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 MeasureTheory.AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup 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.AEStronglyMeasurable DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike 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.AEStronglyMeasurable DFunLike.coe ProbabilityTheory.Kernel Prod.instMeasurableSpace MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike 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.AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup MeasureTheory.integral DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike	—	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.AEStronglyMeasurable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup MeasureTheory.integral DFunLike.coe ProbabilityTheory.Kernel Prod.instMeasurableSpace MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike	—	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.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike 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.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup DFunLike.coe ProbabilityTheory.Kernel Prod.instMeasurableSpace MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike 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.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup MeasureTheory.integral DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike	—	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.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup MeasureTheory.integral DFunLike.coe ProbabilityTheory.Kernel Prod.instMeasurableSpace MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike	—	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	MeasureTheory.Integrable Real UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing MeasureTheory.integral Real.normedAddCommGroup InnerProductSpace.toNormedSpace Real.instRCLike NormedAddCommGroup.toSeminormedAddCommGroup RCLike.toInnerProductSpaceReal DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike 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	MeasureTheory.Integrable Real UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing MeasureTheory.integral Real.normedAddCommGroup InnerProductSpace.toNormedSpace Real.instRCLike NormedAddCommGroup.toSeminormedAddCommGroup RCLike.toInnerProductSpaceReal DFunLike.coe ProbabilityTheory.Kernel Prod.instMeasurableSpace MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike 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.Integrable UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike 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 UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Prod.instMeasurableSpace compProd 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 UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Prod.instMeasurableSpace compProd bind DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike	—	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 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 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	MeasureTheory.Integrable Real UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace Real.pseudoMetricSpace SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup NonUnitalSeminormedRing.toSeminormedAddCommGroup NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing SeminormedCommRing.toNonUnitalSeminormedCommRing NormedCommRing.toSeminormedCommRing Real.normedCommRing MeasureTheory.integral Real.normedAddCommGroup InnerProductSpace.toNormedSpace Real.instRCLike NormedAddCommGroup.toSeminormedAddCommGroup RCLike.toInnerProductSpaceReal DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure ProbabilityTheory.Kernel.instFunLike 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 Prod.instMeasurableSpace compProd 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 Prod.instMeasurableSpace restrict compProd SProd.sprod Set Set.instSProd 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 UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm Prod.instMeasurableSpace 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 UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm Prod.instMeasurableSpace 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 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 UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup 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 UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddGroup.toPseudoMetricSpace SeminormedAddCommGroup.toSeminormedAddGroup NormedAddCommGroup.toSeminormedAddCommGroup 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 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 UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup Prod.instMeasurableSpace DFunLike.coe Kernel MeasureTheory.Measure Kernel.instFunLike Kernel.compProd 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 UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup SeminormedAddGroup.toContinuousENorm SeminormedAddCommGroup.toSeminormedAddGroup DFunLike.coe Kernel MeasureTheory.Measure Kernel.instFunLike Kernel.comp 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 Prod.instMeasurableSpace 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 Prod.instMeasurableSpace DFunLike.coe Kernel MeasureTheory.Measure Kernel.instFunLike Kernel.compProd	—	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 Prod.instMeasurableSpace MeasureTheory.Measure.restrict DFunLike.coe Kernel MeasureTheory.Measure Kernel.instFunLike Kernel.compProd SProd.sprod Set Set.instSProd	—	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 Prod.instMeasurableSpace MeasureTheory.Measure.restrict DFunLike.coe Kernel MeasureTheory.Measure Kernel.instFunLike Kernel.compProd SProd.sprod Set Set.instSProd Set.univ	—	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 Prod.instMeasurableSpace MeasureTheory.Measure.restrict DFunLike.coe Kernel MeasureTheory.Measure Kernel.instFunLike Kernel.compProd SProd.sprod Set Set.instSProd Set.univ	—	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 DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike comp	—	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 DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike Prod.instMeasurableSpace AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup ESeminormedAddCommMonoid.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup 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 DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup ESeminormedAddCommMonoid.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup 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 DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike Prod.instMeasurableSpace 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 DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike 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 DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike Prod.instMeasurableSpace AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup ESeminormedAddCommMonoid.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup 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 DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike Prod.instMeasurableSpace Pi.instAdd AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup ESeminormedAddCommMonoid.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup 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 DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike Pi.instAdd AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup ESeminormedAddCommMonoid.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup 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 DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike AddCommMagma.toAdd AddCommSemigroup.toAddCommMagma AddCommMonoid.toAddCommSemigroup ESeminormedAddCommMonoid.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup 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 DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike Prod.instMeasurableSpace 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 DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike Prod.instMeasurableSpace 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 DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike 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 DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike 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 DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike MeasureTheory.integral Prod.instMeasurableSpace 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 DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike 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 DFunLike.coe ProbabilityTheory.Kernel MeasureTheory.Measure instFunLike comp	—	restrict_apply comp_restrict integral_comp

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