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 ⌀ |
|---|---|---|---|---|---|---|
comap_prod (κ : Kernel β γ) [IsSFiniteKernel κ] (η : Kernel β δ) [IsSFiniteKernel η]
{f : α → β} (hf : Measurable f) :
(κ ×ₖ η).comap f hf = (κ.comap f hf) ×ₖ (η.comap f hf) := by
ext1 x
rw [comap_apply, prod_apply, prod_apply, comap_apply, comap_apply] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | comap_prod | null |
map_prod_map {ε} {mε : MeasurableSpace ε} (κ : Kernel α β) [IsSFiniteKernel κ]
(η : Kernel α δ) [IsSFiniteKernel η] {f : β → γ} (hf : Measurable f) {g : δ → ε}
(hg : Measurable g) : (κ.map f) ×ₖ (η.map g) = (κ ×ₖ η).map (Prod.map f g) := by
ext1 x
rw [map_apply _ (hf.prodMap hg), prod_apply κ, ← Measure.map_prod_map _ _ hf hg, prod_apply,
map_apply _ hf, map_apply _ hg] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | map_prod_map | null |
map_prod_eq (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel α γ) [IsSFiniteKernel η]
{f : β → δ} (hf : Measurable f) : (κ.map f) ×ₖ η = (κ ×ₖ η).map (Prod.map f id) := by
rw [← map_prod_map _ _ hf measurable_id, map_id] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | map_prod_eq | null |
comap_prod_swap (κ : Kernel α β) (η : Kernel γ δ) [IsSFiniteKernel κ] [IsSFiniteKernel η] :
comap (prodMkRight α η ×ₖ prodMkLeft γ κ) Prod.swap measurable_swap
= map (prodMkRight γ κ ×ₖ prodMkLeft α η) Prod.swap := by
rw [ext_fun_iff]
intro x f hf
rw [lintegral_comap, lintegral_map _ measurable_swap _ hf, lintegral_prod _ _ _ hf,
lintegral_prod]
swap; · fun_prop
simp only [prodMkRight_apply, Prod.fst_swap, Prod.swap_prod_mk, lintegral_prodMkLeft,
Prod.snd_swap]
refine (lintegral_lintegral_swap ?_).symm
fun_prop | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | comap_prod_swap | null |
map_prod_swap (κ : Kernel α β) (η : Kernel α γ) [IsSFiniteKernel κ] [IsSFiniteKernel η] :
map (κ ×ₖ η) Prod.swap = η ×ₖ κ := by
rw [ext_fun_iff]
intro x f hf
rw [lintegral_map _ measurable_swap _ hf, lintegral_prod, lintegral_prod _ _ _ hf]
swap; · fun_prop
refine (lintegral_lintegral_swap ?_).symm
fun_prop | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | map_prod_swap | null |
prodComm_prod {κ : Kernel α β} [IsSFiniteKernel κ] {η : Kernel α γ} [IsSFiniteKernel η] :
(κ ×ₖ η).map MeasurableEquiv.prodComm = η ×ₖ κ :=
map_prod_swap κ η
@[simp] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | prodComm_prod | null |
swap_prod {κ : Kernel α β} [IsSFiniteKernel κ] {η : Kernel α γ} [IsSFiniteKernel η] :
(swap β γ) ∘ₖ (κ ×ₖ η) = (η ×ₖ κ) := by
rw [swap_comp_eq_map, map_prod_swap] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | swap_prod | null |
deterministic_prod_deterministic {f : α → β} {g : α → γ}
(hf : Measurable f) (hg : Measurable g) :
deterministic f hf ×ₖ deterministic g hg
= deterministic (fun a ↦ (f a, g a)) (hf.prodMk hg) := by
ext; simp_rw [prod_apply, deterministic_apply, Measure.dirac_prod_dirac] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | deterministic_prod_deterministic | null |
id_prod_eq : @Kernel.id (α × β) inferInstance =
(deterministic Prod.fst measurable_fst) ×ₖ (deterministic Prod.snd measurable_snd) := by
rw [deterministic_prod_deterministic]
rfl | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | id_prod_eq | null |
prodAssoc_prod (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel α γ) [IsSFiniteKernel η]
(ξ : Kernel α δ) [IsSFiniteKernel ξ] :
((κ ×ₖ ξ) ×ₖ η).map MeasurableEquiv.prodAssoc = κ ×ₖ (ξ ×ₖ η) := by
ext1 a
rw [map_apply _ (by fun_prop), prod_apply, prod_apply, Measure.prodAssoc_prod, prod_apply,
prod_apply] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | prodAssoc_prod | null |
prodAssoc_symm_prod (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel α γ) [IsSFiniteKernel η]
(ξ : Kernel α δ) [IsSFiniteKernel ξ] :
(κ ×ₖ (ξ ×ₖ η)).map MeasurableEquiv.prodAssoc.symm = (κ ×ₖ ξ) ×ₖ η := by
rw [← prodAssoc_prod, ← Kernel.map_comp_right _ (by fun_prop) (by fun_prop)]
simp | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | prodAssoc_symm_prod | null |
prod_const_comp {δ} {mδ : MeasurableSpace δ} (κ : Kernel α β) [IsSFiniteKernel κ]
(η : Kernel β γ) [IsSFiniteKernel η] (μ : Measure δ) [SFinite μ] :
(η ×ₖ (const β μ)) ∘ₖ κ = (η ∘ₖ κ) ×ₖ (const α μ) := by
ext x s ms
simp_rw [comp_apply' _ _ _ ms, prod_apply' _ _ _ ms, const_apply,
lintegral_comp _ _ _ (measurable_measure_prodMk_left ms)] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | prod_const_comp | null |
const_prod_comp {δ} {mδ : MeasurableSpace δ} (κ : Kernel α β) [IsSFiniteKernel κ]
(μ : Measure γ) [SFinite μ] (η : Kernel β δ) [IsSFiniteKernel η] :
((const β μ) ×ₖ η) ∘ₖ κ = (const α μ) ×ₖ (η ∘ₖ κ) := by
ext x s ms
simp_rw [comp_apply' _ _ _ ms, prod_apply, Measure.prod_apply_symm ms, const_apply,
lintegral_comp _ _ _ (measurable_measure_prodMk_right ms)] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompMap",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/Prod.lean | const_prod_comp | null |
IsCondKernel : Prop where
disintegrate : ρ.fst ⊗ₘ ρCond = ρ
variable [ρ.IsCondKernel ρCond] | class | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfLIntegral",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Disintegration/Basic.lean | IsCondKernel | A kernel `ρCond` is a conditional kernel for a measure `ρ` if it disintegrates it in the sense
that `ρ.fst ⊗ₘ ρCond = ρ`. |
disintegrate : ρ.fst ⊗ₘ ρCond = ρ := IsCondKernel.disintegrate | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfLIntegral",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Disintegration/Basic.lean | disintegrate | null |
IsCondKernel.isSFiniteKernel (hρ : ρ ≠ 0) : IsSFiniteKernel ρCond := by
contrapose! hρ; rwa [← ρ.disintegrate ρCond, Measure.compProd_of_not_isSFiniteKernel]
variable [IsFiniteMeasure ρ] | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfLIntegral",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Disintegration/Basic.lean | IsCondKernel.isSFiniteKernel | null |
private IsCondKernel.apply_of_ne_zero_of_measurableSet [MeasurableSingletonClass α] {x : α}
(hx : ρ.fst {x} ≠ 0) {s : Set Ω} (hs : MeasurableSet s) :
ρCond x s = (ρ.fst {x})⁻¹ * ρ ({x} ×ˢ s) := by
have := isSFiniteKernel ρ ρCond (by rintro rfl; simp at hx)
nth_rewrite 2 [← ρ.disintegrate ρCond]
rw [Measure.compProd_apply (measurableSet_prod.mpr (Or.inl ⟨measurableSet_singleton x, hs⟩))]
classical
have (a : _) : ρCond a (Prod.mk a ⁻¹' {x} ×ˢ s) = ({x} : Set α).indicator (ρCond · s) a := by
obtain rfl | hax := eq_or_ne a x
· simp only [singleton_prod, mem_singleton_iff, indicator_of_mem]
congr with y
simp
· simp only [singleton_prod, mem_singleton_iff, hax, not_false_eq_true, indicator_of_notMem]
have : Prod.mk a ⁻¹' (Prod.mk x '' s) = ∅ := by ext y; simp [Ne.symm hax]
simp only [this, measure_empty]
simp_rw [this]
rw [MeasureTheory.lintegral_indicator (measurableSet_singleton x)]
simp only [Measure.restrict_singleton, lintegral_smul_measure, lintegral_dirac, smul_eq_mul]
rw [← mul_assoc, ENNReal.inv_mul_cancel hx (measure_ne_top _ _), one_mul] | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfLIntegral",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Disintegration/Basic.lean | IsCondKernel.apply_of_ne_zero_of_measurableSet | Auxiliary lemma for `IsCondKernel.apply_of_ne_zero`. |
IsCondKernel.apply_of_ne_zero [MeasurableSingletonClass α] {x : α}
(hx : ρ.fst {x} ≠ 0) (s : Set Ω) : ρCond x s = (ρ.fst {x})⁻¹ * ρ ({x} ×ˢ s) := by
have : ρCond x s = ((ρ.fst {x})⁻¹ • ρ).comap (fun (y : Ω) ↦ (x, y)) s := by
congr 2 with s hs
simp [IsCondKernel.apply_of_ne_zero_of_measurableSet _ _ hx hs,
(measurableEmbedding_prodMk_left x).comap_apply, Set.singleton_prod]
simp [this, (measurableEmbedding_prodMk_left x).comap_apply, Set.singleton_prod] | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfLIntegral",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Disintegration/Basic.lean | IsCondKernel.apply_of_ne_zero | If the singleton `{x}` has non-zero mass for `ρ.fst`, then for all `s : Set Ω`,
`ρCond x s = (ρ.fst {x})⁻¹ * ρ ({x} ×ˢ s)` . |
IsCondKernel.isProbabilityMeasure [MeasurableSingletonClass α] {a : α} (ha : ρ.fst {a} ≠ 0) :
IsProbabilityMeasure (ρCond a) := by
constructor
rw [IsCondKernel.apply_of_ne_zero _ _ ha, prod_univ, ← Measure.fst_apply
(measurableSet_singleton _), ENNReal.inv_mul_cancel ha (measure_ne_top _ _)] | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfLIntegral",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Disintegration/Basic.lean | IsCondKernel.isProbabilityMeasure | null |
IsCondKernel.isMarkovKernel [MeasurableSingletonClass α] (hρ : ∀ a, ρ.fst {a} ≠ 0) :
IsMarkovKernel ρCond := ⟨fun _ ↦ isProbabilityMeasure _ _ (hρ _)⟩ | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfLIntegral",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Disintegration/Basic.lean | IsCondKernel.isMarkovKernel | null |
IsCondKernel : Prop where
protected disintegrate : κ.fst ⊗ₖ κCond = κ | class | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfLIntegral",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Disintegration/Basic.lean | IsCondKernel | A kernel `κCond` is a conditional kernel for a kernel `κ` if it disintegrates it in the sense
that `κ.fst ⊗ₖ κCond = κ`. |
instIsCondKernel_zero (κCond : Kernel (α × β) Ω) : IsCondKernel 0 κCond where
disintegrate := by simp | instance | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfLIntegral",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Disintegration/Basic.lean | instIsCondKernel_zero | null |
disintegrate [κ.IsCondKernel κCond] : κ.fst ⊗ₖ κCond = κ := IsCondKernel.disintegrate | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfLIntegral",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Disintegration/Basic.lean | disintegrate | null |
IsCondKernel.isProbabilityMeasure_ae [IsFiniteKernel κ.fst] [κ.IsCondKernel κCond] (a : α) :
∀ᵐ b ∂(κ.fst a), IsProbabilityMeasure (κCond (a, b)) := by
have h := disintegrate κ κCond
by_cases h_sfin : IsSFiniteKernel κCond
swap; · rw [Kernel.compProd_of_not_isSFiniteKernel_right _ _ h_sfin] at h; simp [h.symm]
suffices ∀ᵐ b ∂(κ.fst a), κCond (a, b) Set.univ = 1 by
convert this with b
exact ⟨fun _ ↦ measure_univ, fun h ↦ ⟨h⟩⟩
suffices (∀ᵐ b ∂(κ.fst a), κCond (a, b) Set.univ ≤ 1)
∧ (∀ᵐ b ∂(κ.fst a), 1 ≤ κCond (a, b) Set.univ) by
filter_upwards [this.1, this.2] with b h1 h2 using le_antisymm h1 h2
have h_eq s (hs : MeasurableSet s) :
∫⁻ b, s.indicator (fun b ↦ κCond (a, b) Set.univ) b ∂κ.fst a = κ.fst a s := by
conv_rhs => rw [← h]
rw [fst_compProd_apply _ _ _ hs]
have h_meas : Measurable fun b ↦ κCond (a, b) Set.univ :=
(κCond.measurable_coe MeasurableSet.univ).comp measurable_prodMk_left
constructor
· rw [ae_le_const_iff_forall_gt_measure_zero]
intro r hr
let s := {b | r ≤ κCond (a, b) Set.univ}
have hs : MeasurableSet s := h_meas measurableSet_Ici
have h_2_le : s.indicator (fun _ ↦ r) ≤ s.indicator (fun b ↦ (κCond (a, b)) Set.univ) := by
intro b
by_cases hbs : b ∈ s
· simpa [hbs]
· simp [hbs]
have : ∫⁻ b, s.indicator (fun _ ↦ r) b ∂(κ.fst a) ≤ κ.fst a s :=
(lintegral_mono h_2_le).trans_eq (h_eq s hs)
rw [lintegral_indicator_const hs] at this
contrapose! this with h_ne_zero
conv_lhs => rw [← one_mul (κ.fst a s)]
exact ENNReal.mul_lt_mul_right' h_ne_zero (measure_ne_top _ _) hr
· rw [ae_const_le_iff_forall_lt_measure_zero]
intro r hr
let s := {b | κCond (a, b) Set.univ ≤ r}
have hs : MeasurableSet s := h_meas measurableSet_Iic
have h_2_le : s.indicator (fun b ↦ (κCond (a, b)) Set.univ) ≤ s.indicator (fun _ ↦ r) := by
intro b
by_cases hbs : b ∈ s
· simpa [hbs]
· simp [hbs]
have : κ.fst a s ≤ ∫⁻ b, s.indicator (fun _ ↦ r) b ∂(κ.fst a) :=
(h_eq s hs).symm.trans_le (lintegral_mono h_2_le)
rw [lintegral_indicator_const hs] at this
contrapose! this with h_ne_zero
conv_rhs => rw [← one_mul (κ.fst a s)]
exact ENNReal.mul_lt_mul_right' h_ne_zero (measure_ne_top _ _) hr
/-! #### Existence of a disintegrating kernel in a countable space -/ | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfLIntegral",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Disintegration/Basic.lean | IsCondKernel.isProbabilityMeasure_ae | A conditional kernel is almost everywhere a probability measure. |
noncomputable condKernelCountable (h_atom : ∀ x y, x ∈ measurableAtom y → κCond x = κCond y) :
Kernel (α × β) Ω where
toFun p := κCond p.1 p.2
measurable' := by
refine measurable_from_prod_countable_right' (fun a ↦ (κCond a).measurable) fun x y hx hy ↦ ?_
simpa using DFunLike.congr (h_atom _ _ hy) rfl | def | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfLIntegral",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Disintegration/Basic.lean | condKernelCountable | Auxiliary definition for `ProbabilityTheory.Kernel.condKernel`.
A conditional kernel for `κ : Kernel α (β × Ω)` where `α` is countable and `Ω` is a measurable
space. |
condKernelCountable_apply (h_atom : ∀ x y, x ∈ measurableAtom y → κCond x = κCond y)
(p : α × β) : condKernelCountable κCond h_atom p = κCond p.1 p.2 := rfl | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfLIntegral",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Disintegration/Basic.lean | condKernelCountable_apply | null |
condKernelCountable.instIsMarkovKernel [∀ a, IsMarkovKernel (κCond a)]
(h_atom : ∀ x y, x ∈ measurableAtom y → κCond x = κCond y) :
IsMarkovKernel (condKernelCountable κCond h_atom) where
isProbabilityMeasure p := (‹∀ a, IsMarkovKernel (κCond a)› p.1).isProbabilityMeasure p.2 | instance | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfLIntegral",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Disintegration/Basic.lean | condKernelCountable.instIsMarkovKernel | null |
condKernelCountable.instIsCondKernel [∀ a, IsMarkovKernel (κCond a)]
(h_atom : ∀ x y, x ∈ measurableAtom y → κCond x = κCond y) (κ : Kernel α (β × Ω))
[IsSFiniteKernel κ] [∀ a, (κ a).IsCondKernel (κCond a)] :
κ.IsCondKernel (condKernelCountable κCond h_atom) := by
constructor
ext a s hs
conv_rhs => rw [← (κ a).disintegrate (κCond a)]
simp_rw [compProd_apply hs, condKernelCountable_apply, Measure.compProd_apply hs]
congr | instance | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfLIntegral",
"Mathlib.Probability.Kernel.Composition.MeasureCompProd"
] | Mathlib/Probability/Kernel/Disintegration/Basic.lean | condKernelCountable.instIsCondKernel | null |
IsRatCondKernelCDF (f : α × β → ℚ → ℝ) (κ : Kernel α (β × ℝ)) (ν : Kernel α β) :
Prop where
measurable : Measurable f
isRatStieltjesPoint_ae (a : α) : ∀ᵐ b ∂(ν a), IsRatStieltjesPoint f (a, b)
integrable (a : α) (q : ℚ) : Integrable (fun b ↦ f (a, b) q) (ν a)
setIntegral (a : α) {s : Set β} (_hs : MeasurableSet s) (q : ℚ) :
∫ b in s, f (a, b) q ∂(ν a) = (κ a).real (s ×ˢ Iic (q : ℝ)) | structure | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsRatCondKernelCDF | a function `f : α × β → ℚ → ℝ` is called a rational conditional kernel CDF of `κ` with respect
to `ν` if is measurable, if `fun b ↦ f (a, b) x` is `(ν a)`-integrable for all `a : α` and `x : ℝ`
and for all measurable sets `s : Set β`, `∫ b in s, f (a, b) x ∂(ν a) = (κ a).real (s ×ˢ Iic x)`.
Also the `ℚ → ℝ` function `f (a, b)` should satisfy the properties of a Stieltjes function for
`(ν a)`-almost all `b : β`. |
IsRatCondKernelCDF.mono (hf : IsRatCondKernelCDF f κ ν) (a : α) :
∀ᵐ b ∂(ν a), Monotone (f (a, b)) := by
filter_upwards [hf.isRatStieltjesPoint_ae a] with b hb using hb.mono | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsRatCondKernelCDF.mono | null |
IsRatCondKernelCDF.tendsto_atTop_one (hf : IsRatCondKernelCDF f κ ν) (a : α) :
∀ᵐ b ∂(ν a), Tendsto (f (a, b)) atTop (𝓝 1) := by
filter_upwards [hf.isRatStieltjesPoint_ae a] with b hb using hb.tendsto_atTop_one | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsRatCondKernelCDF.tendsto_atTop_one | null |
IsRatCondKernelCDF.tendsto_atBot_zero (hf : IsRatCondKernelCDF f κ ν) (a : α) :
∀ᵐ b ∂(ν a), Tendsto (f (a, b)) atBot (𝓝 0) := by
filter_upwards [hf.isRatStieltjesPoint_ae a] with b hb using hb.tendsto_atBot_zero | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsRatCondKernelCDF.tendsto_atBot_zero | null |
IsRatCondKernelCDF.iInf_rat_gt_eq (hf : IsRatCondKernelCDF f κ ν) (a : α) :
∀ᵐ b ∂(ν a), ∀ q, ⨅ r : Ioi q, f (a, b) r = f (a, b) q := by
filter_upwards [hf.isRatStieltjesPoint_ae a] with b hb using hb.iInf_rat_gt_eq | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsRatCondKernelCDF.iInf_rat_gt_eq | null |
stieltjesOfMeasurableRat_ae_eq (hf : IsRatCondKernelCDF f κ ν) (a : α) (q : ℚ) :
(fun b ↦ stieltjesOfMeasurableRat f hf.measurable (a, b) q) =ᵐ[ν a] fun b ↦ f (a, b) q := by
filter_upwards [hf.isRatStieltjesPoint_ae a] with a ha
rw [stieltjesOfMeasurableRat_eq, toRatCDF_of_isRatStieltjesPoint ha] | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | stieltjesOfMeasurableRat_ae_eq | null |
setIntegral_stieltjesOfMeasurableRat_rat (hf : IsRatCondKernelCDF f κ ν) (a : α) (q : ℚ)
{s : Set β} (hs : MeasurableSet s) :
∫ b in s, stieltjesOfMeasurableRat f hf.measurable (a, b) q ∂(ν a)
= (κ a).real (s ×ˢ Iic (q : ℝ)) := by
rw [setIntegral_congr_ae hs (g := fun b ↦ f (a, b) q) ?_, hf.setIntegral a hs]
filter_upwards [stieltjesOfMeasurableRat_ae_eq hf a q] with b hb using fun _ ↦ hb | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | setIntegral_stieltjesOfMeasurableRat_rat | null |
setLIntegral_stieltjesOfMeasurableRat_rat [IsFiniteKernel κ] (hf : IsRatCondKernelCDF f κ ν)
(a : α) (q : ℚ) {s : Set β} (hs : MeasurableSet s) :
∫⁻ b in s, ENNReal.ofReal (stieltjesOfMeasurableRat f hf.measurable (a, b) q) ∂(ν a)
= κ a (s ×ˢ Iic (q : ℝ)) := by
rw [← ofReal_integral_eq_lintegral_ofReal]
· rw [setIntegral_stieltjesOfMeasurableRat_rat hf a q hs, ofReal_measureReal]
· refine Integrable.restrict ?_
rw [integrable_congr (stieltjesOfMeasurableRat_ae_eq hf a q)]
exact hf.integrable a q
· exact ae_of_all _ (fun x ↦ stieltjesOfMeasurableRat_nonneg _ _ _) | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | setLIntegral_stieltjesOfMeasurableRat_rat | null |
setLIntegral_stieltjesOfMeasurableRat [IsFiniteKernel κ] (hf : IsRatCondKernelCDF f κ ν)
(a : α) (x : ℝ) {s : Set β} (hs : MeasurableSet s) :
∫⁻ b in s, ENNReal.ofReal (stieltjesOfMeasurableRat f hf.measurable (a, b) x) ∂(ν a)
= κ a (s ×ˢ Iic x) := by
by_cases hρ_zero : (ν a).restrict s = 0
· rw [hρ_zero, lintegral_zero_measure]
have ⟨q, hq⟩ := exists_rat_gt x
suffices κ a (s ×ˢ Iic (q : ℝ)) = 0 by
symm
refine measure_mono_null (fun p ↦ ?_) this
simp only [mem_prod, mem_Iic, and_imp]
exact fun h1 h2 ↦ ⟨h1, h2.trans hq.le⟩
suffices (κ a).real (s ×ˢ Iic (q : ℝ)) = 0 by
rw [measureReal_eq_zero_iff] at this
simpa [measure_ne_top] using this
rw [← hf.setIntegral a hs q]
simp [hρ_zero]
have h : ∫⁻ b in s, ENNReal.ofReal (stieltjesOfMeasurableRat f hf.measurable (a, b) x) ∂(ν a)
= ∫⁻ b in s, ⨅ r : { r' : ℚ // x < r' },
ENNReal.ofReal (stieltjesOfMeasurableRat f hf.measurable (a, b) r) ∂(ν a) := by
congr with b : 1
simp_rw [← measure_stieltjesOfMeasurableRat_Iic]
rw [← Monotone.measure_iInter]
· congr with y : 1
simp only [mem_Iic, mem_iInter, Subtype.forall]
refine ⟨fun h a ha ↦ h.trans ?_, fun h ↦ ?_⟩
· exact mod_cast ha.le
· refine le_of_forall_lt_rat_imp_le fun q hq ↦ h q ?_
exact mod_cast hq
· exact fun r r' hrr' ↦ Iic_subset_Iic.mpr <| mod_cast hrr'
· exact fun _ ↦ nullMeasurableSet_Iic
· obtain ⟨q, hq⟩ := exists_rat_gt x
exact ⟨⟨q, hq⟩, measure_ne_top _ _⟩
have h_nonempty : Nonempty { r' : ℚ // x < ↑r' } := by
obtain ⟨r, hrx⟩ := exists_rat_gt x
exact ⟨⟨r, hrx⟩⟩
rw [h, lintegral_iInf_directed_of_measurable hρ_zero fun q : { r' : ℚ // x < ↑r' } ↦ ?_]
rotate_left
· intro b
rw [setLIntegral_stieltjesOfMeasurableRat_rat hf a _ hs]
exact measure_ne_top _ _
· refine Monotone.directed_ge fun i j hij b ↦ ?_
simp_rw [← measure_stieltjesOfMeasurableRat_Iic]
refine measure_mono (Iic_subset_Iic.mpr ?_)
exact mod_cast hij
· refine Measurable.ennreal_ofReal ?_
exact (measurable_stieltjesOfMeasurableRat hf.measurable _).comp measurable_prodMk_left
simp_rw [setLIntegral_stieltjesOfMeasurableRat_rat hf _ _ hs]
rw [← Monotone.measure_iInter]
· rw [← prod_iInter]
congr with y
... | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | setLIntegral_stieltjesOfMeasurableRat | null |
lintegral_stieltjesOfMeasurableRat [IsFiniteKernel κ] (hf : IsRatCondKernelCDF f κ ν)
(a : α) (x : ℝ) :
∫⁻ b, ENNReal.ofReal (stieltjesOfMeasurableRat f hf.measurable (a, b) x) ∂(ν a)
= κ a (univ ×ˢ Iic x) := by
rw [← setLIntegral_univ, setLIntegral_stieltjesOfMeasurableRat hf _ _ MeasurableSet.univ] | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | lintegral_stieltjesOfMeasurableRat | null |
integrable_stieltjesOfMeasurableRat [IsFiniteKernel κ] (hf : IsRatCondKernelCDF f κ ν)
(a : α) (x : ℝ) :
Integrable (fun b ↦ stieltjesOfMeasurableRat f hf.measurable (a, b) x) (ν a) := by
have : (fun b ↦ stieltjesOfMeasurableRat f hf.measurable (a, b) x)
= fun b ↦ (ENNReal.ofReal (stieltjesOfMeasurableRat f hf.measurable (a, b) x)).toReal := by
ext t
rw [ENNReal.toReal_ofReal]
exact stieltjesOfMeasurableRat_nonneg _ _ _
rw [this]
refine integrable_toReal_of_lintegral_ne_top ?_ ?_
· refine (Measurable.ennreal_ofReal ?_).aemeasurable
exact (measurable_stieltjesOfMeasurableRat hf.measurable x).comp measurable_prodMk_left
· rw [lintegral_stieltjesOfMeasurableRat hf]
exact measure_ne_top _ _ | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | integrable_stieltjesOfMeasurableRat | null |
setIntegral_stieltjesOfMeasurableRat [IsFiniteKernel κ] (hf : IsRatCondKernelCDF f κ ν)
(a : α) (x : ℝ) {s : Set β} (hs : MeasurableSet s) :
∫ b in s, stieltjesOfMeasurableRat f hf.measurable (a, b) x ∂(ν a)
= (κ a).real (s ×ˢ Iic x) := by
rw [← ENNReal.ofReal_eq_ofReal_iff, ofReal_measureReal]
rotate_left
· exact setIntegral_nonneg hs (fun _ _ ↦ stieltjesOfMeasurableRat_nonneg _ _ _)
· exact ENNReal.toReal_nonneg
rw [ofReal_integral_eq_lintegral_ofReal, setLIntegral_stieltjesOfMeasurableRat hf _ _ hs]
· exact (integrable_stieltjesOfMeasurableRat hf _ _).restrict
· exact ae_of_all _ (fun _ ↦ stieltjesOfMeasurableRat_nonneg _ _ _) | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | setIntegral_stieltjesOfMeasurableRat | null |
integral_stieltjesOfMeasurableRat [IsFiniteKernel κ] (hf : IsRatCondKernelCDF f κ ν)
(a : α) (x : ℝ) :
∫ b, stieltjesOfMeasurableRat f hf.measurable (a, b) x ∂(ν a)
= (κ a).real (univ ×ˢ Iic x) := by
rw [← setIntegral_univ, setIntegral_stieltjesOfMeasurableRat hf _ _ MeasurableSet.univ] | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | integral_stieltjesOfMeasurableRat | null |
IsRatCondKernelCDFAux (f : α × β → ℚ → ℝ) (κ : Kernel α (β × ℝ)) (ν : Kernel α β) :
Prop where
measurable : Measurable f
mono' (a : α) {q r : ℚ} (_hqr : q ≤ r) : ∀ᵐ c ∂(ν a), f (a, c) q ≤ f (a, c) r
nonneg' (a : α) (q : ℚ) : ∀ᵐ c ∂(ν a), 0 ≤ f (a, c) q
le_one' (a : α) (q : ℚ) : ∀ᵐ c ∂(ν a), f (a, c) q ≤ 1
/- Same as `Tendsto (fun q : ℚ ↦ ∫ c, f (a, c) q ∂(ν a)) atBot (𝓝 0)` but slightly easier
to prove in the current applications of this definition (some integral convergence lemmas
currently apply only to `ℕ`, not `ℚ`) -/
tendsto_integral_of_antitone (a : α) (seq : ℕ → ℚ) (_hs : Antitone seq)
(_hs_tendsto : Tendsto seq atTop atBot) :
Tendsto (fun m ↦ ∫ c, f (a, c) (seq m) ∂(ν a)) atTop (𝓝 0)
/- Same as `Tendsto (fun q : ℚ ↦ ∫ c, f (a, c) q ∂(ν a)) atTop (𝓝 ((ν a).real univ))` but
slightly easier to prove in the current applications of this definition (some integral convergence
lemmas currently apply only to `ℕ`, not `ℚ`) -/
tendsto_integral_of_monotone (a : α) (seq : ℕ → ℚ) (_hs : Monotone seq)
(_hs_tendsto : Tendsto seq atTop atTop) :
Tendsto (fun m ↦ ∫ c, f (a, c) (seq m) ∂(ν a)) atTop (𝓝 ((ν a).real univ))
integrable (a : α) (q : ℚ) : Integrable (fun c ↦ f (a, c) q) (ν a)
setIntegral (a : α) {A : Set β} (_hA : MeasurableSet A) (q : ℚ) :
∫ c in A, f (a, c) q ∂(ν a) = (κ a).real (A ×ˢ Iic ↑q) | structure | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsRatCondKernelCDFAux | This property implies `IsRatCondKernelCDF`. The measurability, integrability and integral
conditions are the same, but the limit properties of `IsRatCondKernelCDF` are replaced by
limits of integrals. |
IsRatCondKernelCDFAux.measurable_right (hf : IsRatCondKernelCDFAux f κ ν) (a : α) (q : ℚ) :
Measurable (fun t ↦ f (a, t) q) := by
let h := hf.measurable
rw [measurable_pi_iff] at h
exact (h q).comp measurable_prodMk_left | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsRatCondKernelCDFAux.measurable_right | null |
IsRatCondKernelCDFAux.mono (hf : IsRatCondKernelCDFAux f κ ν) (a : α) :
∀ᵐ c ∂(ν a), Monotone (f (a, c)) := by
unfold Monotone
simp_rw [ae_all_iff]
exact fun _ _ hqr ↦ hf.mono' a hqr | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsRatCondKernelCDFAux.mono | null |
IsRatCondKernelCDFAux.nonneg (hf : IsRatCondKernelCDFAux f κ ν) (a : α) :
∀ᵐ c ∂(ν a), ∀ q, 0 ≤ f (a, c) q := ae_all_iff.mpr <| hf.nonneg' a | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsRatCondKernelCDFAux.nonneg | null |
IsRatCondKernelCDFAux.le_one (hf : IsRatCondKernelCDFAux f κ ν) (a : α) :
∀ᵐ c ∂(ν a), ∀ q, f (a, c) q ≤ 1 := ae_all_iff.mpr <| hf.le_one' a | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsRatCondKernelCDFAux.le_one | null |
IsRatCondKernelCDFAux.tendsto_zero_of_antitone (hf : IsRatCondKernelCDFAux f κ ν)
[IsFiniteKernel ν] (a : α) (seq : ℕ → ℚ) (hseq : Antitone seq)
(hseq_tendsto : Tendsto seq atTop atBot) :
∀ᵐ c ∂(ν a), Tendsto (fun m ↦ f (a, c) (seq m)) atTop (𝓝 0) := by
refine tendsto_of_integral_tendsto_of_antitone ?_ (integrable_const _) ?_ ?_ ?_
· exact fun n ↦ hf.integrable a (seq n)
· rw [integral_zero]
exact hf.tendsto_integral_of_antitone a seq hseq hseq_tendsto
· filter_upwards [hf.mono a] with t ht using fun n m hnm ↦ ht (hseq hnm)
· filter_upwards [hf.nonneg a] with c hc using fun i ↦ hc (seq i) | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsRatCondKernelCDFAux.tendsto_zero_of_antitone | null |
IsRatCondKernelCDFAux.tendsto_one_of_monotone (hf : IsRatCondKernelCDFAux f κ ν)
[IsFiniteKernel ν] (a : α) (seq : ℕ → ℚ) (hseq : Monotone seq)
(hseq_tendsto : Tendsto seq atTop atTop) :
∀ᵐ c ∂(ν a), Tendsto (fun m ↦ f (a, c) (seq m)) atTop (𝓝 1) := by
refine tendsto_of_integral_tendsto_of_monotone ?_ (integrable_const _) ?_ ?_ ?_
· exact fun n ↦ hf.integrable a (seq n)
· rw [MeasureTheory.integral_const, smul_eq_mul, mul_one]
exact hf.tendsto_integral_of_monotone a seq hseq hseq_tendsto
· filter_upwards [hf.mono a] with t ht using fun n m hnm ↦ ht (hseq hnm)
· filter_upwards [hf.le_one a] with c hc using fun i ↦ hc (seq i) | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsRatCondKernelCDFAux.tendsto_one_of_monotone | null |
IsRatCondKernelCDFAux.tendsto_atTop_one (hf : IsRatCondKernelCDFAux f κ ν) [IsFiniteKernel ν]
(a : α) :
∀ᵐ t ∂(ν a), Tendsto (f (a, t)) atTop (𝓝 1) := by
suffices ∀ᵐ t ∂(ν a), Tendsto (fun (n : ℕ) ↦ f (a, t) n) atTop (𝓝 1) by
filter_upwards [this, hf.mono a] with t ht h_mono
rw [tendsto_iff_tendsto_subseq_of_monotone h_mono tendsto_natCast_atTop_atTop]
exact ht
filter_upwards [hf.tendsto_one_of_monotone a Nat.cast Nat.mono_cast tendsto_natCast_atTop_atTop]
with x hx using hx | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsRatCondKernelCDFAux.tendsto_atTop_one | null |
IsRatCondKernelCDFAux.tendsto_atBot_zero (hf : IsRatCondKernelCDFAux f κ ν) [IsFiniteKernel ν]
(a : α) :
∀ᵐ t ∂(ν a), Tendsto (f (a, t)) atBot (𝓝 0) := by
suffices ∀ᵐ t ∂(ν a), Tendsto (fun q : ℚ ↦ f (a, t) (-q)) atTop (𝓝 0) by
filter_upwards [this] with t ht
have h_eq_neg : f (a, t) = fun q : ℚ ↦ f (a, t) (- -q) := by
simp_rw [neg_neg]
rw [h_eq_neg]
convert ht.comp tendsto_neg_atBot_atTop
simp
suffices ∀ᵐ t ∂(ν a), Tendsto (fun (n : ℕ) ↦ f (a, t) (-n)) atTop (𝓝 0) by
filter_upwards [this, hf.mono a] with t ht h_mono
have h_anti : Antitone (fun q ↦ f (a, t) (-q)) := h_mono.comp_antitone monotone_id.neg
exact (tendsto_iff_tendsto_subseq_of_antitone h_anti tendsto_natCast_atTop_atTop).mpr ht
exact hf.tendsto_zero_of_antitone _ _ Nat.mono_cast.neg
(tendsto_neg_atBot_iff.mpr tendsto_natCast_atTop_atTop) | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsRatCondKernelCDFAux.tendsto_atBot_zero | null |
IsRatCondKernelCDFAux.bddBelow_range (hf : IsRatCondKernelCDFAux f κ ν) (a : α) :
∀ᵐ t ∂(ν a), ∀ q : ℚ, BddBelow (range fun (r : Ioi q) ↦ f (a, t) r) := by
filter_upwards [hf.nonneg a] with c hc
refine fun q ↦ ⟨0, ?_⟩
simp [mem_lowerBounds, hc] | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsRatCondKernelCDFAux.bddBelow_range | null |
IsRatCondKernelCDFAux.integrable_iInf_rat_gt (hf : IsRatCondKernelCDFAux f κ ν)
[IsFiniteKernel ν] (a : α) (q : ℚ) :
Integrable (fun t ↦ ⨅ r : Ioi q, f (a, t) r) (ν a) := by
rw [← memLp_one_iff_integrable]
refine ⟨(Measurable.iInf fun i ↦ hf.measurable_right a _).aestronglyMeasurable, ?_⟩
refine (?_ : _ ≤ (ν a univ : ℝ≥0∞)).trans_lt (measure_lt_top _ _)
refine (eLpNorm_le_of_ae_bound (C := 1) ?_).trans (by simp)
filter_upwards [hf.bddBelow_range a, hf.nonneg a, hf.le_one a]
with t hbdd_below h_nonneg h_le_one
rw [Real.norm_eq_abs, abs_of_nonneg]
· refine ciInf_le_of_le ?_ ?_ ?_
· exact hbdd_below _
· exact ⟨q + 1, by simp⟩
· exact h_le_one _
· exact le_ciInf fun r ↦ h_nonneg _ | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsRatCondKernelCDFAux.integrable_iInf_rat_gt | null |
_root_.MeasureTheory.Measure.iInf_rat_gt_prod_Iic {ρ : Measure (α × ℝ)} [IsFiniteMeasure ρ]
{s : Set α} (hs : MeasurableSet s) (t : ℚ) :
⨅ r : { r' : ℚ // t < r' }, ρ (s ×ˢ Iic (r : ℝ)) = ρ (s ×ˢ Iic (t : ℝ)) := by
rw [← Monotone.measure_iInter]
· rw [← prod_iInter]
congr with x : 1
simp only [mem_iInter, mem_Iic, Subtype.forall]
refine ⟨fun h ↦ ?_, fun h a hta ↦ h.trans ?_⟩
· refine le_of_forall_lt_rat_imp_le fun q htq ↦ h q ?_
exact mod_cast htq
· exact mod_cast hta.le
· exact fun r r' hrr' ↦ prod_mono_right <| by gcongr
· exact fun _ => (hs.prod measurableSet_Iic).nullMeasurableSet
· exact ⟨⟨t + 1, lt_add_one _⟩, measure_ne_top ρ _⟩ | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | _root_.MeasureTheory.Measure.iInf_rat_gt_prod_Iic | null |
IsRatCondKernelCDFAux.setIntegral_iInf_rat_gt (hf : IsRatCondKernelCDFAux f κ ν)
[IsFiniteKernel κ] [IsFiniteKernel ν] (a : α) (q : ℚ) {A : Set β} (hA : MeasurableSet A) :
∫ t in A, ⨅ r : Ioi q, f (a, t) r ∂(ν a) = (κ a).real (A ×ˢ Iic (q : ℝ)) := by
refine le_antisymm ?_ ?_
· have h : ∀ r : Ioi q, ∫ t in A, ⨅ r' : Ioi q, f (a, t) r' ∂(ν a)
≤ (κ a).real (A ×ˢ Iic (r : ℝ)) := by
intro r
rw [← hf.setIntegral a hA]
refine setIntegral_mono_ae ?_ ?_ ?_
· exact (hf.integrable_iInf_rat_gt _ _).integrableOn
· exact (hf.integrable _ _).integrableOn
· filter_upwards [hf.bddBelow_range a] with t ht using ciInf_le (ht _) r
calc ∫ t in A, ⨅ r : Ioi q, f (a, t) r ∂(ν a)
≤ ⨅ r : Ioi q, (κ a).real (A ×ˢ Iic (r : ℝ)) := le_ciInf h
_ = (κ a).real (A ×ˢ Iic (q : ℝ)) := by
rw [measureReal_def, ← Measure.iInf_rat_gt_prod_Iic hA q]
exact (ENNReal.toReal_iInf (fun r ↦ measure_ne_top _ _)).symm
· rw [← hf.setIntegral a hA]
refine setIntegral_mono_ae ?_ ?_ ?_
· exact (hf.integrable _ _).integrableOn
· exact (hf.integrable_iInf_rat_gt _ _).integrableOn
· filter_upwards [hf.mono a] with c h_mono using le_ciInf (fun r ↦ h_mono (le_of_lt r.prop)) | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsRatCondKernelCDFAux.setIntegral_iInf_rat_gt | null |
IsRatCondKernelCDFAux.iInf_rat_gt_eq (hf : IsRatCondKernelCDFAux f κ ν) [IsFiniteKernel κ]
[IsFiniteKernel ν] (a : α) :
∀ᵐ t ∂(ν a), ∀ q : ℚ, ⨅ r : Ioi q, f (a, t) r = f (a, t) q := by
rw [ae_all_iff]
refine fun q ↦ ae_eq_of_forall_setIntegral_eq_of_sigmaFinite (μ := ν a) ?_ ?_ ?_
· exact fun _ _ _ ↦ (hf.integrable_iInf_rat_gt _ _).integrableOn
· exact fun _ _ _ ↦ (hf.integrable a _).integrableOn
· intro s hs _
rw [hf.setIntegral _ hs, hf.setIntegral_iInf_rat_gt _ _ hs] | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsRatCondKernelCDFAux.iInf_rat_gt_eq | null |
IsRatCondKernelCDFAux.isRatStieltjesPoint_ae (hf : IsRatCondKernelCDFAux f κ ν)
[IsFiniteKernel κ] [IsFiniteKernel ν] (a : α) :
∀ᵐ t ∂(ν a), IsRatStieltjesPoint f (a, t) := by
filter_upwards [hf.tendsto_atTop_one a, hf.tendsto_atBot_zero a,
hf.iInf_rat_gt_eq a, hf.mono a] with t ht_top ht_bot ht_iInf h_mono
exact ⟨h_mono, ht_top, ht_bot, ht_iInf⟩ | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsRatCondKernelCDFAux.isRatStieltjesPoint_ae | null |
IsRatCondKernelCDFAux.isRatCondKernelCDF (hf : IsRatCondKernelCDFAux f κ ν) [IsFiniteKernel κ]
[IsFiniteKernel ν] :
IsRatCondKernelCDF f κ ν where
measurable := hf.measurable
isRatStieltjesPoint_ae := hf.isRatStieltjesPoint_ae
integrable := hf.integrable
setIntegral := hf.setIntegral | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsRatCondKernelCDFAux.isRatCondKernelCDF | null |
IsCondKernelCDF (f : α × β → StieltjesFunction) (κ : Kernel α (β × ℝ)) (ν : Kernel α β) :
Prop where
measurable (x : ℝ) : Measurable fun p ↦ f p x
integrable (a : α) (x : ℝ) : Integrable (fun b ↦ f (a, b) x) (ν a)
tendsto_atTop_one (p : α × β) : Tendsto (f p) atTop (𝓝 1)
tendsto_atBot_zero (p : α × β) : Tendsto (f p) atBot (𝓝 0)
setIntegral (a : α) {s : Set β} (_hs : MeasurableSet s) (x : ℝ) :
∫ b in s, f (a, b) x ∂(ν a) = (κ a).real (s ×ˢ Iic x) | structure | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsCondKernelCDF | A function `f : α × β → StieltjesFunction` is called a conditional kernel CDF of `κ` with
respect to `ν` if it is measurable, tends to 0 at -∞ and to 1 at +∞ for all `p : α × β`,
`fun b ↦ f (a, b) x` is `(ν a)`-integrable for all `a : α` and `x : ℝ` and for all
measurable sets `s : Set β`, `∫ b in s, f (a, b) x ∂(ν a) = (κ a).real (s ×ˢ Iic x)`. |
IsCondKernelCDF.nonneg (hf : IsCondKernelCDF f κ ν) (p : α × β) (x : ℝ) : 0 ≤ f p x :=
Monotone.le_of_tendsto (f p).mono (hf.tendsto_atBot_zero p) x | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsCondKernelCDF.nonneg | null |
IsCondKernelCDF.le_one (hf : IsCondKernelCDF f κ ν) (p : α × β) (x : ℝ) : f p x ≤ 1 :=
Monotone.ge_of_tendsto (f p).mono (hf.tendsto_atTop_one p) x | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsCondKernelCDF.le_one | null |
IsCondKernelCDF.integral
{f : α × β → StieltjesFunction} (hf : IsCondKernelCDF f κ ν) (a : α) (x : ℝ) :
∫ b, f (a, b) x ∂(ν a) = (κ a).real (univ ×ˢ Iic x) := by
rw [← hf.setIntegral _ MeasurableSet.univ, Measure.restrict_univ] | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsCondKernelCDF.integral | null |
IsCondKernelCDF.setLIntegral [IsFiniteKernel κ]
{f : α × β → StieltjesFunction} (hf : IsCondKernelCDF f κ ν)
(a : α) {s : Set β} (hs : MeasurableSet s) (x : ℝ) :
∫⁻ b in s, ENNReal.ofReal (f (a, b) x) ∂(ν a) = κ a (s ×ˢ Iic x) := by
rw [← ofReal_integral_eq_lintegral_ofReal (hf.integrable a x).restrict
(ae_of_all _ (fun _ ↦ hf.nonneg _ _)), hf.setIntegral a hs x, ofReal_measureReal] | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsCondKernelCDF.setLIntegral | null |
IsCondKernelCDF.lintegral [IsFiniteKernel κ]
{f : α × β → StieltjesFunction} (hf : IsCondKernelCDF f κ ν) (a : α) (x : ℝ) :
∫⁻ b, ENNReal.ofReal (f (a, b) x) ∂(ν a) = κ a (univ ×ˢ Iic x) := by
rw [← hf.setLIntegral _ MeasurableSet.univ, Measure.restrict_univ] | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsCondKernelCDF.lintegral | null |
isCondKernelCDF_stieltjesOfMeasurableRat {f : α × β → ℚ → ℝ} (hf : IsRatCondKernelCDF f κ ν)
[IsFiniteKernel κ] :
IsCondKernelCDF (stieltjesOfMeasurableRat f hf.measurable) κ ν where
measurable := measurable_stieltjesOfMeasurableRat hf.measurable
integrable := integrable_stieltjesOfMeasurableRat hf
tendsto_atTop_one := tendsto_stieltjesOfMeasurableRat_atTop hf.measurable
tendsto_atBot_zero := tendsto_stieltjesOfMeasurableRat_atBot hf.measurable
setIntegral a _ hs x := setIntegral_stieltjesOfMeasurableRat hf a x hs | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | isCondKernelCDF_stieltjesOfMeasurableRat | null |
noncomputable
IsCondKernelCDF.toKernel (f : α × β → StieltjesFunction) (hf : IsCondKernelCDF f κ ν) :
Kernel (α × β) ℝ where
toFun p := (f p).measure
measurable' := StieltjesFunction.measurable_measure hf.measurable
hf.tendsto_atBot_zero hf.tendsto_atTop_one | def | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsCondKernelCDF.toKernel | A function `f : α × β → StieltjesFunction` with the property `IsCondKernelCDF f κ ν` gives a
Markov kernel from `α × β` to `ℝ`, by taking for each `p : α × β` the measure defined by `f p`. |
IsCondKernelCDF.toKernel_apply {hf : IsCondKernelCDF f κ ν} (p : α × β) :
hf.toKernel f p = (f p).measure := rfl | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsCondKernelCDF.toKernel_apply | null |
instIsMarkovKernel_toKernel {hf : IsCondKernelCDF f κ ν} :
IsMarkovKernel (hf.toKernel f) :=
⟨fun _ ↦ (f _).isProbabilityMeasure (hf.tendsto_atBot_zero _) (hf.tendsto_atTop_one _)⟩ | instance | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | instIsMarkovKernel_toKernel | null |
IsCondKernelCDF.toKernel_Iic {hf : IsCondKernelCDF f κ ν} (p : α × β) (x : ℝ) :
hf.toKernel f p (Iic x) = ENNReal.ofReal (f p x) := by
rw [IsCondKernelCDF.toKernel_apply p, (f p).measure_Iic (hf.tendsto_atBot_zero p)]
simp | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | IsCondKernelCDF.toKernel_Iic | null |
setLIntegral_toKernel_Iic [IsFiniteKernel κ] (hf : IsCondKernelCDF f κ ν)
(a : α) (x : ℝ) {s : Set β} (hs : MeasurableSet s) :
∫⁻ b in s, hf.toKernel f (a, b) (Iic x) ∂(ν a) = κ a (s ×ˢ Iic x) := by
simp_rw [IsCondKernelCDF.toKernel_Iic]
exact hf.setLIntegral _ hs _ | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | setLIntegral_toKernel_Iic | null |
setLIntegral_toKernel_univ [IsFiniteKernel κ] (hf : IsCondKernelCDF f κ ν)
(a : α) {s : Set β} (hs : MeasurableSet s) :
∫⁻ b in s, hf.toKernel f (a, b) univ ∂(ν a) = κ a (s ×ˢ univ) := by
rw [← Real.iUnion_Iic_rat, prod_iUnion]
have h_dir : Directed (fun x y ↦ x ⊆ y) fun q : ℚ ↦ Iic (q : ℝ) := by
refine Monotone.directed_le fun r r' hrr' ↦ Iic_subset_Iic.mpr ?_
exact mod_cast hrr'
have h_dir_prod : Directed (fun x y ↦ x ⊆ y) fun q : ℚ ↦ s ×ˢ Iic (q : ℝ) := by
refine Monotone.directed_le fun i j hij ↦ ?_
refine prod_subset_prod_iff.mpr (Or.inl ⟨subset_rfl, Iic_subset_Iic.mpr ?_⟩)
exact mod_cast hij
simp_rw [h_dir.measure_iUnion, h_dir_prod.measure_iUnion]
rw [lintegral_iSup_directed]
· simp_rw [setLIntegral_toKernel_Iic hf _ _ hs]
· refine fun q ↦ Measurable.aemeasurable ?_
exact (Kernel.measurable_coe _ measurableSet_Iic).comp measurable_prodMk_left
· refine Monotone.directed_le fun i j hij t ↦ measure_mono (Iic_subset_Iic.mpr ?_)
exact mod_cast hij | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | setLIntegral_toKernel_univ | null |
lintegral_toKernel_univ [IsFiniteKernel κ] (hf : IsCondKernelCDF f κ ν) (a : α) :
∫⁻ b, hf.toKernel f (a, b) univ ∂(ν a) = κ a univ := by
rw [← setLIntegral_univ, setLIntegral_toKernel_univ hf a MeasurableSet.univ, univ_prod_univ] | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | lintegral_toKernel_univ | null |
setLIntegral_toKernel_prod [IsFiniteKernel κ] (hf : IsCondKernelCDF f κ ν)
(a : α) {s : Set β} (hs : MeasurableSet s) {t : Set ℝ} (ht : MeasurableSet t) :
∫⁻ b in s, hf.toKernel f (a, b) t ∂(ν a) = κ a (s ×ˢ t) := by
induction t, ht
using MeasurableSpace.induction_on_inter (borel_eq_generateFrom_Iic ℝ) isPiSystem_Iic with
| empty => simp only [measure_empty, lintegral_const, zero_mul, prod_empty]
| basic t ht =>
obtain ⟨q, rfl⟩ := ht
exact setLIntegral_toKernel_Iic hf a _ hs
| compl t ht iht =>
calc ∫⁻ b in s, hf.toKernel f (a, b) tᶜ ∂(ν a)
= ∫⁻ b in s, hf.toKernel f (a, b) univ - hf.toKernel f (a, b) t ∂(ν a) := by
congr with x; rw [measure_compl ht (measure_ne_top (hf.toKernel f (a, x)) _)]
_ = ∫⁻ b in s, hf.toKernel f (a, b) univ ∂(ν a)
- ∫⁻ b in s, hf.toKernel f (a, b) t ∂(ν a) := by
rw [lintegral_sub]
· exact (Kernel.measurable_coe (hf.toKernel f) ht).comp measurable_prodMk_left
· rw [iht]
exact measure_ne_top _ _
· exact Eventually.of_forall fun a ↦ measure_mono (subset_univ _)
_ = κ a (s ×ˢ univ) - κ a (s ×ˢ t) := by
rw [setLIntegral_toKernel_univ hf a hs, iht]
_ = κ a (s ×ˢ tᶜ) := by
rw [← measure_diff _ (hs.prod ht).nullMeasurableSet (measure_ne_top _ _)]
· rw [prod_diff_prod, compl_eq_univ_diff]
simp only [diff_self, empty_prod, union_empty]
· rw [prod_subset_prod_iff]
exact Or.inl ⟨subset_rfl, subset_univ t⟩
| iUnion f hf_disj hf_meas ihf =>
simp_rw [measure_iUnion hf_disj hf_meas]
rw [lintegral_tsum, prod_iUnion, measure_iUnion]
· simp_rw [ihf]
· exact hf_disj.mono fun i j h ↦ h.set_prod_right _ _
· exact fun i ↦ MeasurableSet.prod hs (hf_meas i)
· exact fun i ↦
((Kernel.measurable_coe _ (hf_meas i)).comp measurable_prodMk_left).aemeasurable.restrict
open scoped Function in -- required for scoped `on` notation | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | setLIntegral_toKernel_prod | null |
lintegral_toKernel_mem [IsFiniteKernel κ] (hf : IsCondKernelCDF f κ ν)
(a : α) {s : Set (β × ℝ)} (hs : MeasurableSet s) :
∫⁻ b, hf.toKernel f (a, b) (Prod.mk b ⁻¹' s) ∂(ν a) = κ a s := by
induction s, hs
using MeasurableSpace.induction_on_inter generateFrom_prod.symm isPiSystem_prod with
| empty =>
simp only [preimage_empty, measure_empty, lintegral_const, zero_mul]
| basic s hs =>
rcases hs with ⟨t₁, ht₁, t₂, ht₂, rfl⟩
simp only [mem_setOf_eq] at ht₁ ht₂
rw [← lintegral_add_compl _ ht₁]
have h_eq1 : ∫⁻ x in t₁, hf.toKernel f (a, x) (Prod.mk x ⁻¹' t₁ ×ˢ t₂) ∂(ν a)
= ∫⁻ x in t₁, hf.toKernel f (a, x) t₂ ∂(ν a) := by
refine setLIntegral_congr_fun ht₁ (fun a ha ↦ ?_)
rw [mk_preimage_prod_right ha]
have h_eq2 :
∫⁻ x in t₁ᶜ, hf.toKernel f (a, x) (Prod.mk x ⁻¹' t₁ ×ˢ t₂) ∂(ν a) = 0 := by
suffices h_eq_zero :
∀ x ∈ t₁ᶜ, hf.toKernel f (a, x) (Prod.mk x ⁻¹' t₁ ×ˢ t₂) = 0 by
rw [setLIntegral_congr_fun ht₁.compl h_eq_zero]
simp only [lintegral_const, zero_mul]
intro a hat₁
rw [mem_compl_iff] at hat₁
simp only [hat₁, not_false_eq_true, mk_preimage_prod_right_eq_empty, measure_empty]
rw [h_eq1, h_eq2, add_zero]
exact setLIntegral_toKernel_prod hf a ht₁ ht₂
| compl t ht ht_eq =>
calc ∫⁻ b, hf.toKernel f (a, b) (Prod.mk b ⁻¹' tᶜ) ∂(ν a)
= ∫⁻ b, hf.toKernel f (a, b) (Prod.mk b ⁻¹' t)ᶜ ∂(ν a) := rfl
_ = ∫⁻ b, hf.toKernel f (a, b) univ
- hf.toKernel f (a, b) (Prod.mk b ⁻¹' t) ∂(ν a) := by
congr with x : 1
exact measure_compl (measurable_prodMk_left ht)
(measure_ne_top (hf.toKernel f (a, x)) _)
_ = ∫⁻ x, hf.toKernel f (a, x) univ ∂(ν a) -
∫⁻ x, hf.toKernel f (a, x) (Prod.mk x ⁻¹' t) ∂(ν a) := by
have h_le : (fun x ↦ hf.toKernel f (a, x) (Prod.mk x ⁻¹' t))
≤ᵐ[ν a] fun x ↦ hf.toKernel f (a, x) univ :=
Eventually.of_forall fun _ ↦ measure_mono (subset_univ _)
rw [lintegral_sub _ _ h_le]
· exact Kernel.measurable_kernel_prodMk_left' ht a
refine ((lintegral_mono_ae h_le).trans_lt ?_).ne
rw [lintegral_toKernel_univ hf]
exact measure_lt_top _ univ
_ = κ a univ - κ a t := by rw [ht_eq, lintegral_toKernel_univ hf]
_ = κ a tᶜ := (measure_compl ht (measure_ne_top _ _)).symm
| iUnion f' hf_disj hf_meas hf_eq =>
have h_eq : ∀ a, Prod.mk a ⁻¹' ⋃ i, f' i = ⋃ i, Prod.mk a ⁻¹' f' i := by
simp only [preimage_iUnion, implies_true]
simp_rw [h_eq]
have h_disj : ∀ a, Pairwise (Disjoint on fun i ↦ Prod.mk a ⁻¹' f' i) := by
... | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | lintegral_toKernel_mem | null |
compProd_toKernel [IsFiniteKernel κ] [IsSFiniteKernel ν] (hf : IsCondKernelCDF f κ ν) :
ν ⊗ₖ hf.toKernel f = κ := by
ext a s hs
rw [Kernel.compProd_apply hs, lintegral_toKernel_mem hf a hs] | lemma | Probability | [
"Mathlib.MeasureTheory.Function.AEEqOfIntegral",
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes"
] | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | compProd_toKernel | null |
noncomputable IicSnd (r : ℝ) : Measure α :=
(ρ.restrict (univ ×ˢ Iic r)).fst | def | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | IicSnd | Measure on `α` such that for a measurable set `s`, `ρ.IicSnd r s = ρ (s ×ˢ Iic r)`. |
IicSnd_apply (r : ℝ) {s : Set α} (hs : MeasurableSet s) :
ρ.IicSnd r s = ρ (s ×ˢ Iic r) := by
rw [IicSnd, fst_apply hs, restrict_apply' (MeasurableSet.univ.prod measurableSet_Iic),
univ_prod, Set.prod_eq] | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | IicSnd_apply | null |
IicSnd_univ (r : ℝ) : ρ.IicSnd r univ = ρ (univ ×ˢ Iic r) :=
IicSnd_apply ρ r MeasurableSet.univ
@[gcongr] | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | IicSnd_univ | null |
IicSnd_mono {r r' : ℝ} (h_le : r ≤ r') : ρ.IicSnd r ≤ ρ.IicSnd r' := by
unfold IicSnd; gcongr | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | IicSnd_mono | null |
IicSnd_le_fst (r : ℝ) : ρ.IicSnd r ≤ ρ.fst :=
fst_mono restrict_le_self | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | IicSnd_le_fst | null |
IicSnd_ac_fst (r : ℝ) : ρ.IicSnd r ≪ ρ.fst :=
Measure.absolutelyContinuous_of_le (IicSnd_le_fst ρ r) | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | IicSnd_ac_fst | null |
IsFiniteMeasure.IicSnd {ρ : Measure (α × ℝ)} [IsFiniteMeasure ρ] (r : ℝ) :
IsFiniteMeasure (ρ.IicSnd r) :=
isFiniteMeasure_of_le _ (IicSnd_le_fst ρ _) | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | IsFiniteMeasure.IicSnd | null |
iInf_IicSnd_gt (t : ℚ) {s : Set α} (hs : MeasurableSet s) [IsFiniteMeasure ρ] :
⨅ r : { r' : ℚ // t < r' }, ρ.IicSnd r s = ρ.IicSnd t s := by
simp_rw [ρ.IicSnd_apply _ hs, Measure.iInf_rat_gt_prod_Iic hs] | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | iInf_IicSnd_gt | null |
tendsto_IicSnd_atTop {s : Set α} (hs : MeasurableSet s) :
Tendsto (fun r : ℚ ↦ ρ.IicSnd r s) atTop (𝓝 (ρ.fst s)) := by
simp_rw [ρ.IicSnd_apply _ hs, fst_apply hs, ← prod_univ]
rw [← Real.iUnion_Iic_rat, prod_iUnion]
apply tendsto_measure_iUnion_atTop
exact monotone_const.set_prod Rat.cast_mono.Iic | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | tendsto_IicSnd_atTop | null |
tendsto_IicSnd_atBot [IsFiniteMeasure ρ] {s : Set α} (hs : MeasurableSet s) :
Tendsto (fun r : ℚ ↦ ρ.IicSnd r s) atBot (𝓝 0) := by
simp_rw [ρ.IicSnd_apply _ hs]
have h_empty : ρ (s ×ˢ ∅) = 0 := by simp only [prod_empty, measure_empty]
rw [← h_empty, ← Real.iInter_Iic_rat, prod_iInter]
suffices h_neg :
Tendsto (fun r : ℚ ↦ ρ (s ×ˢ Iic ↑(-r))) atTop (𝓝 (ρ (⋂ r : ℚ, s ×ˢ Iic ↑(-r)))) by
have h_inter_eq : ⋂ r : ℚ, s ×ˢ Iic ↑(-r) = ⋂ r : ℚ, s ×ˢ Iic (r : ℝ) := by
ext1 x
simp only [mem_iInter, mem_prod, mem_Iic]
refine ⟨fun h i ↦ ⟨(h i).1, ?_⟩, fun h i ↦ ⟨(h i).1, ?_⟩⟩ <;> have h' := h (-i)
· rw [neg_neg] at h'; exact h'.2
· exact h'.2
rw [h_inter_eq] at h_neg
have h_fun_eq : (fun r : ℚ ↦ ρ (s ×ˢ Iic (r : ℝ))) = fun r : ℚ ↦ ρ (s ×ˢ Iic ↑(- -r)) := by
simp_rw [neg_neg]
rw [h_fun_eq]
exact h_neg.comp tendsto_neg_atBot_atTop
refine tendsto_measure_iInter_atTop (fun q ↦ (hs.prod measurableSet_Iic).nullMeasurableSet)
?_ ⟨0, measure_ne_top ρ _⟩
refine fun q r hqr ↦ Set.prod_mono subset_rfl fun x hx ↦ ?_
simp only [Rat.cast_neg, mem_Iic] at hx ⊢
refine hx.trans (neg_le_neg ?_)
exact mod_cast hqr | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | tendsto_IicSnd_atBot | null |
noncomputable preCDF (ρ : Measure (α × ℝ)) (r : ℚ) : α → ℝ≥0∞ :=
Measure.rnDeriv (ρ.IicSnd r) ρ.fst | def | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | preCDF | `preCDF` is the Radon-Nikodym derivative of `ρ.IicSnd` with respect to `ρ.fst` at each
`r : ℚ`. This function `ℚ → α → ℝ≥0∞` is such that for almost all `a : α`, the function `ℚ → ℝ≥0∞`
satisfies the properties of a cdf (monotone with limit 0 at -∞ and 1 at +∞, right-continuous).
We define this function on `ℚ` and not `ℝ` because `ℚ` is countable, which allows us to prove
properties of the form `∀ᵐ a ∂ρ.fst, ∀ q, P (preCDF q a)`, instead of the weaker
`∀ q, ∀ᵐ a ∂ρ.fst, P (preCDF q a)`. |
measurable_preCDF {ρ : Measure (α × ℝ)} {r : ℚ} : Measurable (preCDF ρ r) :=
Measure.measurable_rnDeriv _ _ | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | measurable_preCDF | null |
measurable_preCDF' {ρ : Measure (α × ℝ)} :
Measurable fun a r ↦ (preCDF ρ r a).toReal := by
rw [measurable_pi_iff]
exact fun _ ↦ measurable_preCDF.ennreal_toReal | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | measurable_preCDF' | null |
withDensity_preCDF (ρ : Measure (α × ℝ)) (r : ℚ) [IsFiniteMeasure ρ] :
ρ.fst.withDensity (preCDF ρ r) = ρ.IicSnd r :=
Measure.absolutelyContinuous_iff_withDensity_rnDeriv_eq.mp (Measure.IicSnd_ac_fst ρ r) | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | withDensity_preCDF | null |
setLIntegral_preCDF_fst (ρ : Measure (α × ℝ)) (r : ℚ) {s : Set α} (hs : MeasurableSet s)
[IsFiniteMeasure ρ] : ∫⁻ x in s, preCDF ρ r x ∂ρ.fst = ρ.IicSnd r s := by
have : ∀ r, ∫⁻ x in s, preCDF ρ r x ∂ρ.fst = ∫⁻ x in s, (preCDF ρ r * 1) x ∂ρ.fst := by
simp only [mul_one, forall_const]
rw [this, ← setLIntegral_withDensity_eq_setLIntegral_mul _ measurable_preCDF _ hs]
· simp only [withDensity_preCDF ρ r, Pi.one_apply, lintegral_one, Measure.restrict_apply,
MeasurableSet.univ, univ_inter]
· rw [Pi.one_def]
exact measurable_const | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | setLIntegral_preCDF_fst | null |
lintegral_preCDF_fst (ρ : Measure (α × ℝ)) (r : ℚ) [IsFiniteMeasure ρ] :
∫⁻ x, preCDF ρ r x ∂ρ.fst = ρ.IicSnd r univ := by
rw [← setLIntegral_univ, setLIntegral_preCDF_fst ρ r MeasurableSet.univ] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | lintegral_preCDF_fst | null |
monotone_preCDF (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] :
∀ᵐ a ∂ρ.fst, Monotone fun r ↦ preCDF ρ r a := by
simp_rw [Monotone, ae_all_iff]
refine fun r r' hrr' ↦ ae_le_of_forall_setLIntegral_le_of_sigmaFinite measurable_preCDF
fun s hs _ ↦ ?_
rw [setLIntegral_preCDF_fst ρ r hs, setLIntegral_preCDF_fst ρ r' hs]
exact Measure.IicSnd_mono ρ (mod_cast hrr') s | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | monotone_preCDF | null |
preCDF_le_one (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] :
∀ᵐ a ∂ρ.fst, ∀ r, preCDF ρ r a ≤ 1 := by
rw [ae_all_iff]
refine fun r ↦ ae_le_of_forall_setLIntegral_le_of_sigmaFinite measurable_preCDF fun s hs _ ↦ ?_
rw [setLIntegral_preCDF_fst ρ r hs]
simp only [lintegral_one, Measure.restrict_apply, MeasurableSet.univ, univ_inter]
exact Measure.IicSnd_le_fst ρ r s | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | preCDF_le_one | null |
setIntegral_preCDF_fst (ρ : Measure (α × ℝ)) (r : ℚ) {s : Set α} (hs : MeasurableSet s)
[IsFiniteMeasure ρ] :
∫ x in s, (preCDF ρ r x).toReal ∂ρ.fst = (ρ.IicSnd r).real s := by
rw [integral_toReal]
· rw [setLIntegral_preCDF_fst _ _ hs, measureReal_def]
· exact measurable_preCDF.aemeasurable
· refine ae_restrict_of_ae ?_
filter_upwards [preCDF_le_one ρ] with a ha
exact (ha r).trans_lt ENNReal.one_lt_top | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | setIntegral_preCDF_fst | null |
integral_preCDF_fst (ρ : Measure (α × ℝ)) (r : ℚ) [IsFiniteMeasure ρ] :
∫ x, (preCDF ρ r x).toReal ∂ρ.fst = (ρ.IicSnd r).real univ := by
rw [← setIntegral_univ, setIntegral_preCDF_fst ρ _ MeasurableSet.univ] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | integral_preCDF_fst | null |
integrable_preCDF (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] (x : ℚ) :
Integrable (fun a ↦ (preCDF ρ x a).toReal) ρ.fst := by
refine integrable_of_forall_fin_meas_le _ (measure_lt_top ρ.fst univ) ?_ fun t _ _ ↦ ?_
· exact measurable_preCDF.ennreal_toReal.aestronglyMeasurable
· simp_rw [← ofReal_norm_eq_enorm, Real.norm_of_nonneg ENNReal.toReal_nonneg]
rw [← lintegral_one]
refine (setLIntegral_le_lintegral _ _).trans (lintegral_mono_ae ?_)
filter_upwards [preCDF_le_one ρ] with a ha using ENNReal.ofReal_toReal_le.trans (ha _) | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | integrable_preCDF | null |
isRatCondKernelCDFAux_preCDF (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] :
IsRatCondKernelCDFAux (fun p r ↦ (preCDF ρ r p.2).toReal)
(Kernel.const Unit ρ) (Kernel.const Unit ρ.fst) where
measurable := measurable_preCDF'.comp measurable_snd
mono' a r r' hrr' := by
filter_upwards [monotone_preCDF ρ, preCDF_le_one ρ] with a h₁ h₂
exact ENNReal.toReal_mono ((h₂ _).trans_lt ENNReal.one_lt_top).ne (h₁ hrr')
nonneg' _ q := by simp
le_one' a q := by
simp only [Kernel.const_apply]
filter_upwards [preCDF_le_one ρ] with a ha
refine ENNReal.toReal_le_of_le_ofReal zero_le_one ?_
simp [ha]
tendsto_integral_of_antitone a s _ hs_tendsto := by
simp_rw [Kernel.const_apply, integral_preCDF_fst ρ]
have h := ρ.tendsto_IicSnd_atBot MeasurableSet.univ
rw [← ENNReal.toReal_zero]
have h0 : Tendsto ENNReal.toReal (𝓝 0) (𝓝 0) :=
ENNReal.continuousAt_toReal ENNReal.zero_ne_top
exact h0.comp (h.comp hs_tendsto)
tendsto_integral_of_monotone a s _ hs_tendsto := by
simp_rw [Kernel.const_apply, integral_preCDF_fst ρ]
have h := ρ.tendsto_IicSnd_atTop MeasurableSet.univ
have h0 : Tendsto ENNReal.toReal (𝓝 (ρ.fst univ)) (𝓝 (ρ.fst.real univ)) :=
ENNReal.continuousAt_toReal (measure_ne_top _ _)
exact h0.comp (h.comp hs_tendsto)
integrable _ q := integrable_preCDF ρ q
setIntegral a s hs q := by rw [Kernel.const_apply, Kernel.const_apply,
setIntegral_preCDF_fst _ _ hs, measureReal_def, measureReal_def, Measure.IicSnd_apply _ _ hs] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | isRatCondKernelCDFAux_preCDF | null |
isRatCondKernelCDF_preCDF (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] :
IsRatCondKernelCDF (fun p r ↦ (preCDF ρ r p.2).toReal)
(Kernel.const Unit ρ) (Kernel.const Unit ρ.fst) :=
(isRatCondKernelCDFAux_preCDF ρ).isRatCondKernelCDF
/-! ### Conditional cdf -/ | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | isRatCondKernelCDF_preCDF | null |
noncomputable condCDF (ρ : Measure (α × ℝ)) (a : α) : StieltjesFunction :=
stieltjesOfMeasurableRat (fun a r ↦ (preCDF ρ r a).toReal) measurable_preCDF' a | def | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | condCDF | Conditional cdf of the measure given the value on `α`, as a Stieltjes function. |
condCDF_eq_stieltjesOfMeasurableRat_unit_prod (ρ : Measure (α × ℝ)) (a : α) :
condCDF ρ a = stieltjesOfMeasurableRat (fun (p : Unit × α) r ↦ (preCDF ρ r p.2).toReal)
(measurable_preCDF'.comp measurable_snd) ((), a) := by
ext x
rw [condCDF, ← stieltjesOfMeasurableRat_unit_prod] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | condCDF_eq_stieltjesOfMeasurableRat_unit_prod | null |
isCondKernelCDF_condCDF (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] :
IsCondKernelCDF (fun p : Unit × α ↦ condCDF ρ p.2) (Kernel.const Unit ρ)
(Kernel.const Unit ρ.fst) := by
simp_rw [condCDF_eq_stieltjesOfMeasurableRat_unit_prod ρ]
exact isCondKernelCDF_stieltjesOfMeasurableRat (isRatCondKernelCDF_preCDF ρ) | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | isCondKernelCDF_condCDF | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.