fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
comp_congr (h : ∀ᵐ a ∂μ, κ a = η a) : κ ∘ₘ μ = η ∘ₘ μ := bind_congr_right h | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompNotation",
"Mathlib.Probability.Kernel.Composition.KernelLemmas",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureComp.lean | comp_congr | null |
ae_ae_of_ae_comp {p : β → Prop} (h : ∀ᵐ ω ∂(κ ∘ₘ μ), p ω) :
∀ᵐ ω' ∂μ, ∀ᵐ ω ∂(κ ω'), p ω := by
rw [comp_eq_comp_const_apply] at h
exact Kernel.ae_ae_of_ae_comp h | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompNotation",
"Mathlib.Probability.Kernel.Composition.KernelLemmas",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureComp.lean | ae_ae_of_ae_comp | null |
ae_comp_of_ae_ae {p : β → Prop} (hp : MeasurableSet {z | p z})
(h : ∀ᵐ y ∂μ, ∀ᵐ z ∂κ y, p z) : ∀ᵐ z ∂(κ ∘ₘ μ), p z := by
rw [comp_eq_comp_const_apply]
exact Kernel.ae_comp_of_ae_ae hp h | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompNotation",
"Mathlib.Probability.Kernel.Composition.KernelLemmas",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureComp.lean | ae_comp_of_ae_ae | null |
ae_comp_iff {p : β → Prop} (hp : MeasurableSet {z | p z}) :
(∀ᵐ z ∂(κ ∘ₘ μ), p z) ↔ ∀ᵐ y ∂μ, ∀ᵐ z ∂κ y, p z :=
⟨ae_ae_of_ae_comp, ae_comp_of_ae_ae hp⟩ | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompNotation",
"Mathlib.Probability.Kernel.Composition.KernelLemmas",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureComp.lean | ae_comp_iff | null |
@[simp]
_root_.ProbabilityTheory.Kernel.comp_const (κ : Kernel β γ) (μ : Measure β) :
κ ∘ₖ Kernel.const α μ = Kernel.const α (κ ∘ₘ μ) := rfl | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompNotation",
"Mathlib.Probability.Kernel.Composition.KernelLemmas",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureComp.lean | _root_.ProbabilityTheory.Kernel.comp_const | null |
map_comp (μ : Measure α) (κ : Kernel α β) {f : β → γ} (hf : Measurable f) :
(κ ∘ₘ μ).map f = (κ.map f) ∘ₘ μ := by
ext s hs
rw [Measure.map_apply hf hs, Measure.bind_apply (hf hs) κ.aemeasurable,
Measure.bind_apply hs (Kernel.aemeasurable _)]
simp_rw [Kernel.map_apply' _ hf _ hs]
@[simp] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompNotation",
"Mathlib.Probability.Kernel.Composition.KernelLemmas",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureComp.lean | map_comp | null |
discard_comp (μ : Measure α) : Kernel.discard α ∘ₘ μ = μ .univ • Measure.dirac () := by
ext s hs; simp [Measure.bind_apply hs (Kernel.aemeasurable _), mul_comm] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompNotation",
"Mathlib.Probability.Kernel.Composition.KernelLemmas",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureComp.lean | discard_comp | null |
copy_comp_map {f : α → β} (hf : AEMeasurable f μ) :
Kernel.copy β ∘ₘ (μ.map f) = μ.map (fun a ↦ (f a, f a)) := by
rw [Kernel.copy, deterministic_comp_eq_map, AEMeasurable.map_map_of_aemeasurable (by fun_prop) hf]
rfl | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompNotation",
"Mathlib.Probability.Kernel.Composition.KernelLemmas",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureComp.lean | copy_comp_map | null |
compProd_eq_comp_prod (μ : Measure α) [SFinite μ] (κ : Kernel α β) [IsSFiniteKernel κ] :
μ ⊗ₘ κ = (Kernel.id ×ₖ κ) ∘ₘ μ := by
rw [compProd, Kernel.compProd_prodMkLeft_eq_comp]
rfl | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompNotation",
"Mathlib.Probability.Kernel.Composition.KernelLemmas",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureComp.lean | compProd_eq_comp_prod | null |
compProd_id_eq_copy_comp [SFinite μ] : μ ⊗ₘ Kernel.id = Kernel.copy α ∘ₘ μ := by
rw [compProd_id, Kernel.copy, deterministic_comp_eq_map] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompNotation",
"Mathlib.Probability.Kernel.Composition.KernelLemmas",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureComp.lean | compProd_id_eq_copy_comp | null |
comp_compProd_comm {η : Kernel (α × β) γ} [SFinite μ] [IsSFiniteKernel η] :
η ∘ₘ (μ ⊗ₘ κ) = ((κ ⊗ₖ η) ∘ₘ μ).snd := by
by_cases hκ : IsSFiniteKernel κ; swap
· simp [compProd_of_not_isSFiniteKernel _ _ hκ,
Kernel.compProd_of_not_isSFiniteKernel_left _ _ hκ]
ext s hs
rw [Measure.bind_apply hs η.aemeasurable, Measure.snd_apply hs,
Measure.bind_apply _ (Kernel.aemeasurable _), Measure.lintegral_compProd (η.measurable_coe hs)]
swap; · exact measurable_snd hs
congr with a
rw [Kernel.compProd_apply]
· rfl
· exact measurable_snd hs
@[simp] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompNotation",
"Mathlib.Probability.Kernel.Composition.KernelLemmas",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureComp.lean | comp_compProd_comm | null |
prodMkLeft_comp_compProd {η : Kernel β γ} [SFinite μ] [IsSFiniteKernel κ] :
(η.prodMkLeft α) ∘ₘ μ ⊗ₘ κ = η ∘ₘ κ ∘ₘ μ := by
rw [← snd_compProd μ κ, Kernel.prodMkLeft, snd, ← deterministic_comp_eq_map measurable_snd,
comp_assoc, Kernel.comp_deterministic_eq_comap] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompNotation",
"Mathlib.Probability.Kernel.Composition.KernelLemmas",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureComp.lean | prodMkLeft_comp_compProd | null |
compProd_deterministic [SFinite μ] {f : α → β} (hf : Measurable f) :
μ ⊗ₘ Kernel.deterministic f hf = μ.map (fun a ↦ (a, f a)) := by
rw [compProd_eq_comp_prod, Kernel.id, Kernel.deterministic_prod_deterministic,
deterministic_comp_eq_map]
rfl | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompNotation",
"Mathlib.Probability.Kernel.Composition.KernelLemmas",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureComp.lean | compProd_deterministic | null |
@[simp]
comp_add : κ ∘ₘ (μ + ν) = κ ∘ₘ μ + κ ∘ₘ ν := by
simp_rw [comp_eq_comp_const_apply, Kernel.const_add, Kernel.comp_add_right, Kernel.add_apply] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompNotation",
"Mathlib.Probability.Kernel.Composition.KernelLemmas",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureComp.lean | comp_add | null |
add_comp : (κ + η) ∘ₘ μ = κ ∘ₘ μ + η ∘ₘ μ := by
simp_rw [comp_eq_comp_const_apply, Kernel.comp_add_left, Kernel.add_apply] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompNotation",
"Mathlib.Probability.Kernel.Composition.KernelLemmas",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureComp.lean | add_comp | null |
@[simp]
add_comp' : (⇑κ + ⇑η) ∘ₘ μ = κ ∘ₘ μ + η ∘ₘ μ := by rw [← Kernel.coe_add, add_comp]
@[simp] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompNotation",
"Mathlib.Probability.Kernel.Composition.KernelLemmas",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureComp.lean | add_comp' | Same as `add_comp` except that it uses `⇑κ + ⇑η` instead of `⇑(κ + η)` in order to have
a simp-normal form on the left of the equality. |
comp_smul (a : ℝ≥0∞) : κ ∘ₘ (a • μ) = a • (κ ∘ₘ μ) := by
ext s hs
simp only [bind_apply hs κ.aemeasurable, lintegral_smul_measure, smul_apply, smul_eq_mul] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompNotation",
"Mathlib.Probability.Kernel.Composition.KernelLemmas",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureComp.lean | comp_smul | null |
AbsolutelyContinuous.comp_right (hμν : μ ≪ ν) (κ : Kernel α γ) :
κ ∘ₘ μ ≪ κ ∘ₘ ν := by
refine Measure.AbsolutelyContinuous.mk fun s hs hs_zero ↦ ?_
rw [Measure.bind_apply hs (Kernel.aemeasurable _),
lintegral_eq_zero_iff (Kernel.measurable_coe _ hs)] at hs_zero ⊢
exact hμν.ae_eq hs_zero | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompNotation",
"Mathlib.Probability.Kernel.Composition.KernelLemmas",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureComp.lean | AbsolutelyContinuous.comp_right | null |
AbsolutelyContinuous.comp_left (μ : Measure α) (hκη : ∀ᵐ a ∂μ, κ a ≪ η a) :
κ ∘ₘ μ ≪ η ∘ₘ μ := by
refine Measure.AbsolutelyContinuous.mk fun s hs hs_zero ↦ ?_
rw [Measure.bind_apply hs (Kernel.aemeasurable _),
lintegral_eq_zero_iff (Kernel.measurable_coe _ hs)] at hs_zero ⊢
filter_upwards [hs_zero, hκη] with a ha_zero ha_ac using ha_ac ha_zero | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompNotation",
"Mathlib.Probability.Kernel.Composition.KernelLemmas",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureComp.lean | AbsolutelyContinuous.comp_left | null |
AbsolutelyContinuous.comp (hμν : μ ≪ ν) (hκη : ∀ᵐ a ∂μ, κ a ≪ η a) :
κ ∘ₘ μ ≪ η ∘ₘ ν :=
(AbsolutelyContinuous.comp_left μ hκη).trans (hμν.comp_right η) | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompNotation",
"Mathlib.Probability.Kernel.Composition.KernelLemmas",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureComp.lean | AbsolutelyContinuous.comp | null |
absolutelyContinuous_comp_of_countable [Countable α] [MeasurableSingletonClass α] :
∀ᵐ ω ∂μ, κ ω ≪ κ ∘ₘ μ := by
rw [Measure.comp_eq_sum_of_countable, ae_iff_of_countable]
exact fun ω hμω ↦ Measure.absolutelyContinuous_sum_right ω (Measure.absolutelyContinuous_smul hμω) | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompNotation",
"Mathlib.Probability.Kernel.Composition.KernelLemmas",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureComp.lean | absolutelyContinuous_comp_of_countable | null |
@[simp]
Kernel.comp_boolKernel (κ : Kernel α β) (μ ν : Measure α) :
κ ∘ₖ (boolKernel μ ν) = boolKernel (κ ∘ₘ μ) (κ ∘ₘ ν) := by
ext b : 1
rw [comp_apply]
cases b <;> simp | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompNotation",
"Mathlib.Probability.Kernel.Composition.KernelLemmas",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureComp.lean | Kernel.comp_boolKernel | null |
boolKernel_comp_measure (μ ν : Measure α) (π : Measure Bool) :
Kernel.boolKernel μ ν ∘ₘ π = π {true} • ν + π {false} • μ := by
ext s hs
rw [Measure.bind_apply hs (Kernel.aemeasurable _)]
simp [lintegral_fintype, mul_comm] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompNotation",
"Mathlib.Probability.Kernel.Composition.KernelLemmas",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureComp.lean | boolKernel_comp_measure | null |
absolutelyContinuous_boolKernel_comp_left (μ ν : Measure α) (hπ : π {false} ≠ 0) :
μ ≪ Kernel.boolKernel μ ν ∘ₘ π :=
boolKernel_comp_measure _ _ _ ▸ add_comm _ (π {true} • ν) ▸
(Measure.absolutelyContinuous_smul hπ).add_right _ | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompNotation",
"Mathlib.Probability.Kernel.Composition.KernelLemmas",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureComp.lean | absolutelyContinuous_boolKernel_comp_left | null |
absolutelyContinuous_boolKernel_comp_right (μ ν : Measure α) (hπ : π {true} ≠ 0) :
ν ≪ Kernel.boolKernel μ ν ∘ₘ π :=
boolKernel_comp_measure _ _ _ ▸ (Measure.absolutelyContinuous_smul hπ).add_right _ | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompNotation",
"Mathlib.Probability.Kernel.Composition.KernelLemmas",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureComp.lean | absolutelyContinuous_boolKernel_comp_right | null |
noncomputable
compProd (μ : Measure α) (κ : Kernel α β) : Measure (α × β) :=
(Kernel.const Unit μ ⊗ₖ Kernel.prodMkLeft Unit κ) ()
@[inherit_doc]
scoped[ProbabilityTheory] infixl:100 " ⊗ₘ " => MeasureTheory.Measure.compProd
@[simp] | def | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | compProd | The composition-product of a measure and a kernel. |
compProd_of_not_sfinite (μ : Measure α) (κ : Kernel α β) (h : ¬ SFinite μ) :
μ ⊗ₘ κ = 0 := by
rw [compProd, Kernel.compProd_of_not_isSFiniteKernel_left, Kernel.zero_apply]
rwa [Kernel.isSFiniteKernel_const]
@[simp] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | compProd_of_not_sfinite | null |
compProd_of_not_isSFiniteKernel (μ : Measure α) (κ : Kernel α β) (h : ¬ IsSFiniteKernel κ) :
μ ⊗ₘ κ = 0 := by
rw [compProd, Kernel.compProd_of_not_isSFiniteKernel_right, Kernel.zero_apply]
rwa [Kernel.isSFiniteKernel_prodMkLeft_unit] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | compProd_of_not_isSFiniteKernel | null |
compProd_apply [SFinite μ] [IsSFiniteKernel κ] {s : Set (α × β)} (hs : MeasurableSet s) :
(μ ⊗ₘ κ) s = ∫⁻ a, κ a (Prod.mk a ⁻¹' s) ∂μ := by
simp_rw [compProd, Kernel.compProd_apply hs, Kernel.const_apply, Kernel.prodMkLeft_apply']
@[simp] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | compProd_apply | null |
compProd_apply_univ [SFinite μ] [IsMarkovKernel κ] : (μ ⊗ₘ κ) univ = μ univ := by
simp [compProd] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | compProd_apply_univ | null |
compProd_apply_prod [SFinite μ] [IsSFiniteKernel κ]
{s : Set α} {t : Set β} (hs : MeasurableSet s) (ht : MeasurableSet t) :
(μ ⊗ₘ κ) (s ×ˢ t) = ∫⁻ a in s, κ a t ∂μ := by
simp [compProd, Kernel.compProd_apply_prod hs ht] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | compProd_apply_prod | null |
compProd_congr [IsSFiniteKernel κ] [IsSFiniteKernel η] (h : κ =ᵐ[μ] η) :
μ ⊗ₘ κ = μ ⊗ₘ η := by
rw [compProd, compProd]
congr 1
refine Kernel.compProd_congr ?_
simpa
@[simp] lemma compProd_zero_left (κ : Kernel α β) : (0 : Measure α) ⊗ₘ κ = 0 := by simp [compProd]
@[simp] lemma compProd_zero_right (μ : Measure α) : μ ⊗ₘ (0 : Kernel α β) = 0 := by simp [compProd] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | compProd_congr | null |
compProd_eq_zero_iff [SFinite μ] [IsSFiniteKernel κ] :
μ ⊗ₘ κ = 0 ↔ ∀ᵐ a ∂μ, κ a = 0 := by
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· simp_rw [← measure_univ_eq_zero]
refine (lintegral_eq_zero_iff (Kernel.measurable_coe _ .univ)).mp ?_
rw [← setLIntegral_univ, ← compProd_apply_prod .univ .univ, h]
simp
· rw [← compProd_zero_right μ]
exact compProd_congr h | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | compProd_eq_zero_iff | null |
_root_.ProbabilityTheory.Kernel.compProd_apply_eq_compProd_sectR {γ : Type*}
{mγ : MeasurableSpace γ} (κ : Kernel α β) (η : Kernel (α × β) γ)
[IsSFiniteKernel κ] [IsSFiniteKernel η] (a : α) :
(κ ⊗ₖ η) a = (κ a) ⊗ₘ (Kernel.sectR η a) := by
ext s hs
simp_rw [Kernel.compProd_apply hs, compProd_apply hs, Kernel.sectR_apply] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | _root_.ProbabilityTheory.Kernel.compProd_apply_eq_compProd_sectR | null |
compProd_id [SFinite μ] : μ ⊗ₘ Kernel.id = μ.map (fun x ↦ (x, x)) := by
ext s hs
rw [compProd_apply hs, map_apply (measurable_id.prod measurable_id) hs]
have h_meas a : MeasurableSet (Prod.mk a ⁻¹' s) := measurable_prodMk_left hs
simp_rw [Kernel.id_apply, dirac_apply' _ (h_meas _)]
calc ∫⁻ a, (Prod.mk a ⁻¹' s).indicator 1 a ∂μ
_ = ∫⁻ a, ((fun x ↦ (x, x)) ⁻¹' s).indicator 1 a ∂μ := rfl
_ = μ ((fun x ↦ (x, x)) ⁻¹' s) := by
rw [lintegral_indicator_one]
exact (measurable_id.prod measurable_id) hs | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | compProd_id | null |
ae_compProd_of_ae_ae {p : α × β → Prop}
(hp : MeasurableSet {x | p x}) (h : ∀ᵐ a ∂μ, ∀ᵐ b ∂(κ a), p (a, b)) :
∀ᵐ x ∂(μ ⊗ₘ κ), p x :=
Kernel.ae_compProd_of_ae_ae hp h | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | ae_compProd_of_ae_ae | null |
ae_ae_of_ae_compProd [SFinite μ] [IsSFiniteKernel κ] {p : α × β → Prop}
(h : ∀ᵐ x ∂(μ ⊗ₘ κ), p x) :
∀ᵐ a ∂μ, ∀ᵐ b ∂κ a, p (a, b) := by
convert Kernel.ae_ae_of_ae_compProd h -- Much faster with `convert` | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | ae_ae_of_ae_compProd | null |
ae_compProd_iff [SFinite μ] [IsSFiniteKernel κ] {p : α × β → Prop}
(hp : MeasurableSet {x | p x}) :
(∀ᵐ x ∂(μ ⊗ₘ κ), p x) ↔ ∀ᵐ a ∂μ, ∀ᵐ b ∂(κ a), p (a, b) :=
Kernel.ae_compProd_iff hp | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | ae_compProd_iff | null |
@[simp]
compProd_const {ν : Measure β} [SFinite μ] [SFinite ν] :
μ ⊗ₘ (Kernel.const α ν) = μ.prod ν := by
ext s hs
simp_rw [compProd_apply hs, prod_apply hs, Kernel.const_apply] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | compProd_const | The composition product of a measure and a constant kernel is the product between the two
measures. |
compProd_add_left (μ ν : Measure α) [SFinite μ] [SFinite ν] (κ : Kernel α β) :
(μ + ν) ⊗ₘ κ = μ ⊗ₘ κ + ν ⊗ₘ κ := by
by_cases hκ : IsSFiniteKernel κ
· simp_rw [Measure.compProd, Kernel.const_add, Kernel.compProd_add_left, Kernel.add_apply]
· simp [hκ] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | compProd_add_left | null |
compProd_add_right (μ : Measure α) (κ η : Kernel α β)
[IsSFiniteKernel κ] [IsSFiniteKernel η] :
μ ⊗ₘ (κ + η) = μ ⊗ₘ κ + μ ⊗ₘ η := by
by_cases hμ : SFinite μ
· simp_rw [Measure.compProd, Kernel.prodMkLeft_add, Kernel.compProd_add_right, Kernel.add_apply]
· simp [hμ] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | compProd_add_right | null |
compProd_sum_left {ι : Type*} [Countable ι] {μ : ι → Measure α} [∀ i, SFinite (μ i)] :
(sum μ) ⊗ₘ κ = sum (fun i ↦ (μ i) ⊗ₘ κ) := by
rw [compProd, ← Kernel.sum_const, Kernel.compProd_sum_left]
rfl | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | compProd_sum_left | null |
compProd_sum_right {ι : Type*} [Countable ι] {κ : ι → Kernel α β}
[h : ∀ i, IsSFiniteKernel (κ i)] :
μ ⊗ₘ (Kernel.sum κ) = sum (fun i ↦ μ ⊗ₘ (κ i)) := by
rw [compProd, ← Kernel.sum_prodMkLeft, Kernel.compProd_sum_right]
rfl
@[simp] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | compProd_sum_right | null |
fst_compProd (μ : Measure α) [SFinite μ] (κ : Kernel α β) [IsMarkovKernel κ] :
(μ ⊗ₘ κ).fst = μ := by
ext s
rw [compProd, Measure.fst, ← Kernel.fst_apply, Kernel.fst_compProd, Kernel.const_apply] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | fst_compProd | null |
compProd_smul_left (a : ℝ≥0∞) [SFinite μ] [IsSFiniteKernel κ] :
(a • μ) ⊗ₘ κ = a • (μ ⊗ₘ κ) := by
ext s hs
simp only [compProd_apply hs, lintegral_smul_measure, smul_apply, smul_eq_mul] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | compProd_smul_left | null |
lintegral_compProd [SFinite μ] [IsSFiniteKernel κ]
{f : α × β → ℝ≥0∞} (hf : Measurable f) :
∫⁻ x, f x ∂(μ ⊗ₘ κ) = ∫⁻ a, ∫⁻ b, f (a, b) ∂(κ a) ∂μ := by
rw [compProd, Kernel.lintegral_compProd _ _ _ hf]
simp | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | lintegral_compProd | null |
setLIntegral_compProd [SFinite μ] [IsSFiniteKernel κ]
{f : α × β → ℝ≥0∞} (hf : Measurable f)
{s : Set α} (hs : MeasurableSet s) {t : Set β} (ht : MeasurableSet t) :
∫⁻ x in s ×ˢ t, f x ∂(μ ⊗ₘ κ) = ∫⁻ a in s, ∫⁻ b in t, f (a, b) ∂(κ a) ∂μ := by
rw [compProd, Kernel.setLIntegral_compProd _ _ _ hf hs ht]
simp | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | setLIntegral_compProd | null |
dirac_compProd_apply [MeasurableSingletonClass α] {a : α} [IsSFiniteKernel κ]
{s : Set (α × β)} (hs : MeasurableSet s) :
(Measure.dirac a ⊗ₘ κ) s = κ a (Prod.mk a ⁻¹' s) := by
rw [compProd_apply hs, lintegral_dirac] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | dirac_compProd_apply | null |
dirac_unit_compProd (κ : Kernel Unit β) [IsSFiniteKernel κ] :
Measure.dirac () ⊗ₘ κ = (κ ()).map (Prod.mk ()) := by
ext s hs; rw [dirac_compProd_apply hs, Measure.map_apply measurable_prodMk_left hs] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | dirac_unit_compProd | null |
dirac_unit_compProd_const (μ : Measure β) [SFinite μ] :
Measure.dirac () ⊗ₘ Kernel.const Unit μ = μ.map (Prod.mk ()) := by
rw [dirac_unit_compProd, Kernel.const_apply] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | dirac_unit_compProd_const | null |
snd_dirac_unit_compProd_const (μ : Measure β) [SFinite μ] :
snd (Measure.dirac () ⊗ₘ Kernel.const Unit μ) = μ := by simp | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | snd_dirac_unit_compProd_const | null |
@[simp]
compProd_assoc {γ : Type*} {mγ : MeasurableSpace γ} {η : Kernel (α × β) γ} :
(μ ⊗ₘ (κ ⊗ₖ η)).map MeasurableEquiv.prodAssoc.symm = μ ⊗ₘ κ ⊗ₘ η := by
by_cases hμ : SFinite μ
swap; · simp [hμ]
by_cases hκ : IsSFiniteKernel κ
swap; · simp [hκ]
by_cases hη : IsSFiniteKernel η
swap; · simp [hη]
ext s hs
rw [Measure.compProd_apply hs, Measure.map_apply (by fun_prop) hs,
Measure.compProd_apply (hs.preimage (by fun_prop)), Measure.lintegral_compProd]
swap; · exact Kernel.measurable_kernel_prodMk_left hs
congr with a
rw [Kernel.compProd_apply]
· congr
· exact hs.preimage (by fun_prop) | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | compProd_assoc | `Measure.compProd` is associative. We have to insert `MeasurableEquiv.prodAssoc`
because the products of types `α × β × γ` and `(α × β) × γ` are different. |
@[simp]
compProd_assoc' {γ : Type*} {mγ : MeasurableSpace γ} {η : Kernel (α × β) γ} :
(μ ⊗ₘ κ ⊗ₘ η).map MeasurableEquiv.prodAssoc = μ ⊗ₘ (κ ⊗ₖ η) := by
simp [← Measure.compProd_assoc] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | compProd_assoc' | `Measure.compProd` is associative. We have to insert `MeasurableEquiv.prodAssoc`
because the products of types `α × β × γ` and `(α × β) × γ` are different. |
AbsolutelyContinuous.compProd_left [SFinite ν] (hμν : μ ≪ ν) (κ : Kernel α β) :
μ ⊗ₘ κ ≪ ν ⊗ₘ κ := by
by_cases hκ : IsSFiniteKernel κ
· have : SFinite μ := sFinite_of_absolutelyContinuous hμν
refine Measure.AbsolutelyContinuous.mk fun s hs hs_zero ↦ ?_
rw [Measure.compProd_apply hs, lintegral_eq_zero_iff (Kernel.measurable_kernel_prodMk_left hs)]
at hs_zero ⊢
exact hμν.ae_eq hs_zero
· simp [compProd_of_not_isSFiniteKernel _ _ hκ] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | AbsolutelyContinuous.compProd_left | null |
AbsolutelyContinuous.compProd_right [SFinite μ] [IsSFiniteKernel η]
(hκη : ∀ᵐ a ∂μ, κ a ≪ η a) :
μ ⊗ₘ κ ≪ μ ⊗ₘ η := by
by_cases hκ : IsSFiniteKernel κ
· refine Measure.AbsolutelyContinuous.mk fun s hs hs_zero ↦ ?_
rw [Measure.compProd_apply hs, lintegral_eq_zero_iff (Kernel.measurable_kernel_prodMk_left hs)]
at hs_zero ⊢
filter_upwards [hs_zero, hκη] with a ha_zero ha_ac using ha_ac ha_zero
· simp [compProd_of_not_isSFiniteKernel _ _ hκ] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | AbsolutelyContinuous.compProd_right | null |
AbsolutelyContinuous.compProd [SFinite ν] [IsSFiniteKernel η]
(hμν : μ ≪ ν) (hκη : ∀ᵐ a ∂μ, κ a ≪ η a) :
μ ⊗ₘ κ ≪ ν ⊗ₘ η :=
have : SFinite μ := sFinite_of_absolutelyContinuous hμν
(Measure.AbsolutelyContinuous.compProd_right hκη).trans (hμν.compProd_left _) | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | AbsolutelyContinuous.compProd | null |
absolutelyContinuous_of_compProd [SFinite μ] [IsSFiniteKernel κ] [h_zero : ∀ a, NeZero (κ a)]
(h : μ ⊗ₘ κ ≪ ν ⊗ₘ η) :
μ ≪ ν := by
refine Measure.AbsolutelyContinuous.mk (fun s hs hs0 ↦ ?_)
have h1 : (ν ⊗ₘ η) (s ×ˢ univ) = 0 := by
by_cases hν : SFinite ν
swap; · simp [compProd_of_not_sfinite _ _ hν]
by_cases hη : IsSFiniteKernel η
swap; · simp [compProd_of_not_isSFiniteKernel _ _ hη]
rw [Measure.compProd_apply_prod hs MeasurableSet.univ]
exact setLIntegral_measure_zero _ _ hs0
have h2 : (μ ⊗ₘ κ) (s ×ˢ univ) = 0 := h h1
rw [Measure.compProd_apply_prod hs MeasurableSet.univ, lintegral_eq_zero_iff] at h2
swap; · exact Kernel.measurable_coe _ MeasurableSet.univ
by_contra hμs
have : Filter.NeBot (ae (μ.restrict s)) := by simp [hμs]
obtain ⟨a, ha⟩ : ∃ a, κ a univ = 0 := h2.exists
refine absurd ha ?_
simp only [Measure.measure_univ_eq_zero]
exact (h_zero a).out | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | absolutelyContinuous_of_compProd | null |
absolutelyContinuous_compProd_left_iff [SFinite μ] [SFinite ν]
[IsSFiniteKernel κ] [∀ a, NeZero (κ a)] :
μ ⊗ₘ κ ≪ ν ⊗ₘ κ ↔ μ ≪ ν :=
⟨absolutelyContinuous_of_compProd, fun h ↦ h.compProd_left κ⟩ | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | absolutelyContinuous_compProd_left_iff | null |
AbsolutelyContinuous.compProd_of_compProd [SFinite ν] [IsSFiniteKernel η]
(hμν : μ ≪ ν) (hκη : μ ⊗ₘ κ ≪ μ ⊗ₘ η) :
μ ⊗ₘ κ ≪ ν ⊗ₘ η := by
by_cases hμ : SFinite μ
swap; · rw [compProd_of_not_sfinite _ _ hμ]; simp
refine AbsolutelyContinuous.mk fun s hs hs_zero ↦ ?_
suffices (μ ⊗ₘ η) s = 0 from hκη this
rw [measure_eq_zero_iff_ae_notMem, ae_compProd_iff hs.compl] at hs_zero ⊢
exact hμν.ae_le hs_zero | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | AbsolutelyContinuous.compProd_of_compProd | null |
MutuallySingular.compProd_of_left (hμν : μ ⟂ₘ ν) (κ η : Kernel α β) :
μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η := by
by_cases hμ : SFinite μ
swap; · rw [compProd_of_not_sfinite _ _ hμ]; simp
by_cases hν : SFinite ν
swap; · rw [compProd_of_not_sfinite _ _ hν]; simp
by_cases hκ : IsSFiniteKernel κ
swap; · rw [compProd_of_not_isSFiniteKernel _ _ hκ]; simp
by_cases hη : IsSFiniteKernel η
swap; · rw [compProd_of_not_isSFiniteKernel _ _ hη]; simp
refine ⟨hμν.nullSet ×ˢ univ, hμν.measurableSet_nullSet.prod .univ, ?_⟩
rw [compProd_apply_prod hμν.measurableSet_nullSet .univ, compl_prod_eq_union]
simp only [MutuallySingular.restrict_nullSet, lintegral_zero_measure, compl_univ,
prod_empty, union_empty, true_and]
rw [compProd_apply_prod hμν.measurableSet_nullSet.compl .univ]
simp | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | MutuallySingular.compProd_of_left | null |
mutuallySingular_of_mutuallySingular_compProd {ξ : Measure α}
[SFinite μ] [SFinite ν] [IsSFiniteKernel κ] [IsSFiniteKernel η]
(h : μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η) (hμ : ξ ≪ μ) (hν : ξ ≪ ν) :
∀ᵐ x ∂ξ, κ x ⟂ₘ η x := by
have hs : MeasurableSet h.nullSet := h.measurableSet_nullSet
have hμ_zero : (μ ⊗ₘ κ) h.nullSet = 0 := h.measure_nullSet
have hν_zero : (ν ⊗ₘ η) h.nullSetᶜ = 0 := h.measure_compl_nullSet
rw [compProd_apply, lintegral_eq_zero_iff'] at hμ_zero hν_zero
· filter_upwards [hμ hμ_zero, hν hν_zero] with x hxμ hxν
exact ⟨Prod.mk x ⁻¹' h.nullSet, measurable_prodMk_left hs, ⟨hxμ, hxν⟩⟩
· exact (Kernel.measurable_kernel_prodMk_left hs.compl).aemeasurable
· exact (Kernel.measurable_kernel_prodMk_left hs).aemeasurable
· exact hs.compl
· exact hs | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | mutuallySingular_of_mutuallySingular_compProd | null |
mutuallySingular_compProd_left_iff [SFinite μ] [SigmaFinite ν]
[IsSFiniteKernel κ] [hκ : ∀ x, NeZero (κ x)] :
μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ κ ↔ μ ⟂ₘ ν := by
refine ⟨fun h ↦ ?_, fun h ↦ h.compProd_of_left _ _⟩
rw [← withDensity_rnDeriv_eq_zero]
have hh := mutuallySingular_of_mutuallySingular_compProd h ?_ ?_
(ξ := ν.withDensity (μ.rnDeriv ν))
rotate_left
· exact absolutelyContinuous_of_le (μ.withDensity_rnDeriv_le ν)
· exact withDensity_absolutelyContinuous _ _
simp_rw [MutuallySingular.self_iff, (hκ _).ne] at hh
exact ae_eq_bot.mp (Filter.eventually_false_iff_eq_bot.mp hh) | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | mutuallySingular_compProd_left_iff | null |
AbsolutelyContinuous.mutuallySingular_compProd_iff [SigmaFinite μ] [SigmaFinite ν]
(hμν : μ ≪ ν) :
μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η ↔ μ ⊗ₘ κ ⟂ₘ μ ⊗ₘ η := by
conv_lhs => rw [ν.haveLebesgueDecomposition_add μ]
rw [compProd_add_left, MutuallySingular.add_right_iff]
simp only [(mutuallySingular_singularPart ν μ).symm.compProd_of_left κ η, true_and]
refine ⟨fun h ↦ h.mono_ac .rfl ?_, fun h ↦ h.mono_ac .rfl ?_⟩
· exact (absolutelyContinuous_withDensity_rnDeriv hμν).compProd_left _
· exact (withDensity_absolutelyContinuous μ (ν.rnDeriv μ)).compProd_left _ | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | AbsolutelyContinuous.mutuallySingular_compProd_iff | null |
mutuallySingular_compProd_iff [SigmaFinite μ] [SigmaFinite ν] :
μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η ↔ ∀ ξ, SFinite ξ → ξ ≪ μ → ξ ≪ ν → ξ ⊗ₘ κ ⟂ₘ ξ ⊗ₘ η := by
conv_lhs => rw [μ.haveLebesgueDecomposition_add ν]
rw [compProd_add_left, MutuallySingular.add_left_iff]
simp only [(mutuallySingular_singularPart μ ν).compProd_of_left κ η, true_and]
rw [(withDensity_absolutelyContinuous ν (μ.rnDeriv ν)).mutuallySingular_compProd_iff]
refine ⟨fun h ξ hξ hξμ hξν ↦ ?_, fun h ↦ ?_⟩
· exact h.mono_ac ((hξμ.withDensity_rnDeriv hξν).compProd_left _)
((hξμ.withDensity_rnDeriv hξν).compProd_left _)
· refine h _ ?_ ?_ ?_
· infer_instance
· exact absolutelyContinuous_of_le (withDensity_rnDeriv_le _ _)
· exact withDensity_absolutelyContinuous ν (μ.rnDeriv ν) | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | mutuallySingular_compProd_iff | null |
absolutelyContinuous_compProd_of_compProd [SigmaFinite μ] [SigmaFinite ν]
(hκη : μ ⊗ₘ κ ≪ ν ⊗ₘ η) :
μ ⊗ₘ κ ≪ μ ⊗ₘ η := by
rw [ν.haveLebesgueDecomposition_add μ, compProd_add_left, add_comm] at hκη
have h := absolutelyContinuous_of_add_of_mutuallySingular hκη
((mutuallySingular_singularPart _ _).symm.compProd_of_left _ _)
refine h.trans (AbsolutelyContinuous.compProd_left ?_ _)
exact withDensity_absolutelyContinuous _ _ | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | absolutelyContinuous_compProd_of_compProd | null |
absolutelyContinuous_compProd_iff
[SigmaFinite μ] [SigmaFinite ν] [IsSFiniteKernel κ] [IsSFiniteKernel η] [∀ x, NeZero (κ x)] :
μ ⊗ₘ κ ≪ ν ⊗ₘ η ↔ μ ≪ ν ∧ μ ⊗ₘ κ ≪ μ ⊗ₘ η :=
⟨fun h ↦ ⟨absolutelyContinuous_of_compProd h, absolutelyContinuous_compProd_of_compProd h⟩,
fun h ↦ h.1.compProd_of_compProd h.2⟩ | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.CompProd"
] | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | absolutelyContinuous_compProd_iff | null |
@[simp]
parallelComp_of_not_isSFiniteKernel_left (η : Kernel γ δ) (h : ¬ IsSFiniteKernel κ) :
κ ∥ₖ η = 0 := by
rw [parallelComp, dif_neg (not_and_of_not_left _ h)]
@[simp] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Kernel.MeasurableLIntegral"
] | Mathlib/Probability/Kernel/Composition/ParallelComp.lean | parallelComp_of_not_isSFiniteKernel_left | null |
parallelComp_of_not_isSFiniteKernel_right (κ : Kernel α β) (h : ¬ IsSFiniteKernel η) :
κ ∥ₖ η = 0 := by
rw [parallelComp, dif_neg (not_and_of_not_right _ h)] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Kernel.MeasurableLIntegral"
] | Mathlib/Probability/Kernel/Composition/ParallelComp.lean | parallelComp_of_not_isSFiniteKernel_right | null |
parallelComp_apply (κ : Kernel α β) [IsSFiniteKernel κ]
(η : Kernel γ δ) [IsSFiniteKernel η] (x : α × γ) :
(κ ∥ₖ η) x = (κ x.1).prod (η x.2) := by
rw [parallelComp, dif_pos ⟨inferInstance, inferInstance⟩, coe_mk] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Kernel.MeasurableLIntegral"
] | Mathlib/Probability/Kernel/Composition/ParallelComp.lean | parallelComp_apply | null |
parallelComp_apply' [IsSFiniteKernel κ] [IsSFiniteKernel η]
{s : Set (β × δ)} (hs : MeasurableSet s) :
(κ ∥ₖ η) x s = ∫⁻ b, η x.2 (Prod.mk b ⁻¹' s) ∂κ x.1 := by
rw [parallelComp_apply, Measure.prod_apply hs] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Kernel.MeasurableLIntegral"
] | Mathlib/Probability/Kernel/Composition/ParallelComp.lean | parallelComp_apply' | null |
parallelComp_apply_prod [IsSFiniteKernel κ] [IsSFiniteKernel η] (s : Set β) (t : Set δ) :
(κ ∥ₖ η) x (s ×ˢ t) = (κ x.1 s) * (η x.2 t) := by
rw [parallelComp_apply, Measure.prod_prod]
@[simp] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Kernel.MeasurableLIntegral"
] | Mathlib/Probability/Kernel/Composition/ParallelComp.lean | parallelComp_apply_prod | null |
parallelComp_apply_univ [IsSFiniteKernel κ] [IsSFiniteKernel η] :
(κ ∥ₖ η) x Set.univ = κ x.1 Set.univ * η x.2 Set.univ := by
rw [parallelComp_apply, Measure.prod_apply .univ, mul_comm]
simp
@[simp] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Kernel.MeasurableLIntegral"
] | Mathlib/Probability/Kernel/Composition/ParallelComp.lean | parallelComp_apply_univ | null |
parallelComp_zero_left (η : Kernel γ δ) : (0 : Kernel α β) ∥ₖ η = 0 := by
by_cases h : IsSFiniteKernel η
· ext; simp [parallelComp_apply]
· exact parallelComp_of_not_isSFiniteKernel_right _ h
@[simp] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Kernel.MeasurableLIntegral"
] | Mathlib/Probability/Kernel/Composition/ParallelComp.lean | parallelComp_zero_left | null |
parallelComp_zero_right (κ : Kernel α β) : κ ∥ₖ (0 : Kernel γ δ) = 0 := by
by_cases h : IsSFiniteKernel κ
· ext; simp [parallelComp_apply]
· exact parallelComp_of_not_isSFiniteKernel_left _ h | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Kernel.MeasurableLIntegral"
] | Mathlib/Probability/Kernel/Composition/ParallelComp.lean | parallelComp_zero_right | null |
deterministic_parallelComp_deterministic
{f : α → γ} {g : β → δ} (hf : Measurable f) (hg : Measurable g) :
(deterministic f hf) ∥ₖ (deterministic g hg)
= deterministic (Prod.map f g) (hf.prodMap hg) := by
ext x : 1
simp_rw [parallelComp_apply, deterministic_apply, Prod.map, Measure.dirac_prod_dirac] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Kernel.MeasurableLIntegral"
] | Mathlib/Probability/Kernel/Composition/ParallelComp.lean | deterministic_parallelComp_deterministic | null |
lintegral_parallelComp [IsSFiniteKernel κ] [IsSFiniteKernel η]
(ac : α × γ) {g : β × δ → ℝ≥0∞} (hg : Measurable g) :
∫⁻ bd, g bd ∂(κ ∥ₖ η) ac = ∫⁻ b, ∫⁻ d, g (b, d) ∂η ac.2 ∂κ ac.1 := by
rw [parallelComp_apply, MeasureTheory.lintegral_prod _ hg.aemeasurable] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Kernel.MeasurableLIntegral"
] | Mathlib/Probability/Kernel/Composition/ParallelComp.lean | lintegral_parallelComp | null |
lintegral_parallelComp_symm [IsSFiniteKernel κ] [IsSFiniteKernel η]
(ac : α × γ) {g : β × δ → ℝ≥0∞} (hg : Measurable g) :
∫⁻ bd, g bd ∂(κ ∥ₖ η) ac = ∫⁻ d, ∫⁻ b, g (b, d) ∂κ ac.1 ∂η ac.2 := by
rw [parallelComp_apply, MeasureTheory.lintegral_prod_symm _ hg.aemeasurable] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Kernel.MeasurableLIntegral"
] | Mathlib/Probability/Kernel/Composition/ParallelComp.lean | lintegral_parallelComp_symm | null |
parallelComp_sum_left {ι : Type*} [Countable ι] (κ : ι → Kernel α β)
[∀ i, IsSFiniteKernel (κ i)] (η : Kernel γ δ) :
Kernel.sum κ ∥ₖ η = Kernel.sum fun i ↦ κ i ∥ₖ η := by
by_cases h : IsSFiniteKernel η
swap; · simp [h]
ext x
simp_rw [Kernel.sum_apply, parallelComp_apply, Kernel.sum_apply, Measure.prod_sum_left] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Kernel.MeasurableLIntegral"
] | Mathlib/Probability/Kernel/Composition/ParallelComp.lean | parallelComp_sum_left | null |
parallelComp_sum_right {ι : Type*} [Countable ι] (κ : Kernel α β)
(η : ι → Kernel γ δ) [∀ i, IsSFiniteKernel (η i)] :
κ ∥ₖ Kernel.sum η = Kernel.sum fun i ↦ κ ∥ₖ η i := by
by_cases h : IsSFiniteKernel κ
swap; · simp [h]
ext x
simp_rw [Kernel.sum_apply, parallelComp_apply, Kernel.sum_apply, Measure.prod_sum_right] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Kernel.MeasurableLIntegral"
] | Mathlib/Probability/Kernel/Composition/ParallelComp.lean | parallelComp_sum_right | null |
noncomputable prod (κ : Kernel α β) (η : Kernel α γ) : Kernel α (β × γ) :=
(κ ∥ₖ η) ∘ₖ copy α
@[inherit_doc]
scoped[ProbabilityTheory] infixl:100 " ×ₖ " => ProbabilityTheory.Kernel.prod | def | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | prod | Product of two kernels. This is meaningful only when the kernels are s-finite. |
parallelComp_comp_copy (κ : Kernel α β) (η : Kernel α γ) :
(κ ∥ₖ η) ∘ₖ copy α = κ ×ₖ η := rfl
@[simp] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | parallelComp_comp_copy | null |
zero_prod (η : Kernel α γ) : (0 : Kernel α β) ×ₖ η = 0 := by simp [prod]
@[simp] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | zero_prod | null |
prod_zero (κ : Kernel α β) : κ ×ₖ (0 : Kernel α γ) = 0 := by simp [prod]
@[simp] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | prod_zero | null |
prod_of_not_isSFiniteKernel_left {κ : Kernel α β} (η : Kernel α γ) (h : ¬ IsSFiniteKernel κ) :
κ ×ₖ η = 0 := by
simp [prod, h]
@[simp] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | prod_of_not_isSFiniteKernel_left | null |
prod_of_not_isSFiniteKernel_right (κ : Kernel α β) {η : Kernel α γ}
(h : ¬ IsSFiniteKernel η) :
κ ×ₖ η = 0 := by
simp [prod, h] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | prod_of_not_isSFiniteKernel_right | null |
prod_apply' (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel α γ) [IsSFiniteKernel η]
(a : α) {s : Set (β × γ)} (hs : MeasurableSet s) :
(κ ×ₖ η) a s = ∫⁻ b : β, (η a) (Prod.mk b ⁻¹' s) ∂κ a := by
simp_rw [prod, comp_apply, copy_apply, Measure.dirac_bind (Kernel.measurable _) (a, a),
parallelComp_apply, Measure.prod_apply hs] | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | prod_apply' | null |
prod_apply (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel α γ) [IsSFiniteKernel η]
(a : α) :
(κ ×ₖ η) a = (κ a).prod (η a) := by
ext s hs
rw [prod_apply' _ _ _ hs, Measure.prod_apply hs] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | prod_apply | null |
prod_apply_prod {κ : Kernel α β} {η : Kernel α γ}
[IsSFiniteKernel κ] [IsSFiniteKernel η] {s : Set β} {t : Set γ} {a : α} :
(κ ×ₖ η) a (s ×ˢ t) = (κ a s) * (η a t) := by
rw [prod_apply, Measure.prod_prod] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | prod_apply_prod | null |
prod_const (μ : Measure β) [SFinite μ] (ν : Measure γ) [SFinite ν] :
const α μ ×ₖ const α ν = const α (μ.prod ν) := by
ext x
rw [const_apply, prod_apply, const_apply, const_apply] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | prod_const | null |
lintegral_prod (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel α γ) [IsSFiniteKernel η]
(a : α) {g : β × γ → ℝ≥0∞} (hg : Measurable g) :
∫⁻ c, g c ∂(κ ×ₖ η) a = ∫⁻ b, ∫⁻ c, g (b, c) ∂η a ∂κ a := by
simp_rw [prod, lintegral_comp _ _ _ hg, copy_apply]
rw [lintegral_dirac' _ (by fun_prop)]
simp_rw [parallelComp_apply, MeasureTheory.lintegral_prod _ hg.aemeasurable] | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | lintegral_prod | null |
lintegral_prod_symm (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel α γ)
[IsSFiniteKernel η] (a : α) {g : β × γ → ℝ≥0∞} (hg : Measurable g) :
∫⁻ c, g c ∂(κ ×ₖ η) a = ∫⁻ c, ∫⁻ b, g (b, c) ∂κ a ∂η a := by
rw [prod_apply, MeasureTheory.lintegral_prod_symm _ hg.aemeasurable] | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | lintegral_prod_symm | null |
lintegral_deterministic_prod {f : α → β} (hf : Measurable f) (κ : Kernel α γ)
[IsSFiniteKernel κ] (a : α) {g : (β × γ) → ℝ≥0∞} (hg : Measurable g) :
∫⁻ p, g p ∂((deterministic f hf) ×ₖ κ) a = ∫⁻ c, g (f a, c) ∂κ a := by
rw [lintegral_prod _ _ _ hg, lintegral_deterministic' _ hg.lintegral_prod_right'] | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | lintegral_deterministic_prod | null |
lintegral_prod_deterministic {f : α → γ} (hf : Measurable f) (κ : Kernel α β)
[IsSFiniteKernel κ] (a : α) {g : (β × γ) → ℝ≥0∞} (hg : Measurable g) :
∫⁻ p, g p ∂(κ ×ₖ (deterministic f hf)) a = ∫⁻ b, g (b, f a) ∂κ a := by
rw [lintegral_prod_symm _ _ _ hg, lintegral_deterministic' _ hg.lintegral_prod_left'] | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | lintegral_prod_deterministic | null |
lintegral_id_prod {f : (α × β) → ℝ≥0∞} (hf : Measurable f) (κ : Kernel α β)
[IsSFiniteKernel κ] (a : α) :
∫⁻ p, f p ∂(Kernel.id ×ₖ κ) a = ∫⁻ b, f (a, b) ∂κ a := by
rw [Kernel.id, lintegral_deterministic_prod _ _ _ hf, id_eq] | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | lintegral_id_prod | null |
lintegral_prod_id {f : (α × β) → ℝ≥0∞} (hf : Measurable f) (κ : Kernel β α)
[IsSFiniteKernel κ] (b : β) :
∫⁻ p, f p ∂(κ ×ₖ Kernel.id) b = ∫⁻ a, f (a, b) ∂κ b := by
rw [Kernel.id, lintegral_prod_deterministic _ _ _ hf, id_eq] | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | lintegral_prod_id | null |
deterministic_prod_apply' {f : α → β} (mf : Measurable f) (κ : Kernel α γ)
[IsSFiniteKernel κ] (a : α) {s : Set (β × γ)} (hs : MeasurableSet s) :
((Kernel.deterministic f mf) ×ₖ κ) a s = κ a (Prod.mk (f a) ⁻¹' s) := by
rw [prod_apply' _ _ _ hs, lintegral_deterministic']
exact measurable_measure_prodMk_left hs | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | deterministic_prod_apply' | null |
id_prod_apply' (κ : Kernel α β) [IsSFiniteKernel κ] (a : α) {s : Set (α × β)}
(hs : MeasurableSet s) : (Kernel.id ×ₖ κ) a s = κ a (Prod.mk a ⁻¹' s) := by
rw [Kernel.id, deterministic_prod_apply' _ _ _ hs, id_eq] | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | id_prod_apply' | null |
IsMarkovKernel.prod (κ : Kernel α β) [IsMarkovKernel κ] (η : Kernel α γ)
[IsMarkovKernel η] : IsMarkovKernel (κ ×ₖ η) := by rw [Kernel.prod]; infer_instance
nonrec instance IsZeroOrMarkovKernel.prod (κ : Kernel α β) [h : IsZeroOrMarkovKernel κ]
(η : Kernel α γ) [IsZeroOrMarkovKernel η] : IsZeroOrMarkovKernel (κ ×ₖ η) := by
rcases eq_zero_or_isMarkovKernel κ with rfl | h
· simp only [prod]; infer_instance
rcases eq_zero_or_isMarkovKernel η with rfl | h'
· simp only [prod]; infer_instance
infer_instance | instance | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | IsMarkovKernel.prod | null |
IsFiniteKernel.prod (κ : Kernel α β) [IsFiniteKernel κ] (η : Kernel α γ)
[IsFiniteKernel η] : IsFiniteKernel (κ ×ₖ η) := by rw [Kernel.prod]; infer_instance | instance | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | IsFiniteKernel.prod | null |
IsSFiniteKernel.prod (κ : Kernel α β) (η : Kernel α γ) :
IsSFiniteKernel (κ ×ₖ η) := by rw [Kernel.prod]; infer_instance
@[simp] lemma fst_prod (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel α γ) [IsMarkovKernel η] :
fst (κ ×ₖ η) = κ := by
rw [prod, fst_comp]
ext a : 1
rw [comp_apply, copy_apply, Measure.dirac_bind (by fun_prop), fst_apply, parallelComp_apply]
simp
@[simp] lemma snd_prod (κ : Kernel α β) [IsMarkovKernel κ] (η : Kernel α γ) [IsSFiniteKernel η] :
snd (κ ×ₖ η) = η := by
ext x; simp [snd_apply, prod_apply] | instance | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | IsSFiniteKernel.prod | null |
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