IntegralCompProd

📁 Source: Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean

Statistics

Metric	Count
Definitions	0
Theorems ae_of_compProd, comp, compProd_mk_left, integral_kernel_comp, integral_kernel_compProd, ae_of_comp, ae_of_compProd, integral_comp, integral_compProd, integral_norm_comp, integral_norm_compProd, ae_integrable_of_integrable_comp, integrable_compProd_iff, integrable_compProd_snd_iff, integrable_comp_iff, integrable_integral_norm_of_integrable_comp, integral_compProd, setIntegral_compProd, continuous_integral_integral, continuous_integral_integral_comp, integral_comp, integral_fn_integral_add, integral_fn_integral_add_comp, integral_fn_integral_sub, integral_fn_integral_sub_comp, integral_integral_add, integral_integral_add', integral_integral_add'_comp, integral_integral_add_comp, integral_integral_sub, integral_integral_sub', integral_integral_sub'_comp, integral_integral_sub_comp, lintegral_fn_integral_sub, lintegral_fn_integral_sub_comp, setIntegral_comp, hasFiniteIntegral_compProd_iff, hasFiniteIntegral_compProd_iff', hasFiniteIntegral_comp_iff, hasFiniteIntegral_comp_iff', hasFiniteIntegral_prodMk_left, integrable_compProd_iff, integrable_comp_iff, integrable_kernel_prodMk_left, integral_compProd, setIntegral_compProd, setIntegral_compProd_univ_left, setIntegral_compProd_univ_right	48
Total	48

MeasureTheory.AEStronglyMeasurable

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
ae_of_compProd 📖	mathematical	MeasureTheory.AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup Prod.instMeasurableSpace MeasureTheory.Measure.compProd	Filter.Eventually DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike MeasureTheory.ae MeasureTheory.Measure.instFunLike MeasureTheory.Measure.instOuterMeasureClass	—	compProd_mk_left ProbabilityTheory.Kernel.const.instIsSFiniteKernel ProbabilityTheory.Kernel.IsSFiniteKernel.prodMkLeft
comp 📖	mathematical	MeasureTheory.AEStronglyMeasurable DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike ProbabilityTheory.Kernel.comp	Filter.Eventually MeasureTheory.ae MeasureTheory.Measure.instFunLike MeasureTheory.Measure.instOuterMeasureClass	—	Filter.mp_mem MeasureTheory.Measure.instOuterMeasureClass ProbabilityTheory.Kernel.ae_ae_of_ae_comp Filter.univ_mem'
compProd_mk_left 📖	mathematical	MeasureTheory.AEStronglyMeasurable Prod.instMeasurableSpace DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike ProbabilityTheory.Kernel.compProd	Filter.Eventually MeasureTheory.ae MeasureTheory.Measure.instFunLike MeasureTheory.Measure.instOuterMeasureClass	—	Filter.mp_mem MeasureTheory.Measure.instOuterMeasureClass ProbabilityTheory.Kernel.ae_ae_of_ae_compProd Filter.univ_mem' MeasureTheory.StronglyMeasurable.comp_measurable measurable_prodMk_left
integral_kernel_comp 📖	mathematical	MeasureTheory.AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike ProbabilityTheory.Kernel.comp	MeasureTheory.integral	—	MeasureTheory.Measure.instOuterMeasureClass MeasureTheory.StronglyMeasurable.integral_kernel Filter.mp_mem ProbabilityTheory.Kernel.ae_ae_of_ae_comp Filter.univ_mem' MeasureTheory.integral_congr_ae
integral_kernel_compProd 📖	mathematical	MeasureTheory.AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup Prod.instMeasurableSpace DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike ProbabilityTheory.Kernel.compProd	MeasureTheory.integral	—	MeasureTheory.Measure.instOuterMeasureClass MeasureTheory.StronglyMeasurable.integral_kernel_prod_right'' Filter.mp_mem ProbabilityTheory.Kernel.ae_ae_of_ae_compProd Filter.univ_mem' MeasureTheory.integral_congr_ae

MeasureTheory.Integrable

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
ae_of_comp 📖	mathematical	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike ProbabilityTheory.Kernel.comp	Filter.Eventually MeasureTheory.ae MeasureTheory.Measure.instFunLike MeasureTheory.Measure.instOuterMeasureClass	—	MeasureTheory.Measure.instOuterMeasureClass ProbabilityTheory.integrable_comp_iff
ae_of_compProd 📖	mathematical	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Prod.instMeasurableSpace DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike ProbabilityTheory.Kernel.compProd	Filter.Eventually MeasureTheory.ae MeasureTheory.Measure.instFunLike MeasureTheory.Measure.instOuterMeasureClass	—	MeasureTheory.Measure.instOuterMeasureClass ProbabilityTheory.integrable_compProd_iff aestronglyMeasurable
integral_comp 📖	mathematical	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike ProbabilityTheory.Kernel.comp	MeasureTheory.integral	—	mono integral_norm_comp MeasureTheory.AEStronglyMeasurable.integral_kernel_comp MeasureTheory.ae_of_all MeasureTheory.Measure.instOuterMeasureClass LE.le.trans_eq MeasureTheory.norm_integral_le_integral_norm Real.norm_of_nonneg MeasureTheory.integral_nonneg_of_ae Real.instIsOrderedAddMonoid IsStrictOrderedModule.toIsOrderedModule IsStrictOrderedRing.toIsStrictOrderedModule Real.instIsStrictOrderedRing instClosedIciTopology HasSolidNorm.orderClosedTopology instHasSolidNormReal norm_nonneg
integral_compProd 📖	mathematical	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Prod.instMeasurableSpace DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike ProbabilityTheory.Kernel.compProd	MeasureTheory.integral	—	mono integral_norm_compProd MeasureTheory.AEStronglyMeasurable.integral_kernel_compProd aestronglyMeasurable Filter.Eventually.of_forall MeasureTheory.Measure.instOuterMeasureClass LE.le.trans_eq MeasureTheory.norm_integral_le_integral_norm Real.norm_of_nonneg MeasureTheory.integral_nonneg_of_ae Real.instIsOrderedAddMonoid IsStrictOrderedModule.toIsOrderedModule IsStrictOrderedRing.toIsStrictOrderedModule Real.instIsStrictOrderedRing instClosedIciTopology HasSolidNorm.orderClosedTopology instHasSolidNormReal norm_nonneg
integral_norm_comp 📖	mathematical	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike ProbabilityTheory.Kernel.comp	Real Real.pseudoMetricSpace NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing MeasureTheory.integral Real.normedAddCommGroup InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal Norm.norm NormedAddCommGroup.toNorm	—	MeasureTheory.Measure.instOuterMeasureClass ProbabilityTheory.integrable_comp_iff
integral_norm_compProd 📖	mathematical	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Prod.instMeasurableSpace DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike ProbabilityTheory.Kernel.compProd	Real Real.pseudoMetricSpace NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing MeasureTheory.integral Real.normedAddCommGroup InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal Norm.norm NormedAddCommGroup.toNorm	—	MeasureTheory.Measure.instOuterMeasureClass ProbabilityTheory.integrable_compProd_iff aestronglyMeasurable

MeasureTheory.Measure

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
ae_integrable_of_integrable_comp 📖	mathematical	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup bind DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike	Filter.Eventually MeasureTheory.ae instFunLike instOuterMeasureClass	—	instOuterMeasureClass ProbabilityTheory.integrable_comp_iff comp_eq_comp_const_apply
integrable_compProd_iff 📖	mathematical	MeasureTheory.AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup Prod.instMeasurableSpace compProd	MeasureTheory.Integrable SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Filter.Eventually DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike MeasureTheory.ae instFunLike instOuterMeasureClass Real Real.pseudoMetricSpace NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing MeasureTheory.integral Real.normedAddCommGroup InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal Norm.norm NormedAddCommGroup.toNorm	—	instOuterMeasureClass ProbabilityTheory.integrable_compProd_iff ProbabilityTheory.Kernel.const.instIsSFiniteKernel ProbabilityTheory.Kernel.IsSFiniteKernel.prodMkLeft
integrable_compProd_snd_iff 📖	mathematical	MeasureTheory.AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup bind DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike	MeasureTheory.Integrable SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Prod.instMeasurableSpace compProd	—	snd_compProd snd.eq_1 MeasureTheory.integrable_map_measure Measurable.aemeasurable measurable_snd
integrable_comp_iff 📖	mathematical	MeasureTheory.AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup bind DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike	MeasureTheory.Integrable SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Filter.Eventually MeasureTheory.ae instFunLike instOuterMeasureClass Real Real.pseudoMetricSpace NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing MeasureTheory.integral Real.normedAddCommGroup InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal Norm.norm NormedAddCommGroup.toNorm	—	instOuterMeasureClass comp_eq_comp_const_apply ProbabilityTheory.integrable_comp_iff
integrable_integral_norm_of_integrable_comp 📖	mathematical	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup bind DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike	Real Real.pseudoMetricSpace NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing MeasureTheory.integral Real.normedAddCommGroup InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal Norm.norm NormedAddCommGroup.toNorm	—	instOuterMeasureClass ProbabilityTheory.integrable_comp_iff comp_eq_comp_const_apply
integral_compProd 📖	mathematical	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Prod.instMeasurableSpace compProd	MeasureTheory.integral DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike	—	compProd.eq_1 ProbabilityTheory.integral_compProd ProbabilityTheory.Kernel.const.instIsSFiniteKernel ProbabilityTheory.Kernel.IsSFiniteKernel.prodMkLeft
setIntegral_compProd 📖	mathematical	MeasurableSet MeasureTheory.IntegrableOn Prod.instMeasurableSpace UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup SProd.sprod Set Set.instSProd compProd	MeasureTheory.integral restrict DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike	—	compProd.eq_1 ProbabilityTheory.setIntegral_compProd ProbabilityTheory.Kernel.const.instIsSFiniteKernel ProbabilityTheory.Kernel.IsSFiniteKernel.prodMkLeft

ProbabilityTheory

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
hasFiniteIntegral_compProd_iff 📖	mathematical	MeasureTheory.StronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup Prod.instMeasurableSpace	MeasureTheory.HasFiniteIntegral ContinuousENorm.toENorm SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup SeminormedAddGroup.toContinuousENorm DFunLike.coe Kernel MeasureTheory.Measure Kernel.instFunLike Kernel.compProd Filter.Eventually MeasureTheory.ae MeasureTheory.Measure.instFunLike MeasureTheory.Measure.instOuterMeasureClass Real NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing MeasureTheory.integral Real.normedAddCommGroup InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal Norm.norm NormedAddCommGroup.toNorm	—	MeasureTheory.Measure.instOuterMeasureClass Kernel.lintegral_compProd MeasureTheory.StronglyMeasurable.enorm Filter.Eventually.of_forall norm_nonneg MeasureTheory.integral_eq_lintegral_of_nonneg_ae MeasureTheory.StronglyMeasurable.aestronglyMeasurable MeasureTheory.StronglyMeasurable.comp_measurable MeasureTheory.StronglyMeasurable.norm measurable_prodMk_left Real.enorm_eq_ofReal ENNReal.toReal_nonneg ofReal_norm_eq_enorm MeasureTheory.ae_lt_top Measurable.lintegral_kernel_prod_right'' LT.lt.ne MeasureTheory.lintegral_congr_ae Filter.mp_mem Filter.univ_mem' ENNReal.ofReal_toReal ne_top_of_lt
hasFiniteIntegral_compProd_iff' 📖	mathematical	MeasureTheory.AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup Prod.instMeasurableSpace DFunLike.coe Kernel MeasureTheory.Measure Kernel.instFunLike Kernel.compProd	MeasureTheory.HasFiniteIntegral ContinuousENorm.toENorm SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup SeminormedAddGroup.toContinuousENorm Filter.Eventually MeasureTheory.ae MeasureTheory.Measure.instFunLike MeasureTheory.Measure.instOuterMeasureClass Real NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing MeasureTheory.integral Real.normedAddCommGroup InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal Norm.norm NormedAddCommGroup.toNorm	—	MeasureTheory.Measure.instOuterMeasureClass MeasureTheory.hasFiniteIntegral_congr hasFiniteIntegral_compProd_iff Filter.eventually_congr Filter.mp_mem Kernel.ae_ae_of_ae_compProd Filter.EventuallyEq.symm Filter.univ_mem' MeasureTheory.integral_congr_ae Filter.EventuallyEq.fun_comp
hasFiniteIntegral_comp_iff 📖	mathematical	MeasureTheory.StronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup	MeasureTheory.HasFiniteIntegral ContinuousENorm.toENorm SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup SeminormedAddGroup.toContinuousENorm DFunLike.coe Kernel MeasureTheory.Measure Kernel.instFunLike Kernel.comp Filter.Eventually MeasureTheory.ae MeasureTheory.Measure.instFunLike MeasureTheory.Measure.instOuterMeasureClass Real NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing MeasureTheory.integral Real.normedAddCommGroup InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal Norm.norm NormedAddCommGroup.toNorm	—	MeasureTheory.Measure.instOuterMeasureClass Kernel.lintegral_comp MeasureTheory.StronglyMeasurable.enorm MeasureTheory.integral_eq_lintegral_of_nonneg_ae MeasureTheory.ae_of_all norm_nonneg MeasureTheory.StronglyMeasurable.aestronglyMeasurable MeasureTheory.StronglyMeasurable.norm Real.enorm_eq_ofReal ENNReal.toReal_nonneg ofReal_norm_eq_enorm MeasureTheory.ae_lt_top Measurable.lintegral_kernel LT.lt.ne MeasureTheory.lintegral_congr_ae Filter.mp_mem Filter.univ_mem' ENNReal.ofReal_toReal ne_top_of_lt
hasFiniteIntegral_comp_iff' 📖	mathematical	MeasureTheory.AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup DFunLike.coe Kernel MeasureTheory.Measure Kernel.instFunLike Kernel.comp	MeasureTheory.HasFiniteIntegral ContinuousENorm.toENorm SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup SeminormedAddGroup.toContinuousENorm Filter.Eventually MeasureTheory.ae MeasureTheory.Measure.instFunLike MeasureTheory.Measure.instOuterMeasureClass Real NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing MeasureTheory.integral Real.normedAddCommGroup InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal Norm.norm NormedAddCommGroup.toNorm	—	MeasureTheory.Measure.instOuterMeasureClass MeasureTheory.hasFiniteIntegral_congr hasFiniteIntegral_comp_iff Filter.eventually_congr Filter.mp_mem Kernel.ae_ae_of_ae_comp Filter.EventuallyEq.symm Filter.univ_mem' MeasureTheory.integral_congr_ae Filter.EventuallyEq.fun_comp
hasFiniteIntegral_prodMk_left 📖	mathematical	—	MeasureTheory.HasFiniteIntegral Real ContinuousENorm.toENorm UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing SeminormedAddGroup.toContinuousENorm MeasureTheory.Measure.real DFunLike.coe Kernel Prod.instMeasurableSpace MeasureTheory.Measure Kernel.instFunLike Set.preimage	—	Real.enorm_eq_ofReal ENNReal.toReal_nonneg MeasureTheory.lintegral_mono_ae Filter.mp_mem MeasureTheory.Measure.instOuterMeasureClass Kernel.ae_kernel_lt_top Filter.univ_mem' ENNReal.ofReal_toReal LT.lt.ne MeasureTheory.measure_mono Set.preimage_mono MeasureTheory.subset_toMeasurable Kernel.le_compProd_apply MeasureTheory.measure_toMeasurable Ne.lt_top
integrable_compProd_iff 📖	mathematical	MeasureTheory.AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup Prod.instMeasurableSpace DFunLike.coe Kernel MeasureTheory.Measure Kernel.instFunLike Kernel.compProd	MeasureTheory.Integrable SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Filter.Eventually MeasureTheory.ae MeasureTheory.Measure.instFunLike MeasureTheory.Measure.instOuterMeasureClass Real Real.pseudoMetricSpace NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing MeasureTheory.integral Real.normedAddCommGroup InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal Norm.norm NormedAddCommGroup.toNorm	—	MeasureTheory.Measure.instOuterMeasureClass hasFiniteIntegral_compProd_iff' MeasureTheory.AEStronglyMeasurable.compProd_mk_left MeasureTheory.AEStronglyMeasurable.integral_kernel_compProd MeasureTheory.AEStronglyMeasurable.norm
integrable_comp_iff 📖	mathematical	MeasureTheory.AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup DFunLike.coe Kernel MeasureTheory.Measure Kernel.instFunLike Kernel.comp	MeasureTheory.Integrable SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Filter.Eventually MeasureTheory.ae MeasureTheory.Measure.instFunLike MeasureTheory.Measure.instOuterMeasureClass Real Real.pseudoMetricSpace NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing MeasureTheory.integral Real.normedAddCommGroup InnerProductSpace.toNormedSpace Real.instRCLike RCLike.toInnerProductSpaceReal Norm.norm NormedAddCommGroup.toNorm	—	MeasureTheory.Measure.instOuterMeasureClass hasFiniteIntegral_comp_iff' MeasureTheory.AEStronglyMeasurable.comp MeasureTheory.AEStronglyMeasurable.integral_kernel_comp MeasureTheory.AEStronglyMeasurable.norm
integrable_kernel_prodMk_left 📖	mathematical	MeasurableSet Prod.instMeasurableSpace	MeasureTheory.Integrable Real UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing MeasureTheory.Measure.real DFunLike.coe Kernel MeasureTheory.Measure Kernel.instFunLike Set.preimage	—	Measurable.aestronglyMeasurable PseudoEMetricSpace.pseudoMetrizableSpace instSecondCountableTopologyReal BorelSpace.opensMeasurable Real.borelSpace Measurable.ennreal_toReal Kernel.measurable_kernel_prodMk_left' hasFiniteIntegral_prodMk_left
integral_compProd 📖	mathematical	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Prod.instMeasurableSpace DFunLike.coe Kernel MeasureTheory.Measure Kernel.instFunLike Kernel.compProd	MeasureTheory.integral	—	MeasureTheory.Integrable.induction MeasureTheory.integral_indicator measurable_prodMk_left MeasureTheory.setIntegral_const integral_smul_const MeasureTheory.integral_toReal Measurable.aemeasurable Kernel.measurable_kernel_prodMk_left' Kernel.ae_kernel_lt_top LT.lt.ne Kernel.compProd_apply MeasureTheory.integral_add' Kernel.integral_integral_add' isClosed_eq SeminormedAddCommGroup.toIsTopologicalAddGroup fact_one_le_one_ennreal TopologicalSpace.t2Space_of_metrizableSpace EMetricSpace.metrizableSpace MeasureTheory.continuous_integral Kernel.continuous_integral_integral MeasureTheory.Measure.instOuterMeasureClass MeasureTheory.integral_congr_ae Filter.EventuallyEq.symm Filter.mp_mem Kernel.ae_ae_of_ae_compProd Filter.univ_mem' MeasureTheory.ae_eq_symm MeasureTheory.integral_def MeasureTheory.integral_zero
setIntegral_compProd 📖	mathematical	MeasurableSet MeasureTheory.IntegrableOn Prod.instMeasurableSpace UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup SProd.sprod Set Set.instSProd DFunLike.coe Kernel MeasureTheory.Measure Kernel.instFunLike Kernel.compProd	MeasureTheory.integral MeasureTheory.Measure.restrict	—	MeasurableSet.prod Kernel.restrict_apply Kernel.compProd_restrict integral_compProd Kernel.IsSFiniteKernel.restrict
setIntegral_compProd_univ_left 📖	mathematical	MeasurableSet MeasureTheory.IntegrableOn Prod.instMeasurableSpace UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup SProd.sprod Set Set.instSProd Set.univ DFunLike.coe Kernel MeasureTheory.Measure Kernel.instFunLike Kernel.compProd	MeasureTheory.integral MeasureTheory.Measure.restrict	—	setIntegral_compProd MeasurableSet.univ MeasureTheory.Measure.restrict_univ
setIntegral_compProd_univ_right 📖	mathematical	MeasurableSet MeasureTheory.IntegrableOn Prod.instMeasurableSpace UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup SProd.sprod Set Set.instSProd Set.univ DFunLike.coe Kernel MeasureTheory.Measure Kernel.instFunLike Kernel.compProd	MeasureTheory.integral MeasureTheory.Measure.restrict	—	setIntegral_compProd MeasurableSet.univ MeasureTheory.Measure.restrict_univ

ProbabilityTheory.Kernel

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
continuous_integral_integral 📖	mathematical	—	Continuous MeasureTheory.AEEqFun Prod.instMeasurableSpace UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike compProd AddSubgroup MeasureTheory.AEEqFun.instAddGroup NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup SeminormedAddCommGroup.toIsTopologicalAddGroup SetLike.instMembership AddSubgroup.instSetLike MeasureTheory.Lp ENNReal AddMonoidWithOne.toOne AddCommMonoidWithOne.toAddMonoidWithOne instAddCommMonoidWithOneENNReal MeasureTheory.Lp.instNormedAddCommGroup fact_one_le_one_ennreal MeasureTheory.integral MeasureTheory.AEEqFun.cast	—	SeminormedAddCommGroup.toIsTopologicalAddGroup fact_one_le_one_ennreal continuous_iff_continuousAt MeasureTheory.tendsto_integral_of_L1 MeasureTheory.Integrable.integral_compProd MeasureTheory.L1.integrable_coeFn Filter.Eventually.of_forall lintegral_fn_integral_sub tendsto_of_tendsto_of_tendsto_of_le_of_le ENNReal.instOrderTopology tendsto_const_nhds MeasureTheory.StronglyMeasurable.enorm MeasureTheory.StronglyMeasurable.sub IsTopologicalAddGroup.to_continuousSub MeasureTheory.Lp.stronglyMeasurable lintegral_compProd Filter.Tendsto.comp Continuous.tendsto ENNReal.continuous_ofReal tendsto_iff_norm_sub_tendsto_zero Filter.tendsto_id zero_le ENNReal.instCanonicallyOrderedAdd MeasureTheory.lintegral_mono MeasureTheory.enorm_integral_le_lintegral_enorm
continuous_integral_integral_comp 📖	mathematical	—	Continuous MeasureTheory.AEEqFun UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike comp AddSubgroup MeasureTheory.AEEqFun.instAddGroup NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup SeminormedAddCommGroup.toIsTopologicalAddGroup SetLike.instMembership AddSubgroup.instSetLike MeasureTheory.Lp ENNReal AddMonoidWithOne.toOne AddCommMonoidWithOne.toAddMonoidWithOne instAddCommMonoidWithOneENNReal MeasureTheory.Lp.instNormedAddCommGroup fact_one_le_one_ennreal MeasureTheory.integral MeasureTheory.AEEqFun.cast	—	SeminormedAddCommGroup.toIsTopologicalAddGroup fact_one_le_one_ennreal continuous_iff_continuousAt MeasureTheory.tendsto_integral_of_L1 MeasureTheory.Integrable.integral_comp MeasureTheory.L1.integrable_coeFn Filter.Eventually.of_forall lintegral_fn_integral_sub_comp tendsto_of_tendsto_of_tendsto_of_le_of_le ENNReal.instOrderTopology tendsto_const_nhds MeasureTheory.StronglyMeasurable.enorm MeasureTheory.StronglyMeasurable.sub IsTopologicalAddGroup.to_continuousSub MeasureTheory.Lp.stronglyMeasurable lintegral_comp Filter.Tendsto.comp Continuous.tendsto ENNReal.continuous_ofReal tendsto_iff_norm_sub_tendsto_zero Filter.tendsto_id zero_le ENNReal.instCanonicallyOrderedAdd MeasureTheory.lintegral_mono MeasureTheory.enorm_integral_le_lintegral_enorm
integral_comp 📖	mathematical	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike comp	MeasureTheory.integral	—	MeasureTheory.Integrable.induction MeasureTheory.integral_indicator MeasureTheory.setIntegral_const integral_smul_const MeasureTheory.integral_toReal Measurable.aemeasurable measurable_coe ae_lt_top_of_comp_ne_top LT.lt.ne comp_apply' MeasureTheory.integral_add' integral_integral_add'_comp isClosed_eq SeminormedAddCommGroup.toIsTopologicalAddGroup fact_one_le_one_ennreal TopologicalSpace.t2Space_of_metrizableSpace EMetricSpace.metrizableSpace MeasureTheory.continuous_integral continuous_integral_integral_comp MeasureTheory.Measure.instOuterMeasureClass MeasureTheory.integral_congr_ae Filter.EventuallyEq.symm Filter.mp_mem ae_ae_of_ae_comp Filter.univ_mem' MeasureTheory.ae_eq_symm MeasureTheory.integral_def MeasureTheory.integral_zero
integral_fn_integral_add 📖	mathematical	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Prod.instMeasurableSpace DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike compProd	MeasureTheory.integral AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid	—	MeasureTheory.integral_congr_ae Filter.mp_mem MeasureTheory.Measure.instOuterMeasureClass MeasureTheory.Integrable.ae_of_compProd Filter.univ_mem' MeasureTheory.integral_add
integral_fn_integral_add_comp 📖	mathematical	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike comp	MeasureTheory.integral AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid	—	MeasureTheory.integral_congr_ae Filter.mp_mem MeasureTheory.Measure.instOuterMeasureClass MeasureTheory.Integrable.ae_of_comp Filter.univ_mem' MeasureTheory.integral_add
integral_fn_integral_sub 📖	mathematical	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Prod.instMeasurableSpace DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike compProd	MeasureTheory.integral SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup	—	MeasureTheory.integral_congr_ae Filter.mp_mem MeasureTheory.Measure.instOuterMeasureClass MeasureTheory.Integrable.ae_of_compProd Filter.univ_mem' MeasureTheory.integral_sub
integral_fn_integral_sub_comp 📖	mathematical	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike comp	MeasureTheory.integral SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup	—	MeasureTheory.integral_congr_ae Filter.mp_mem MeasureTheory.Measure.instOuterMeasureClass MeasureTheory.Integrable.ae_of_comp Filter.univ_mem' MeasureTheory.integral_sub
integral_integral_add 📖	mathematical	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Prod.instMeasurableSpace DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike compProd	MeasureTheory.integral AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid	—	integral_fn_integral_add MeasureTheory.integral_add MeasureTheory.Integrable.integral_compProd
integral_integral_add' 📖	mathematical	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Prod.instMeasurableSpace DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike compProd	MeasureTheory.integral Pi.instAdd AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid	—	integral_integral_add
integral_integral_add'_comp 📖	mathematical	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike comp	MeasureTheory.integral Pi.instAdd AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid	—	integral_integral_add_comp
integral_integral_add_comp 📖	mathematical	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike comp	MeasureTheory.integral AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid	—	integral_fn_integral_add_comp MeasureTheory.integral_add MeasureTheory.Integrable.integral_comp
integral_integral_sub 📖	mathematical	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Prod.instMeasurableSpace DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike compProd	MeasureTheory.integral SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup	—	integral_fn_integral_sub MeasureTheory.integral_sub MeasureTheory.Integrable.integral_compProd
integral_integral_sub' 📖	mathematical	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Prod.instMeasurableSpace DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike compProd	MeasureTheory.integral Pi.instSub SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup	—	integral_integral_sub
integral_integral_sub'_comp 📖	mathematical	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike comp	MeasureTheory.integral Pi.instSub SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup	—	integral_integral_sub_comp
integral_integral_sub_comp 📖	mathematical	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike comp	MeasureTheory.integral SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup	—	integral_fn_integral_sub_comp MeasureTheory.integral_sub MeasureTheory.Integrable.integral_comp
lintegral_fn_integral_sub 📖	mathematical	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Prod.instMeasurableSpace DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike compProd	MeasureTheory.lintegral MeasureTheory.integral SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup	—	MeasureTheory.lintegral_congr_ae Filter.mp_mem MeasureTheory.Measure.instOuterMeasureClass MeasureTheory.Integrable.ae_of_compProd Filter.univ_mem' MeasureTheory.integral_sub
lintegral_fn_integral_sub_comp 📖	mathematical	MeasureTheory.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike comp	MeasureTheory.lintegral MeasureTheory.integral SubNegMonoid.toSub AddGroup.toSubNegMonoid NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup	—	MeasureTheory.lintegral_congr_ae Filter.mp_mem MeasureTheory.Measure.instOuterMeasureClass MeasureTheory.Integrable.ae_of_comp Filter.univ_mem' MeasureTheory.integral_sub
setIntegral_comp 📖	mathematical	MeasurableSet MeasureTheory.IntegrableOn UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike comp	MeasureTheory.integral MeasureTheory.Measure.restrict	—	restrict_apply comp_restrict integral_comp

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