCompleteSpace
MulZeroClass.mul_zero
moment_zero_measure ๐Ÿ“–mathematicalโ€”moment
MeasureTheory.Measure
MeasureTheory.Measure.instZero
Real
Real.instZero
โ€”MeasureTheory.integral_zero_measure

ProbabilityTheory.IndepFun

Theorems

NameKindAssumesProvesValidatesDepends On
cgf_add ๐Ÿ“–mathematicalProbabilityTheory.IndepFun
Real
Real.measurableSpace
MeasureTheory.Integrable
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
SeminormedAddGroup.toContinuousENorm
SeminormedAddCommGroup.toSeminormedAddGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
Real.exp
Real.instMul
ProbabilityTheory.cgf
Pi.instAdd
Real
Real.instAdd
โ€”ProbabilityTheory.cgf_zero_measure
add_zero
mgf_add
MeasureTheory.Integrable.aestronglyMeasurable
Real.log_mul
LT.lt.ne'
ProbabilityTheory.mgf_pos'
exp_mul ๐Ÿ“–mathematicalProbabilityTheory.IndepFun
Real
Real.measurableSpace
ProbabilityTheory.IndepFun
Real
Real.measurableSpace
Real.exp
Real.instMul
โ€”Measurable.exp
Measurable.const_mul
ContinuousMul.measurableMul
Real.borelSpace
IsSemitopologicalSemiring.toSeparatelyContinuousMul
IsSemitopologicalRing.toIsSemitopologicalSemiring
IsTopologicalRing.toIsSemitopologicalRing
instIsTopologicalRingReal
measurable_id'
comp
integrable_exp_mul_add ๐Ÿ“–mathematicalProbabilityTheory.IndepFun
Real
Real.measurableSpace
MeasureTheory.Integrable
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
SeminormedAddGroup.toContinuousENorm
SeminormedAddCommGroup.toSeminormedAddGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
Real.exp
Real.instMul
MeasureTheory.Integrable
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
SeminormedAddGroup.toContinuousENorm
SeminormedAddCommGroup.toSeminormedAddGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
Real.exp
Real.instMul
Pi.instAdd
Real.instAdd
โ€”mul_add
Distrib.leftDistribClass
Real.exp_add
integrable_mul
Real.borelSpace
exp_mul
mgf_add ๐Ÿ“–mathematicalProbabilityTheory.IndepFun
Real
Real.measurableSpace
MeasureTheory.AEStronglyMeasurable
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
Real.exp
Real.instMul
ProbabilityTheory.mgf
Pi.instAdd
Real
Real.instAdd
Real.instMul
โ€”mul_add
Distrib.leftDistribClass
Real.exp_add
integral_mul_eq_mul_integral
exp_mul
mgf_add' ๐Ÿ“–mathematicalProbabilityTheory.IndepFun
Real
Real.measurableSpace
MeasureTheory.AEStronglyMeasurable
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
ProbabilityTheory.mgf
Pi.instAdd
Real
Real.instAdd
Real.instMul
โ€”Continuous.rexp
continuous_const_mul
IsSemitopologicalSemiring.toSeparatelyContinuousMul
IsSemitopologicalRing.toIsSemitopologicalSemiring
IsTopologicalRing.toIsSemitopologicalRing
instIsTopologicalRingReal
MeasureTheory.AEStronglyMeasurable.comp_aemeasurable
Continuous.aestronglyMeasurable
BorelSpace.opensMeasurable
Real.borelSpace
PseudoEMetricSpace.pseudoMetrizableSpace
secondCountableTopologyEither_of_right
instSecondCountableTopologyReal
MeasureTheory.AEStronglyMeasurable.aemeasurable
mgf_add

ProbabilityTheory.iIndepFun

Theorems

NameKindAssumesProvesValidatesDepends On
cgf_sum ๐Ÿ“–mathematicalProbabilityTheory.iIndepFun
Real
Real.measurableSpace
Measurable
MeasureTheory.Integrable
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
SeminormedAddGroup.toContinuousENorm
SeminormedAddCommGroup.toSeminormedAddGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
Real.exp
Real.instMul
ProbabilityTheory.cgf
Finset.sum
Pi.addCommMonoid
Real
Real.instAddCommMonoid
โ€”cgf_sumโ‚€
Measurable.aemeasurable
cgf_sumโ‚€ ๐Ÿ“–mathematicalProbabilityTheory.iIndepFun
Real
Real.measurableSpace
AEMeasurable
MeasureTheory.Integrable
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
SeminormedAddGroup.toContinuousENorm
SeminormedAddCommGroup.toSeminormedAddGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
Real.exp
Real.instMul
ProbabilityTheory.cgf
Finset.sum
Pi.addCommMonoid
Real
Real.instAddCommMonoid
โ€”isProbabilityMeasure
Finset.sum_congr
Real.log_prod
LT.lt.ne'
ProbabilityTheory.mgf_pos
mgf_sumโ‚€
integrable_exp_mul_sum ๐Ÿ“–mathematicalProbabilityTheory.iIndepFun
Real
Real.measurableSpace
Measurable
MeasureTheory.Integrable
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
SeminormedAddGroup.toContinuousENorm
SeminormedAddCommGroup.toSeminormedAddGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
Real.exp
Real.instMul
MeasureTheory.Integrable
Real
UniformSpace.toTopologicalSpace
PseudoMetricSpace.toUniformSpace
Real.pseudoMetricSpace
SeminormedAddGroup.toContinuousENorm
SeminormedAddCommGroup.toSeminormedAddGroup
NonUnitalSeminormedRing.toSeminormedAddCommGroup
NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing
SeminormedCommRing.toNonUnitalSeminormedCommRing
NormedCommRing.toSeminormedCommRing
Real.normedCommRing
Real.exp
Real.instMul
Finset.sum
Pi.addCommMonoid
Real.instAddCommMonoid
โ€”Finset.induction_on
Finset.sum_apply
MulZeroClass.mul_zero
Real.exp_zero
MeasureTheory.integrable_const
Finset.mem_insert_of_mem
Finset.sum_insert
ProbabilityTheory.IndepFun.integrable_exp_mul_add
ProbabilityTheory.IndepFun.symm
indepFun_finset_sum_of_notMem
ContinuousAdd.measurableMulโ‚‚
Real.borelSpace
instSecondCountableTopologyReal
IsSemitopologicalSemiring.toContinuousAdd
IsSemitopologicalRing.toIsSemitopologicalSemiring
IsTopologicalRing.toIsSemitopologicalRing
instIsTopologicalRingReal
Finset.mem_insert_self
mgf_sum ๐Ÿ“–mathematicalProbabilityTheory.iIndepFun
Real
Real.measurableSpace
Measurable
ProbabilityTheory.mgf
Finset.sum
Pi.addCommMonoid
Real
Real.instAddCommMonoid
Finset.prod
Real.instCommMonoid
โ€”mgf_sumโ‚€
Measurable.aemeasurable
mgf_sumโ‚€ ๐Ÿ“–mathematicalProbabilityTheory.iIndepFun
Real
Real.measurableSpace
AEMeasurable
ProbabilityTheory.mgf
Finset.sum
Pi.addCommMonoid
Real
Real.instAddCommMonoid
Finset.prod
Real.instCommMonoid
โ€”isProbabilityMeasure
Finset.induction_on
ProbabilityTheory.mgf_zero_fun
MeasureTheory.probReal_univ
AEMeasurable.aestronglyMeasurable
PseudoEMetricSpace.pseudoMetrizableSpace
BorelSpace.opensMeasurable
Real.borelSpace
instSecondCountableTopologyReal
AEMeasurable.exp
AEMeasurable.const_mul
ContinuousMul.measurableMul
IsSemitopologicalSemiring.toSeparatelyContinuousMul
IsSemitopologicalRing.toIsSemitopologicalSemiring
IsTopologicalRing.toIsSemitopologicalRing
instIsTopologicalRingReal
Finset.sum_insert
ProbabilityTheory.IndepFun.mgf_add
ProbabilityTheory.IndepFun.symm
indepFun_finset_sum_of_notMemโ‚€
ContinuousAdd.measurableMulโ‚‚
IsSemitopologicalSemiring.toContinuousAdd
ProbabilityTheory.aestronglyMeasurable_exp_mul_sum
Finset.prod_insert

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