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 ⌀ |
|---|---|---|---|---|---|---|
Integrable.norm_integral_condKernel {f : α × Ω → E} (hf_int : Integrable f ρ) :
Integrable (fun x ↦ ‖∫ y, f (x, y) ∂ρ.condKernel x‖) ρ.fst := by
refine hf_int.integral_norm_condKernel.mono hf_int.1.integral_condKernel.norm ?_
refine Filter.Eventually.of_forall fun x ↦ ?_
rw [norm_norm]
refine (norm_integral_le_integral_norm _).trans_eq (Real.norm_of_nonneg ?_).symm
exact integral_nonneg_of_ae (Filter.Eventually.of_forall fun y ↦ norm_nonneg _) | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Disintegration.StandardBorel"
] | Mathlib/Probability/Kernel/Disintegration/Integral.lean | Integrable.norm_integral_condKernel | null |
Integrable.integral_condKernel {f : α × Ω → E} (hf_int : Integrable f ρ) :
Integrable (fun x ↦ ∫ y, f (x, y) ∂ρ.condKernel x) ρ.fst :=
(integrable_norm_iff hf_int.1.integral_condKernel).mp hf_int.norm_integral_condKernel | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Disintegration.StandardBorel"
] | Mathlib/Probability/Kernel/Disintegration/Integral.lean | Integrable.integral_condKernel | null |
StieltjesFunction.measurable_measure {α : Type*} {_ : MeasurableSpace α}
{f : α → StieltjesFunction} (hf : ∀ q, Measurable fun a ↦ f a q)
(hf_bot : ∀ a, Tendsto (f a) atBot (𝓝 0))
(hf_top : ∀ a, Tendsto (f a) atTop (𝓝 1)) :
Measurable fun a ↦ (f a).measure :=
have : ∀ a, IsProbabilityMeasure (f a).measure :=
fun a ↦ (f a).isProbabilityMeasure (hf_bot a) (hf_top a)
.measure_of_isPiSystem_of_isProbabilityMeasure (borel_eq_generateFrom_Iic ℝ) isPiSystem_Iic <| by
simp_rw [forall_mem_range, StieltjesFunction.measure_Iic (f _) (hf_bot _), sub_zero]
exact fun _ ↦ (hf _).ennreal_ofReal | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | StieltjesFunction.measurable_measure | A measurable function `α → StieltjesFunction` with limits 0 at -∞ and 1 at +∞ gives a measurable
function `α → Measure ℝ` by taking `StieltjesFunction.measure` at each point. |
IsRatStieltjesPoint (f : α → ℚ → ℝ) (a : α) : Prop where
mono : Monotone (f a)
tendsto_atTop_one : Tendsto (f a) atTop (𝓝 1)
tendsto_atBot_zero : Tendsto (f a) atBot (𝓝 0)
iInf_rat_gt_eq : ∀ t : ℚ, ⨅ r : Ioi t, f a r = f a t | structure | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | IsRatStieltjesPoint | `a : α` is a Stieltjes point for `f : α → ℚ → ℝ` if `f a` is monotone with limit 0 at -∞
and 1 at +∞ and satisfies a continuity property. |
isRatStieltjesPoint_unit_prod_iff (f : α → ℚ → ℝ) (a : α) :
IsRatStieltjesPoint (fun p : Unit × α ↦ f p.2) ((), a)
↔ IsRatStieltjesPoint f a := by
constructor <;>
exact fun h ↦ ⟨h.mono, h.tendsto_atTop_one, h.tendsto_atBot_zero, h.iInf_rat_gt_eq⟩ | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | isRatStieltjesPoint_unit_prod_iff | null |
measurableSet_isRatStieltjesPoint [MeasurableSpace α] (hf : Measurable f) :
MeasurableSet {a | IsRatStieltjesPoint f a} := by
have h1 : MeasurableSet {a | Monotone (f a)} := by
change MeasurableSet {a | ∀ q r (_ : q ≤ r), f a q ≤ f a r}
simp_rw [Set.setOf_forall]
refine MeasurableSet.iInter (fun q ↦ ?_)
refine MeasurableSet.iInter (fun r ↦ ?_)
refine MeasurableSet.iInter (fun _ ↦ ?_)
exact measurableSet_le hf.eval hf.eval
have h2 : MeasurableSet {a | Tendsto (f a) atTop (𝓝 1)} :=
measurableSet_tendsto _ (fun q ↦ hf.eval)
have h3 : MeasurableSet {a | Tendsto (f a) atBot (𝓝 0)} :=
measurableSet_tendsto _ (fun q ↦ hf.eval)
have h4 : MeasurableSet {a | ∀ t : ℚ, ⨅ r : Ioi t, f a r = f a t} := by
rw [Set.setOf_forall]
refine MeasurableSet.iInter (fun q ↦ ?_)
exact measurableSet_eq_fun (.iInf fun _ ↦ hf.eval) hf.eval
suffices {a | IsRatStieltjesPoint f a}
= ({a | Monotone (f a)} ∩ {a | Tendsto (f a) atTop (𝓝 1)} ∩ {a | Tendsto (f a) atBot (𝓝 0)}
∩ {a | ∀ t : ℚ, ⨅ r : Ioi t, f a r = f a t}) by
rw [this]
exact (((h1.inter h2).inter h3).inter h4)
ext a
simp only [mem_setOf_eq, mem_inter_iff]
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· exact ⟨⟨⟨h.mono, h.tendsto_atTop_one⟩, h.tendsto_atBot_zero⟩, h.iInf_rat_gt_eq⟩
· exact ⟨h.1.1.1, h.1.1.2, h.1.2, h.2⟩ | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | measurableSet_isRatStieltjesPoint | null |
IsRatStieltjesPoint.ite {f g : α → ℚ → ℝ} {a : α} (p : α → Prop) [DecidablePred p]
(hf : p a → IsRatStieltjesPoint f a) (hg : ¬ p a → IsRatStieltjesPoint g a) :
IsRatStieltjesPoint (fun a ↦ if p a then f a else g a) a where
mono := by split_ifs with h; exacts [(hf h).mono, (hg h).mono]
tendsto_atTop_one := by
split_ifs with h; exacts [(hf h).tendsto_atTop_one, (hg h).tendsto_atTop_one]
tendsto_atBot_zero := by
split_ifs with h; exacts [(hf h).tendsto_atBot_zero, (hg h).tendsto_atBot_zero]
iInf_rat_gt_eq := by split_ifs with h; exacts [(hf h).iInf_rat_gt_eq, (hg h).iInf_rat_gt_eq]
variable [MeasurableSpace α] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | IsRatStieltjesPoint.ite | null |
IsMeasurableRatCDF (f : α → ℚ → ℝ) : Prop where
isRatStieltjesPoint : ∀ a, IsRatStieltjesPoint f a
measurable : Measurable f | structure | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | IsMeasurableRatCDF | A function `f : α → ℚ → ℝ` is a (kernel) rational cumulative distribution function if it is
measurable in the first argument and if `f a` satisfies a list of properties for all `a : α`:
monotonicity between 0 at -∞ and 1 at +∞ and a form of continuity.
A function with these properties can be extended to a measurable function `α → StieltjesFunction`.
See `ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunction`. |
IsMeasurableRatCDF.nonneg {f : α → ℚ → ℝ} (hf : IsMeasurableRatCDF f) (a : α) (q : ℚ) :
0 ≤ f a q :=
Monotone.le_of_tendsto (hf.isRatStieltjesPoint a).mono
(hf.isRatStieltjesPoint a).tendsto_atBot_zero q | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | IsMeasurableRatCDF.nonneg | null |
IsMeasurableRatCDF.le_one {f : α → ℚ → ℝ} (hf : IsMeasurableRatCDF f) (a : α) (q : ℚ) :
f a q ≤ 1 :=
Monotone.ge_of_tendsto (hf.isRatStieltjesPoint a).mono
(hf.isRatStieltjesPoint a).tendsto_atTop_one q | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | IsMeasurableRatCDF.le_one | null |
IsMeasurableRatCDF.tendsto_atTop_one {f : α → ℚ → ℝ} (hf : IsMeasurableRatCDF f) (a : α) :
Tendsto (f a) atTop (𝓝 1) := (hf.isRatStieltjesPoint a).tendsto_atTop_one | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | IsMeasurableRatCDF.tendsto_atTop_one | null |
IsMeasurableRatCDF.tendsto_atBot_zero {f : α → ℚ → ℝ} (hf : IsMeasurableRatCDF f) (a : α) :
Tendsto (f a) atBot (𝓝 0) := (hf.isRatStieltjesPoint a).tendsto_atBot_zero | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | IsMeasurableRatCDF.tendsto_atBot_zero | null |
IsMeasurableRatCDF.iInf_rat_gt_eq {f : α → ℚ → ℝ} (hf : IsMeasurableRatCDF f) (a : α)
(q : ℚ) :
⨅ r : Ioi q, f a r = f a q := (hf.isRatStieltjesPoint a).iInf_rat_gt_eq q | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | IsMeasurableRatCDF.iInf_rat_gt_eq | null |
defaultRatCDF (q : ℚ) := if q < 0 then (0 : ℝ) else 1 | def | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | defaultRatCDF | A function with the property `IsMeasurableRatCDF`.
Used in a piecewise construction to convert a function which only satisfies the properties
defining `IsMeasurableRatCDF` on some set into a true `IsMeasurableRatCDF`. |
monotone_defaultRatCDF : Monotone defaultRatCDF := by
unfold defaultRatCDF
intro x y hxy
dsimp only
split_ifs with h_1 h_2 h_2
exacts [le_rfl, zero_le_one, absurd (hxy.trans_lt h_2) h_1, le_rfl] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | monotone_defaultRatCDF | null |
defaultRatCDF_nonneg (q : ℚ) : 0 ≤ defaultRatCDF q := by
unfold defaultRatCDF
split_ifs
exacts [le_rfl, zero_le_one] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | defaultRatCDF_nonneg | null |
defaultRatCDF_le_one (q : ℚ) : defaultRatCDF q ≤ 1 := by
unfold defaultRatCDF
split_ifs <;> simp | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | defaultRatCDF_le_one | null |
tendsto_defaultRatCDF_atTop : Tendsto defaultRatCDF atTop (𝓝 1) := by
refine (tendsto_congr' ?_).mp tendsto_const_nhds
rw [EventuallyEq, eventually_atTop]
exact ⟨0, fun q hq => (if_neg (not_lt.mpr hq)).symm⟩ | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | tendsto_defaultRatCDF_atTop | null |
tendsto_defaultRatCDF_atBot : Tendsto defaultRatCDF atBot (𝓝 0) := by
refine (tendsto_congr' ?_).mp tendsto_const_nhds
rw [EventuallyEq, eventually_atBot]
refine ⟨-1, fun q hq => (if_pos (hq.trans_lt ?_)).symm⟩
linarith | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | tendsto_defaultRatCDF_atBot | null |
iInf_rat_gt_defaultRatCDF (t : ℚ) :
⨅ r : Ioi t, defaultRatCDF r = defaultRatCDF t := by
simp only [defaultRatCDF]
have h_bdd : BddBelow (range fun r : ↥(Ioi t) ↦ ite ((r : ℚ) < 0) (0 : ℝ) 1) := by
refine ⟨0, fun x hx ↦ ?_⟩
obtain ⟨y, rfl⟩ := mem_range.mpr hx
dsimp only
split_ifs
exacts [le_rfl, zero_le_one]
split_ifs with h
· refine le_antisymm ?_ (le_ciInf fun x ↦ ?_)
· obtain ⟨q, htq, hq_neg⟩ : ∃ q, t < q ∧ q < 0 := ⟨t / 2, by linarith, by linarith⟩
refine (ciInf_le h_bdd ⟨q, htq⟩).trans ?_
rw [if_pos]
rwa [Subtype.coe_mk]
· split_ifs
exacts [le_rfl, zero_le_one]
· refine le_antisymm ?_ ?_
· refine (ciInf_le h_bdd ⟨t + 1, lt_add_one t⟩).trans ?_
split_ifs
exacts [zero_le_one, le_rfl]
· refine le_ciInf fun x ↦ ?_
rw [if_neg]
rw [not_lt] at h ⊢
exact h.trans (mem_Ioi.mp x.prop).le | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | iInf_rat_gt_defaultRatCDF | null |
isRatStieltjesPoint_defaultRatCDF (a : α) :
IsRatStieltjesPoint (fun (_ : α) ↦ defaultRatCDF) a where
mono := monotone_defaultRatCDF
tendsto_atTop_one := tendsto_defaultRatCDF_atTop
tendsto_atBot_zero := tendsto_defaultRatCDF_atBot
iInf_rat_gt_eq := iInf_rat_gt_defaultRatCDF | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | isRatStieltjesPoint_defaultRatCDF | null |
IsMeasurableRatCDF_defaultRatCDF (α : Type*) [MeasurableSpace α] :
IsMeasurableRatCDF (fun (_ : α) (q : ℚ) ↦ defaultRatCDF q) where
isRatStieltjesPoint := isRatStieltjesPoint_defaultRatCDF
measurable := measurable_const | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | IsMeasurableRatCDF_defaultRatCDF | null |
noncomputable
toRatCDF (f : α → ℚ → ℝ) : α → ℚ → ℝ := fun a ↦
if IsRatStieltjesPoint f a then f a else defaultRatCDF | def | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | toRatCDF | Turn a function `f : α → ℚ → ℝ` into another with the property `IsRatStieltjesPoint f a`
everywhere. At `a` that does not satisfy that property, `f a` is replaced by an arbitrary suitable
function.
Mainly useful when `f` satisfies the property `IsRatStieltjesPoint f a` almost everywhere with
respect to some measure. |
toRatCDF_of_isRatStieltjesPoint {a : α} (h : IsRatStieltjesPoint f a) (q : ℚ) :
toRatCDF f a q = f a q := by
rw [toRatCDF, if_pos h] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | toRatCDF_of_isRatStieltjesPoint | null |
toRatCDF_unit_prod (a : α) :
toRatCDF (fun (p : Unit × α) ↦ f p.2) ((), a) = toRatCDF f a := by
unfold toRatCDF
rw [isRatStieltjesPoint_unit_prod_iff]
variable [MeasurableSpace α] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | toRatCDF_unit_prod | null |
measurable_toRatCDF (hf : Measurable f) : Measurable (toRatCDF f) :=
Measurable.ite (measurableSet_isRatStieltjesPoint hf) hf measurable_const | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | measurable_toRatCDF | null |
isMeasurableRatCDF_toRatCDF (hf : Measurable f) :
IsMeasurableRatCDF (toRatCDF f) where
isRatStieltjesPoint a := by
classical
exact IsRatStieltjesPoint.ite (IsRatStieltjesPoint f) id
(fun _ ↦ isRatStieltjesPoint_defaultRatCDF a)
measurable := measurable_toRatCDF hf | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | isMeasurableRatCDF_toRatCDF | null |
noncomputable IsMeasurableRatCDF.stieltjesFunction (a : α) : StieltjesFunction where
toFun := stieltjesFunctionAux f a
mono' := monotone_stieltjesFunctionAux hf a
right_continuous' x := continuousWithinAt_stieltjesFunctionAux_Ici hf a x | def | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | IsMeasurableRatCDF.stieltjesFunction | Auxiliary definition for `IsMeasurableRatCDF.stieltjesFunction`: turn `f : α → ℚ → ℝ` into
a function `α → ℝ → ℝ` by assigning to `f a x` the infimum of `f a q` over `q : ℚ` with `x < q`. -/
noncomputable irreducible_def IsMeasurableRatCDF.stieltjesFunctionAux (f : α → ℚ → ℝ) :
α → ℝ → ℝ :=
fun a x ↦ ⨅ q : { q' : ℚ // x < q' }, f a q
lemma IsMeasurableRatCDF.stieltjesFunctionAux_def' (f : α → ℚ → ℝ) (a : α) :
IsMeasurableRatCDF.stieltjesFunctionAux f a
= fun (t : ℝ) ↦ ⨅ r : { r' : ℚ // t < r' }, f a r := by
ext t; exact IsMeasurableRatCDF.stieltjesFunctionAux_def f a t
lemma IsMeasurableRatCDF.stieltjesFunctionAux_unit_prod {f : α → ℚ → ℝ} (a : α) :
IsMeasurableRatCDF.stieltjesFunctionAux (fun (p : Unit × α) ↦ f p.2) ((), a)
= IsMeasurableRatCDF.stieltjesFunctionAux f a := by
simp_rw [IsMeasurableRatCDF.stieltjesFunctionAux_def']
variable {f : α → ℚ → ℝ} [MeasurableSpace α] (hf : IsMeasurableRatCDF f)
include hf
lemma IsMeasurableRatCDF.stieltjesFunctionAux_eq (a : α) (r : ℚ) :
IsMeasurableRatCDF.stieltjesFunctionAux f a r = f a r := by
rw [← hf.iInf_rat_gt_eq a r, IsMeasurableRatCDF.stieltjesFunctionAux]
refine Equiv.iInf_congr ?_ ?_
· exact
{ toFun := fun t ↦ ⟨t.1, mod_cast t.2⟩
invFun := fun t ↦ ⟨t.1, mod_cast t.2⟩
left_inv := fun t ↦ by simp only [Subtype.coe_eta]
right_inv := fun t ↦ by simp only [Subtype.coe_eta] }
· intro t
simp only [Equiv.coe_fn_mk, Subtype.coe_mk]
lemma IsMeasurableRatCDF.stieltjesFunctionAux_nonneg (a : α) (r : ℝ) :
0 ≤ IsMeasurableRatCDF.stieltjesFunctionAux f a r := by
have : Nonempty { r' : ℚ // r < ↑r' } := by
obtain ⟨r, hrx⟩ := exists_rat_gt r
exact ⟨⟨r, hrx⟩⟩
rw [IsMeasurableRatCDF.stieltjesFunctionAux_def]
exact le_ciInf fun r' ↦ hf.nonneg a _
lemma IsMeasurableRatCDF.monotone_stieltjesFunctionAux (a : α) :
Monotone (IsMeasurableRatCDF.stieltjesFunctionAux f a) := by
intro x y hxy
have : Nonempty { r' : ℚ // y < ↑r' } := by
obtain ⟨r, hrx⟩ := exists_rat_gt y
exact ⟨⟨r, hrx⟩⟩
simp_rw [IsMeasurableRatCDF.stieltjesFunctionAux_def]
refine le_ciInf fun r ↦ (ciInf_le ?_ ?_).trans_eq ?_
· refine ⟨0, fun z ↦ ?_⟩; rintro ⟨u, rfl⟩; exact hf.nonneg a _
· exact ⟨r.1, hxy.trans_lt r.prop⟩
· rfl
lemma IsMeasurableRatCDF.continuousWithinAt_stieltjesFunctionAux_Ici (a : α) (x : ℝ) :
ContinuousWithinAt (IsMeasurableRatCDF.stieltjesFunctionAux f a) (Ici x) x := by
rw [← continuousWithinAt_Ioi_iff_Ici]
convert Monotone.tendsto_nhdsGT (monotone_stieltjesFunctionAux hf a) x
rw [sInf_image']
have h' : ⨅ r : Ioi x, stieltjesFunctionAux f a r
= ⨅ r : { r' : ℚ // x < r' }, stieltjesFunctionAux f a r := by
refine Real.iInf_Ioi_eq_iInf_rat_gt x ?_ (monotone_stieltjesFunctionAux hf a)
refine ⟨0, fun z ↦ ?_⟩
rintro ⟨u, -, rfl⟩
exact stieltjesFunctionAux_nonneg hf a u
have h'' :
⨅ r : { r' : ℚ // x < r' }, stieltjesFunctionAux f a r =
⨅ r : { r' : ℚ // x < r' }, f a r := by
congr with r
exact stieltjesFunctionAux_eq hf a r
rw [h', h'', ContinuousWithinAt]
congr!
rw [stieltjesFunctionAux_def]
/-- Extend a function `f : α → ℚ → ℝ` with property `IsMeasurableRatCDF` from `ℚ` to `ℝ`,
to a function `α → StieltjesFunction`. |
IsMeasurableRatCDF.stieltjesFunction_eq (a : α) (r : ℚ) : hf.stieltjesFunction a r = f a r :=
stieltjesFunctionAux_eq hf a r | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | IsMeasurableRatCDF.stieltjesFunction_eq | null |
IsMeasurableRatCDF.stieltjesFunction_nonneg (a : α) (r : ℝ) : 0 ≤ hf.stieltjesFunction a r :=
stieltjesFunctionAux_nonneg hf a r | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | IsMeasurableRatCDF.stieltjesFunction_nonneg | null |
IsMeasurableRatCDF.stieltjesFunction_le_one (a : α) (x : ℝ) :
hf.stieltjesFunction a x ≤ 1 := by
obtain ⟨r, hrx⟩ := exists_rat_gt x
rw [← StieltjesFunction.iInf_rat_gt_eq]
simp_rw [IsMeasurableRatCDF.stieltjesFunction_eq]
refine ciInf_le_of_le ?_ ?_ (hf.le_one _ _)
· refine ⟨0, fun z ↦ ?_⟩; rintro ⟨u, rfl⟩; exact hf.nonneg a _
· exact ⟨r, hrx⟩ | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | IsMeasurableRatCDF.stieltjesFunction_le_one | null |
IsMeasurableRatCDF.tendsto_stieltjesFunction_atBot (a : α) :
Tendsto (hf.stieltjesFunction a) atBot (𝓝 0) := by
have h_exists : ∀ x : ℝ, ∃ q : ℚ, x < q ∧ ↑q < x + 1 := fun x ↦ exists_rat_btwn (lt_add_one x)
let qs : ℝ → ℚ := fun x ↦ (h_exists x).choose
have hqs_tendsto : Tendsto qs atBot atBot := by
rw [tendsto_atBot_atBot]
refine fun q ↦ ⟨q - 1, fun y hy ↦ ?_⟩
have h_le : ↑(qs y) ≤ (q : ℝ) - 1 + 1 :=
(h_exists y).choose_spec.2.le.trans (add_le_add hy le_rfl)
rw [sub_add_cancel] at h_le
exact mod_cast h_le
refine tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds
((hf.tendsto_atBot_zero a).comp hqs_tendsto) (stieltjesFunction_nonneg hf a) fun x ↦ ?_
rw [Function.comp_apply, ← stieltjesFunction_eq hf]
exact (hf.stieltjesFunction a).mono (h_exists x).choose_spec.1.le | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | IsMeasurableRatCDF.tendsto_stieltjesFunction_atBot | null |
IsMeasurableRatCDF.tendsto_stieltjesFunction_atTop (a : α) :
Tendsto (hf.stieltjesFunction a) atTop (𝓝 1) := by
have h_exists : ∀ x : ℝ, ∃ q : ℚ, x - 1 < q ∧ ↑q < x := fun x ↦ exists_rat_btwn (sub_one_lt x)
let qs : ℝ → ℚ := fun x ↦ (h_exists x).choose
have hqs_tendsto : Tendsto qs atTop atTop := by
rw [tendsto_atTop_atTop]
refine fun q ↦ ⟨q + 1, fun y hy ↦ ?_⟩
have h_le : y - 1 ≤ qs y := (h_exists y).choose_spec.1.le
rw [sub_le_iff_le_add] at h_le
exact_mod_cast le_of_add_le_add_right (hy.trans h_le)
refine tendsto_of_tendsto_of_tendsto_of_le_of_le ((hf.tendsto_atTop_one a).comp hqs_tendsto)
tendsto_const_nhds ?_ (stieltjesFunction_le_one hf a)
intro x
rw [Function.comp_apply, ← stieltjesFunction_eq hf]
exact (hf.stieltjesFunction a).mono (le_of_lt (h_exists x).choose_spec.2) | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | IsMeasurableRatCDF.tendsto_stieltjesFunction_atTop | null |
IsMeasurableRatCDF.measurable_stieltjesFunction (x : ℝ) :
Measurable fun a ↦ hf.stieltjesFunction a x := by
have : (fun a ↦ hf.stieltjesFunction a x) = fun a ↦ ⨅ r : { r' : ℚ // x < r' }, f a ↑r := by
ext1 a
rw [← StieltjesFunction.iInf_rat_gt_eq]
congr with q
rw [stieltjesFunction_eq]
rw [this]
exact .iInf (fun q ↦ hf.measurable.eval) | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | IsMeasurableRatCDF.measurable_stieltjesFunction | null |
IsMeasurableRatCDF.stronglyMeasurable_stieltjesFunction (x : ℝ) :
StronglyMeasurable fun a ↦ hf.stieltjesFunction a x :=
(measurable_stieltjesFunction hf x).stronglyMeasurable | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | IsMeasurableRatCDF.stronglyMeasurable_stieltjesFunction | null |
IsMeasurableRatCDF.measure_stieltjesFunction_Iic (a : α) (x : ℝ) :
(hf.stieltjesFunction a).measure (Iic x) = ENNReal.ofReal (hf.stieltjesFunction a x) := by
rw [← sub_zero (hf.stieltjesFunction a x)]
exact (hf.stieltjesFunction a).measure_Iic (tendsto_stieltjesFunction_atBot hf a) _ | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | IsMeasurableRatCDF.measure_stieltjesFunction_Iic | null |
IsMeasurableRatCDF.measure_stieltjesFunction_univ (a : α) :
(hf.stieltjesFunction a).measure univ = 1 := by
rw [← ENNReal.ofReal_one, ← sub_zero (1 : ℝ)]
exact StieltjesFunction.measure_univ _ (tendsto_stieltjesFunction_atBot hf a)
(tendsto_stieltjesFunction_atTop hf a) | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | IsMeasurableRatCDF.measure_stieltjesFunction_univ | null |
IsMeasurableRatCDF.instIsProbabilityMeasure_stieltjesFunction (a : α) :
IsProbabilityMeasure (hf.stieltjesFunction a).measure :=
⟨measure_stieltjesFunction_univ hf a⟩ | instance | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | IsMeasurableRatCDF.instIsProbabilityMeasure_stieltjesFunction | null |
IsMeasurableRatCDF.measurable_measure_stieltjesFunction :
Measurable fun a ↦ (hf.stieltjesFunction a).measure := by
apply_rules [StieltjesFunction.measurable_measure, measurable_stieltjesFunction,
tendsto_stieltjesFunction_atBot, tendsto_stieltjesFunction_atTop] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | IsMeasurableRatCDF.measurable_measure_stieltjesFunction | null |
noncomputable
stieltjesOfMeasurableRat (f : α → ℚ → ℝ) (hf : Measurable f) : α → StieltjesFunction :=
(isMeasurableRatCDF_toRatCDF hf).stieltjesFunction | def | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | stieltjesOfMeasurableRat | Turn a measurable function `f : α → ℚ → ℝ` into a measurable function `α → StieltjesFunction`.
Composition of `toRatCDF` and `IsMeasurableRatCDF.stieltjesFunction`. |
stieltjesOfMeasurableRat_eq (hf : Measurable f) (a : α) (r : ℚ) :
stieltjesOfMeasurableRat f hf a r = toRatCDF f a r :=
IsMeasurableRatCDF.stieltjesFunction_eq _ a r | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | stieltjesOfMeasurableRat_eq | null |
stieltjesOfMeasurableRat_unit_prod (hf : Measurable f) (a : α) :
stieltjesOfMeasurableRat (fun (p : Unit × α) ↦ f p.2) (hf.comp measurable_snd) ((), a)
= stieltjesOfMeasurableRat f hf a := by
simp_rw [stieltjesOfMeasurableRat, IsMeasurableRatCDF.stieltjesFunction,
← IsMeasurableRatCDF.stieltjesFunctionAux_unit_prod a]
congr 1 with x
congr 1 with p : 1
cases p with
| mk _ b => rw [← toRatCDF_unit_prod b] | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | stieltjesOfMeasurableRat_unit_prod | null |
stieltjesOfMeasurableRat_nonneg (hf : Measurable f) (a : α) (r : ℝ) :
0 ≤ stieltjesOfMeasurableRat f hf a r := IsMeasurableRatCDF.stieltjesFunction_nonneg _ a r | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | stieltjesOfMeasurableRat_nonneg | null |
stieltjesOfMeasurableRat_le_one (hf : Measurable f) (a : α) (x : ℝ) :
stieltjesOfMeasurableRat f hf a x ≤ 1 := IsMeasurableRatCDF.stieltjesFunction_le_one _ a x | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | stieltjesOfMeasurableRat_le_one | null |
tendsto_stieltjesOfMeasurableRat_atBot (hf : Measurable f) (a : α) :
Tendsto (stieltjesOfMeasurableRat f hf a) atBot (𝓝 0) :=
IsMeasurableRatCDF.tendsto_stieltjesFunction_atBot _ a | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | tendsto_stieltjesOfMeasurableRat_atBot | null |
tendsto_stieltjesOfMeasurableRat_atTop (hf : Measurable f) (a : α) :
Tendsto (stieltjesOfMeasurableRat f hf a) atTop (𝓝 1) :=
IsMeasurableRatCDF.tendsto_stieltjesFunction_atTop _ a | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | tendsto_stieltjesOfMeasurableRat_atTop | null |
measurable_stieltjesOfMeasurableRat (hf : Measurable f) (x : ℝ) :
Measurable fun a ↦ stieltjesOfMeasurableRat f hf a x :=
IsMeasurableRatCDF.measurable_stieltjesFunction _ x | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | measurable_stieltjesOfMeasurableRat | null |
stronglyMeasurable_stieltjesOfMeasurableRat (hf : Measurable f) (x : ℝ) :
StronglyMeasurable fun a ↦ stieltjesOfMeasurableRat f hf a x :=
IsMeasurableRatCDF.stronglyMeasurable_stieltjesFunction _ x | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | stronglyMeasurable_stieltjesOfMeasurableRat | null |
measure_stieltjesOfMeasurableRat_Iic (hf : Measurable f) (a : α) (x : ℝ) :
(stieltjesOfMeasurableRat f hf a).measure (Iic x)
= ENNReal.ofReal (stieltjesOfMeasurableRat f hf a x) :=
IsMeasurableRatCDF.measure_stieltjesFunction_Iic _ _ _ | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | measure_stieltjesOfMeasurableRat_Iic | null |
measure_stieltjesOfMeasurableRat_univ (hf : Measurable f) (a : α) :
(stieltjesOfMeasurableRat f hf a).measure univ = 1 :=
IsMeasurableRatCDF.measure_stieltjesFunction_univ _ _ | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | measure_stieltjesOfMeasurableRat_univ | null |
instIsProbabilityMeasure_stieltjesOfMeasurableRat
(hf : Measurable f) (a : α) :
IsProbabilityMeasure (stieltjesOfMeasurableRat f hf a).measure :=
IsMeasurableRatCDF.instIsProbabilityMeasure_stieltjesFunction _ _ | instance | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | instIsProbabilityMeasure_stieltjesOfMeasurableRat | null |
measurable_measure_stieltjesOfMeasurableRat (hf : Measurable f) :
Measurable fun a ↦ (stieltjesOfMeasurableRat f hf a).measure :=
IsMeasurableRatCDF.measurable_measure_stieltjesFunction _ | lemma | Probability | [
"Mathlib.MeasureTheory.Measure.GiryMonad",
"Mathlib.MeasureTheory.Measure.Stieltjes",
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic"
] | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | measurable_measure_stieltjesOfMeasurableRat | null |
isRatCondKernelCDFAux_density_Iic (κ : Kernel α (γ × ℝ)) [IsFiniteKernel κ] :
IsRatCondKernelCDFAux (fun (p : α × γ) q ↦ density κ (fst κ) p.1 p.2 (Iic q)) κ (fst κ) where
measurable := measurable_pi_iff.mpr fun _ ↦ measurable_density κ (fst κ) measurableSet_Iic
mono' a q r hqr :=
ae_of_all _ fun c ↦ density_mono_set le_rfl a c (Iic_subset_Iic.mpr (by exact_mod_cast hqr))
nonneg' _ _ := ae_of_all _ fun _ ↦ density_nonneg le_rfl _ _ _
le_one' _ _ := ae_of_all _ fun _ ↦ density_le_one le_rfl _ _ _
tendsto_integral_of_antitone a s hs_anti hs_tendsto := by
let s' : ℕ → Set ℝ := fun n ↦ Iic (s n)
refine tendsto_integral_density_of_antitone le_rfl a s' ?_ ?_ (fun _ ↦ measurableSet_Iic)
· refine fun i j hij ↦ Iic_subset_Iic.mpr ?_
exact mod_cast hs_anti hij
· ext x
simp only [mem_iInter, mem_Iic, mem_empty_iff_false, iff_false, not_forall, not_le, s']
rw [tendsto_atTop_atBot] at hs_tendsto
have ⟨q, hq⟩ := exists_rat_lt x
obtain ⟨i, hi⟩ := hs_tendsto q
refine ⟨i, lt_of_le_of_lt ?_ hq⟩
exact mod_cast hi i le_rfl
tendsto_integral_of_monotone a s hs_mono hs_tendsto := by
rw [fst_real_apply _ _ MeasurableSet.univ]
let s' : ℕ → Set ℝ := fun n ↦ Iic (s n)
refine tendsto_integral_density_of_monotone (le_rfl : fst κ ≤ fst κ)
a s' ?_ ?_ (fun _ ↦ measurableSet_Iic)
· exact fun i j hij ↦ Iic_subset_Iic.mpr (by exact mod_cast hs_mono hij)
· ext x
simp only [mem_iUnion, mem_univ, iff_true]
rw [tendsto_atTop_atTop] at hs_tendsto
have ⟨q, hq⟩ := exists_rat_gt x
obtain ⟨i, hi⟩ := hs_tendsto q
refine ⟨i, hq.le.trans ?_⟩
exact mod_cast hi i le_rfl
integrable a _ := integrable_density le_rfl a measurableSet_Iic
setIntegral a _ hA _ := setIntegral_density le_rfl a measurableSet_Iic hA | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureCompProd",
"Mathlib.Probability.Kernel.Disintegration.Basic",
"Mathlib.Probability.Kernel.Disintegration.CondCDF",
"Mathlib.Probability.Kernel.Disintegration.Density",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel",
"Mathlib.MeasureTheory.Constructio... | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | isRatCondKernelCDFAux_density_Iic | null |
isRatCondKernelCDF_density_Iic (κ : Kernel α (γ × ℝ)) [IsFiniteKernel κ] :
IsRatCondKernelCDF (fun (p : α × γ) q ↦ density κ (fst κ) p.1 p.2 (Iic q)) κ (fst κ) :=
(isRatCondKernelCDFAux_density_Iic κ).isRatCondKernelCDF | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureCompProd",
"Mathlib.Probability.Kernel.Disintegration.Basic",
"Mathlib.Probability.Kernel.Disintegration.CondCDF",
"Mathlib.Probability.Kernel.Disintegration.Density",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel",
"Mathlib.MeasureTheory.Constructio... | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | isRatCondKernelCDF_density_Iic | Taking the kernel density of intervals `Iic q` for `q : ℚ` gives a function with the property
`isRatCondKernelCDF`. |
noncomputable
condKernelCDF (κ : Kernel α (γ × ℝ)) [IsFiniteKernel κ] : α × γ → StieltjesFunction :=
stieltjesOfMeasurableRat (fun (p : α × γ) q ↦ density κ (fst κ) p.1 p.2 (Iic q))
(isRatCondKernelCDF_density_Iic κ).measurable | def | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureCompProd",
"Mathlib.Probability.Kernel.Disintegration.Basic",
"Mathlib.Probability.Kernel.Disintegration.CondCDF",
"Mathlib.Probability.Kernel.Disintegration.Density",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel",
"Mathlib.MeasureTheory.Constructio... | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | condKernelCDF | The conditional kernel CDF of a kernel `κ : Kernel α (γ × ℝ)`, where `γ` is countably generated. |
isCondKernelCDF_condKernelCDF (κ : Kernel α (γ × ℝ)) [IsFiniteKernel κ] :
IsCondKernelCDF (condKernelCDF κ) κ (fst κ) :=
isCondKernelCDF_stieltjesOfMeasurableRat (isRatCondKernelCDF_density_Iic κ) | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureCompProd",
"Mathlib.Probability.Kernel.Disintegration.Basic",
"Mathlib.Probability.Kernel.Disintegration.CondCDF",
"Mathlib.Probability.Kernel.Disintegration.Density",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel",
"Mathlib.MeasureTheory.Constructio... | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | isCondKernelCDF_condKernelCDF | null |
noncomputable
condKernelReal (κ : Kernel α (γ × ℝ)) [IsFiniteKernel κ] : Kernel (α × γ) ℝ :=
(isCondKernelCDF_condKernelCDF κ).toKernel | def | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureCompProd",
"Mathlib.Probability.Kernel.Disintegration.Basic",
"Mathlib.Probability.Kernel.Disintegration.CondCDF",
"Mathlib.Probability.Kernel.Disintegration.Density",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel",
"Mathlib.MeasureTheory.Constructio... | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | condKernelReal | Auxiliary definition for `ProbabilityTheory.Kernel.condKernel`.
A conditional kernel for `κ : Kernel α (γ × ℝ)` where `γ` is countably generated. |
instIsMarkovKernelCondKernelReal (κ : Kernel α (γ × ℝ)) [IsFiniteKernel κ] :
IsMarkovKernel (condKernelReal κ) := by
rw [condKernelReal]
infer_instance | instance | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureCompProd",
"Mathlib.Probability.Kernel.Disintegration.Basic",
"Mathlib.Probability.Kernel.Disintegration.CondCDF",
"Mathlib.Probability.Kernel.Disintegration.Density",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel",
"Mathlib.MeasureTheory.Constructio... | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | instIsMarkovKernelCondKernelReal | null |
compProd_fst_condKernelReal (κ : Kernel α (γ × ℝ)) [IsFiniteKernel κ] :
fst κ ⊗ₖ condKernelReal κ = κ := by
rw [condKernelReal, compProd_toKernel] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureCompProd",
"Mathlib.Probability.Kernel.Disintegration.Basic",
"Mathlib.Probability.Kernel.Disintegration.CondCDF",
"Mathlib.Probability.Kernel.Disintegration.Density",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel",
"Mathlib.MeasureTheory.Constructio... | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | compProd_fst_condKernelReal | null |
noncomputable
condKernelUnitReal (κ : Kernel Unit (α × ℝ)) [IsFiniteKernel κ] : Kernel (Unit × α) ℝ :=
(isCondKernelCDF_condCDF (κ ())).toKernel | def | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureCompProd",
"Mathlib.Probability.Kernel.Disintegration.Basic",
"Mathlib.Probability.Kernel.Disintegration.CondCDF",
"Mathlib.Probability.Kernel.Disintegration.Density",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel",
"Mathlib.MeasureTheory.Constructio... | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | condKernelUnitReal | Auxiliary definition for `MeasureTheory.Measure.condKernel` and
`ProbabilityTheory.Kernel.condKernel`.
A conditional kernel for `κ : Kernel Unit (α × ℝ)`. |
instIsMarkovKernelCondKernelUnitReal (κ : Kernel Unit (α × ℝ)) [IsFiniteKernel κ] :
IsMarkovKernel (condKernelUnitReal κ) := by
rw [condKernelUnitReal]
infer_instance | instance | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureCompProd",
"Mathlib.Probability.Kernel.Disintegration.Basic",
"Mathlib.Probability.Kernel.Disintegration.CondCDF",
"Mathlib.Probability.Kernel.Disintegration.Density",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel",
"Mathlib.MeasureTheory.Constructio... | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | instIsMarkovKernelCondKernelUnitReal | null |
condKernelUnitReal.instIsCondKernel (κ : Kernel Unit (α × ℝ)) [IsFiniteKernel κ] :
κ.IsCondKernel κ.condKernelUnitReal where
disintegrate := by rw [condKernelUnitReal, compProd_toKernel]; ext; simp | instance | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureCompProd",
"Mathlib.Probability.Kernel.Disintegration.Basic",
"Mathlib.Probability.Kernel.Disintegration.CondCDF",
"Mathlib.Probability.Kernel.Disintegration.Density",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel",
"Mathlib.MeasureTheory.Constructio... | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | condKernelUnitReal.instIsCondKernel | null |
noncomputable
borelMarkovFromReal (Ω : Type*) [Nonempty Ω] [MeasurableSpace Ω] [StandardBorelSpace Ω]
(η : Kernel α ℝ) :
Kernel α Ω :=
have he := measurableEmbedding_embeddingReal Ω
let x₀ := (range_nonempty (embeddingReal Ω)).choose
comapRight
(piecewise ((Kernel.measurable_coe η he.measurableSet_range.compl) (measurableSet_singleton 0) :
MeasurableSet {a | η a (range (embeddingReal Ω))ᶜ = 0})
η (deterministic (fun _ ↦ x₀) measurable_const)) he | def | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureCompProd",
"Mathlib.Probability.Kernel.Disintegration.Basic",
"Mathlib.Probability.Kernel.Disintegration.CondCDF",
"Mathlib.Probability.Kernel.Disintegration.Density",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel",
"Mathlib.MeasureTheory.Constructio... | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | borelMarkovFromReal | Auxiliary definition for `ProbabilityTheory.Kernel.condKernel`.
A Borel space `Ω` embeds measurably into `ℝ` (with embedding `e`), hence we can get a `Kernel α Ω`
from a `Kernel α ℝ` by taking the comap by `e`.
Here we take the comap of a modification of `η : Kernel α ℝ`, useful when `η a` is a probability
measure with all its mass on `range e` almost everywhere with respect to some measure and we want to
ensure that the comap is a Markov kernel.
We thus take the comap by `e` of a kernel defined piecewise: `η` when
`η a (range (embeddingReal Ω))ᶜ = 0`, and an arbitrary deterministic kernel otherwise. |
borelMarkovFromReal_apply (Ω : Type*) [Nonempty Ω] [MeasurableSpace Ω] [StandardBorelSpace Ω]
(η : Kernel α ℝ) (a : α) :
borelMarkovFromReal Ω η a
= if η a (range (embeddingReal Ω))ᶜ = 0 then (η a).comap (embeddingReal Ω)
else (Measure.dirac (range_nonempty (embeddingReal Ω)).choose).comap (embeddingReal Ω) := by
classical
rw [borelMarkovFromReal, comapRight_apply, piecewise_apply, deterministic_apply]
simp only [mem_preimage, mem_singleton_iff]
split_ifs <;> rfl | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureCompProd",
"Mathlib.Probability.Kernel.Disintegration.Basic",
"Mathlib.Probability.Kernel.Disintegration.CondCDF",
"Mathlib.Probability.Kernel.Disintegration.Density",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel",
"Mathlib.MeasureTheory.Constructio... | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | borelMarkovFromReal_apply | null |
borelMarkovFromReal_apply' (Ω : Type*) [Nonempty Ω] [MeasurableSpace Ω] [StandardBorelSpace Ω]
(η : Kernel α ℝ) (a : α) {s : Set Ω} (hs : MeasurableSet s) :
borelMarkovFromReal Ω η a s
= if η a (range (embeddingReal Ω))ᶜ = 0 then η a (embeddingReal Ω '' s)
else (embeddingReal Ω '' s).indicator 1 (range_nonempty (embeddingReal Ω)).choose := by
have he := measurableEmbedding_embeddingReal Ω
rw [borelMarkovFromReal_apply]
split_ifs with h
· rw [Measure.comap_apply _ he.injective he.measurableSet_image' _ hs]
· rw [Measure.comap_apply _ he.injective he.measurableSet_image' _ hs, Measure.dirac_apply] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureCompProd",
"Mathlib.Probability.Kernel.Disintegration.Basic",
"Mathlib.Probability.Kernel.Disintegration.CondCDF",
"Mathlib.Probability.Kernel.Disintegration.Density",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel",
"Mathlib.MeasureTheory.Constructio... | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | borelMarkovFromReal_apply' | null |
instIsSFiniteKernelBorelMarkovFromReal (η : Kernel α ℝ) [IsSFiniteKernel η] :
IsSFiniteKernel (borelMarkovFromReal Ω η) :=
IsSFiniteKernel.comapRight _ (measurableEmbedding_embeddingReal Ω) | instance | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureCompProd",
"Mathlib.Probability.Kernel.Disintegration.Basic",
"Mathlib.Probability.Kernel.Disintegration.CondCDF",
"Mathlib.Probability.Kernel.Disintegration.Density",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel",
"Mathlib.MeasureTheory.Constructio... | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | instIsSFiniteKernelBorelMarkovFromReal | When `η` is an s-finite kernel, `borelMarkovFromReal Ω η` is an s-finite kernel. |
instIsFiniteKernelBorelMarkovFromReal (η : Kernel α ℝ) [IsFiniteKernel η] :
IsFiniteKernel (borelMarkovFromReal Ω η) :=
IsFiniteKernel.comapRight _ (measurableEmbedding_embeddingReal Ω) | instance | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureCompProd",
"Mathlib.Probability.Kernel.Disintegration.Basic",
"Mathlib.Probability.Kernel.Disintegration.CondCDF",
"Mathlib.Probability.Kernel.Disintegration.Density",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel",
"Mathlib.MeasureTheory.Constructio... | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | instIsFiniteKernelBorelMarkovFromReal | When `η` is a finite kernel, `borelMarkovFromReal Ω η` is a finite kernel. |
instIsMarkovKernelBorelMarkovFromReal (η : Kernel α ℝ) [IsMarkovKernel η] :
IsMarkovKernel (borelMarkovFromReal Ω η) := by
refine IsMarkovKernel.comapRight _ (measurableEmbedding_embeddingReal Ω) (fun a ↦ ?_)
classical
rw [piecewise_apply]
split_ifs with h
· rwa [← prob_compl_eq_zero_iff (measurableEmbedding_embeddingReal Ω).measurableSet_range]
· rw [deterministic_apply]
simp [(range_nonempty (embeddingReal Ω)).choose_spec] | instance | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureCompProd",
"Mathlib.Probability.Kernel.Disintegration.Basic",
"Mathlib.Probability.Kernel.Disintegration.CondCDF",
"Mathlib.Probability.Kernel.Disintegration.Density",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel",
"Mathlib.MeasureTheory.Constructio... | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | instIsMarkovKernelBorelMarkovFromReal | When `η` is a Markov kernel, `borelMarkovFromReal Ω η` is a Markov kernel. |
compProd_fst_borelMarkovFromReal_eq_comapRight_compProd
(κ : Kernel α (β × Ω)) [IsSFiniteKernel κ] (η : Kernel (α × β) ℝ) [IsSFiniteKernel η]
(hη : (fst (map κ (Prod.map (id : β → β) (embeddingReal Ω)))) ⊗ₖ η
= map κ (Prod.map (id : β → β) (embeddingReal Ω))) :
fst κ ⊗ₖ borelMarkovFromReal Ω η
= comapRight (fst (map κ (Prod.map (id : β → β) (embeddingReal Ω))) ⊗ₖ η)
(MeasurableEmbedding.id.prodMap (measurableEmbedding_embeddingReal Ω)) := by
let e := embeddingReal Ω
let he := measurableEmbedding_embeddingReal Ω
let κ' := map κ (Prod.map (id : β → β) e)
have hη' : fst κ' ⊗ₖ η = κ' := hη
have h_prod_embed : MeasurableEmbedding (Prod.map (id : β → β) e) :=
MeasurableEmbedding.id.prodMap he
change fst κ ⊗ₖ borelMarkovFromReal Ω η = comapRight (fst κ' ⊗ₖ η) h_prod_embed
rw [comapRight_compProd_id_prod _ _ he]
have h_fst : fst κ' = fst κ := by
ext a u
unfold κ'
rw [fst_apply, map_apply _ (by fun_prop),
Measure.map_map measurable_fst h_prod_embed.measurable, fst_apply]
congr
rw [h_fst]
ext a t ht : 2
simp_rw [compProd_apply ht]
refine lintegral_congr_ae ?_
have h_ae : ∀ᵐ t ∂(fst κ a), (a, t) ∈ {p : α × β | η p (range e)ᶜ = 0} := by
rw [← h_fst]
have h_compProd : κ' a (univ ×ˢ range e)ᶜ = 0 := by
unfold κ'
rw [map_apply' _ (by fun_prop)]
swap; · exact (MeasurableSet.univ.prod he.measurableSet_range).compl
suffices Prod.map id e ⁻¹' (univ ×ˢ range e)ᶜ = ∅ by rw [this]; simp
ext x
simp
rw [← hη', compProd_null] at h_compProd
swap; · exact (MeasurableSet.univ.prod he.measurableSet_range).compl
simp only [preimage_compl, mem_univ, mk_preimage_prod_right] at h_compProd
exact h_compProd
filter_upwards [h_ae] with a ha
rw [borelMarkovFromReal, comapRight_apply', comapRight_apply']
rotate_left
· exact measurable_prodMk_left ht
· exact measurable_prodMk_left ht
classical
rw [piecewise_apply, if_pos]
exact ha | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureCompProd",
"Mathlib.Probability.Kernel.Disintegration.Basic",
"Mathlib.Probability.Kernel.Disintegration.CondCDF",
"Mathlib.Probability.Kernel.Disintegration.Density",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel",
"Mathlib.MeasureTheory.Constructio... | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | compProd_fst_borelMarkovFromReal_eq_comapRight_compProd | For `κ' := map κ (Prod.map (id : β → β) e)`, the hypothesis `hη` is `fst κ' ⊗ₖ η = κ'`.
The conclusion of the lemma is `fst κ ⊗ₖ borelMarkovFromReal Ω η = comapRight (fst κ' ⊗ₖ η) _`. |
compProd_fst_borelMarkovFromReal (κ : Kernel α (β × Ω)) [IsSFiniteKernel κ]
(η : Kernel (α × β) ℝ) [IsSFiniteKernel η]
(hη : (fst (map κ (Prod.map (id : β → β) (embeddingReal Ω)))) ⊗ₖ η
= map κ (Prod.map (id : β → β) (embeddingReal Ω))) :
fst κ ⊗ₖ borelMarkovFromReal Ω η = κ := by
let e := embeddingReal Ω
let he := measurableEmbedding_embeddingReal Ω
let κ' := map κ (Prod.map (id : β → β) e)
have hη' : fst κ' ⊗ₖ η = κ' := hη
have h_prod_embed : MeasurableEmbedding (Prod.map (id : β → β) e) :=
MeasurableEmbedding.id.prodMap he
have : κ = comapRight κ' h_prod_embed := by
ext c t : 2
unfold κ'
rw [comapRight_apply, map_apply _ (by fun_prop), h_prod_embed.comap_map]
conv_rhs => rw [this, ← hη']
exact compProd_fst_borelMarkovFromReal_eq_comapRight_compProd κ η hη | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureCompProd",
"Mathlib.Probability.Kernel.Disintegration.Basic",
"Mathlib.Probability.Kernel.Disintegration.CondCDF",
"Mathlib.Probability.Kernel.Disintegration.Density",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel",
"Mathlib.MeasureTheory.Constructio... | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | compProd_fst_borelMarkovFromReal | For `κ' := map κ (Prod.map (id : β → β) e)`, the hypothesis `hη` is `fst κ' ⊗ₖ η = κ'`.
With that hypothesis, `fst κ ⊗ₖ borelMarkovFromReal κ η = κ`. |
noncomputable
condKernelBorel (κ : Kernel α (γ × Ω)) [IsFiniteKernel κ] : Kernel (α × γ) Ω :=
let κ' := map κ (Prod.map (id : γ → γ) (embeddingReal Ω))
borelMarkovFromReal Ω (condKernelReal κ') | def | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureCompProd",
"Mathlib.Probability.Kernel.Disintegration.Basic",
"Mathlib.Probability.Kernel.Disintegration.CondCDF",
"Mathlib.Probability.Kernel.Disintegration.Density",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel",
"Mathlib.MeasureTheory.Constructio... | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | condKernelBorel | Auxiliary definition for `ProbabilityTheory.Kernel.condKernel`.
A conditional kernel for `κ : Kernel α (γ × Ω)` where `γ` is countably generated and `Ω` is
standard Borel. |
instIsMarkovKernelCondKernelBorel (κ : Kernel α (γ × Ω)) [IsFiniteKernel κ] :
IsMarkovKernel (condKernelBorel κ) := by
rw [condKernelBorel]
infer_instance | instance | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureCompProd",
"Mathlib.Probability.Kernel.Disintegration.Basic",
"Mathlib.Probability.Kernel.Disintegration.CondCDF",
"Mathlib.Probability.Kernel.Disintegration.Density",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel",
"Mathlib.MeasureTheory.Constructio... | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | instIsMarkovKernelCondKernelBorel | null |
condKernelBorel.instIsCondKernel (κ : Kernel α (γ × Ω)) [IsFiniteKernel κ] :
κ.IsCondKernel κ.condKernelBorel where
disintegrate := by
rw [condKernelBorel, compProd_fst_borelMarkovFromReal _ _ (compProd_fst_condKernelReal _)] | instance | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureCompProd",
"Mathlib.Probability.Kernel.Disintegration.Basic",
"Mathlib.Probability.Kernel.Disintegration.CondCDF",
"Mathlib.Probability.Kernel.Disintegration.Density",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel",
"Mathlib.MeasureTheory.Constructio... | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | condKernelBorel.instIsCondKernel | null |
noncomputable
condKernelUnitBorel : Kernel (Unit × α) Ω :=
let κ' := map κ (Prod.map (id : α → α) (embeddingReal Ω))
borelMarkovFromReal Ω (condKernelUnitReal κ') | def | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureCompProd",
"Mathlib.Probability.Kernel.Disintegration.Basic",
"Mathlib.Probability.Kernel.Disintegration.CondCDF",
"Mathlib.Probability.Kernel.Disintegration.Density",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel",
"Mathlib.MeasureTheory.Constructio... | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | condKernelUnitBorel | Auxiliary definition for `MeasureTheory.Measure.condKernel` and
`ProbabilityTheory.Kernel.condKernel`.
A conditional kernel for `κ : Kernel Unit (α × Ω)` where `Ω` is standard Borel. |
instIsMarkovKernelCondKernelUnitBorel : IsMarkovKernel κ.condKernelUnitBorel := by
rw [condKernelUnitBorel]
infer_instance | instance | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureCompProd",
"Mathlib.Probability.Kernel.Disintegration.Basic",
"Mathlib.Probability.Kernel.Disintegration.CondCDF",
"Mathlib.Probability.Kernel.Disintegration.Density",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel",
"Mathlib.MeasureTheory.Constructio... | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | instIsMarkovKernelCondKernelUnitBorel | null |
condKernelUnitBorel.instIsCondKernel : κ.IsCondKernel κ.condKernelUnitBorel where
disintegrate := by
rw [condKernelUnitBorel, compProd_fst_borelMarkovFromReal _ _ (disintegrate _ _)] | instance | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureCompProd",
"Mathlib.Probability.Kernel.Disintegration.Basic",
"Mathlib.Probability.Kernel.Disintegration.CondCDF",
"Mathlib.Probability.Kernel.Disintegration.Density",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel",
"Mathlib.MeasureTheory.Constructio... | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | condKernelUnitBorel.instIsCondKernel | null |
_root_.MeasureTheory.Measure.condKernel_apply_of_ne_zero [MeasurableSingletonClass α]
{x : α} (hx : ρ.fst {x} ≠ 0) (s : Set Ω) :
ρ.condKernel x s = (ρ.fst {x})⁻¹ * ρ ({x} ×ˢ s) :=
Measure.IsCondKernel.apply_of_ne_zero _ _ hx _ | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureCompProd",
"Mathlib.Probability.Kernel.Disintegration.Basic",
"Mathlib.Probability.Kernel.Disintegration.CondCDF",
"Mathlib.Probability.Kernel.Disintegration.Density",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel",
"Mathlib.MeasureTheory.Constructio... | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | _root_.MeasureTheory.Measure.condKernel_apply_of_ne_zero | Conditional kernel of a measure on a product space: a Markov kernel such that
`ρ = ρ.fst ⊗ₘ ρ.condKernel` (see `MeasureTheory.Measure.compProd_fst_condKernel`). -/
noncomputable
irreducible_def _root_.MeasureTheory.Measure.condKernel (ρ : Measure (α × Ω)) [IsFiniteMeasure ρ] :
Kernel α Ω :=
comap (condKernelUnitBorel (const Unit ρ)) (fun a ↦ ((), a)) measurable_prodMk_left
lemma _root_.MeasureTheory.Measure.condKernel_apply (ρ : Measure (α × Ω)) [IsFiniteMeasure ρ]
(a : α) :
ρ.condKernel a = condKernelUnitBorel (const Unit ρ) ((), a) := by
rw [Measure.condKernel]; rfl
instance _root_.MeasureTheory.Measure.condKernel.instIsCondKernel (ρ : Measure (α × Ω))
[IsFiniteMeasure ρ] : ρ.IsCondKernel ρ.condKernel where
disintegrate := by
have h1 : const Unit (Measure.fst ρ) = fst (const Unit ρ) := by
ext
simp only [fst_apply, Measure.fst, const_apply]
have h2 : prodMkLeft Unit (Measure.condKernel ρ) = condKernelUnitBorel (const Unit ρ) := by
ext
simp only [prodMkLeft_apply, Measure.condKernel_apply]
rw [Measure.compProd, h1, h2, disintegrate]
simp
instance _root_.MeasureTheory.Measure.instIsMarkovKernelCondKernel
(ρ : Measure (α × Ω)) [IsFiniteMeasure ρ] : IsMarkovKernel ρ.condKernel := by
rw [Measure.condKernel]
infer_instance
/-- If the singleton `{x}` has non-zero mass for `ρ.fst`, then for all `s : Set Ω`,
`ρ.condKernel x s = (ρ.fst {x})⁻¹ * ρ ({x} ×ˢ s)` . |
instIsMarkovKernelCondKernel : IsMarkovKernel (condKernel κ) := by
rw [condKernel_def]
split_ifs <;> infer_instance | instance | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureCompProd",
"Mathlib.Probability.Kernel.Disintegration.Basic",
"Mathlib.Probability.Kernel.Disintegration.CondCDF",
"Mathlib.Probability.Kernel.Disintegration.Density",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel",
"Mathlib.MeasureTheory.Constructio... | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | instIsMarkovKernelCondKernel | Conditional kernel of a kernel `κ : Kernel α (β × Ω)`: a Markov kernel such that
`fst κ ⊗ₖ condKernel κ = κ` (see `MeasureTheory.Measure.compProd_fst_condKernel`).
It exists whenever `Ω` is standard Borel and either `α` is countable
or `β` is countably generated. -/
noncomputable
irreducible_def condKernel : Kernel (α × β) Ω :=
if hα : Countable α then
condKernelCountable (fun a ↦ (κ a).condKernel)
fun x y h ↦ by simp [apply_congr_of_mem_measurableAtom _ h]
else letI := h.countableOrCountablyGenerated.resolve_left hα; condKernelBorel κ
/-- `condKernel κ` is a Markov kernel. |
condKernel.instIsCondKernel : κ.IsCondKernel κ.condKernel where
disintegrate := by rw [condKernel_def]; split_ifs with hα <;> exact disintegrate _ _ | instance | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureCompProd",
"Mathlib.Probability.Kernel.Disintegration.Basic",
"Mathlib.Probability.Kernel.Disintegration.CondCDF",
"Mathlib.Probability.Kernel.Disintegration.Density",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel",
"Mathlib.MeasureTheory.Constructio... | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | condKernel.instIsCondKernel | null |
eq_condKernel_of_measure_eq_compProd' (κ : Kernel α Ω) [IsSFiniteKernel κ]
(hκ : ρ = ρ.fst ⊗ₘ κ) {s : Set Ω} (hs : MeasurableSet s) :
∀ᵐ x ∂ρ.fst, κ x s = ρ.condKernel x s := by
refine ae_eq_of_forall_setLIntegral_eq_of_sigmaFinite
(Kernel.measurable_coe κ hs) (Kernel.measurable_coe ρ.condKernel hs) (fun t ht _ ↦ ?_)
conv_rhs => rw [Measure.setLIntegral_condKernel_eq_measure_prod ht hs, hκ]
simp only [Measure.compProd_apply (ht.prod hs), ← lintegral_indicator ht]
congr with x
by_cases hx : x ∈ t <;> simp [hx] | theorem | Probability | [
"Mathlib.Probability.Kernel.Disintegration.Integral"
] | Mathlib/Probability/Kernel/Disintegration/Unique.lean | eq_condKernel_of_measure_eq_compProd' | A s-finite kernel which satisfy the disintegration property of the given measure `ρ` is almost
everywhere equal to the disintegration kernel of `ρ` when evaluated on a measurable set.
This theorem in the case of finite kernels is weaker than `eq_condKernel_of_measure_eq_compProd`
which asserts that the kernels are equal almost everywhere and not just on a given measurable
set. |
eq_condKernel_of_measure_eq_compProd_real {ρ : Measure (α × ℝ)} [IsFiniteMeasure ρ]
(κ : Kernel α ℝ) [IsFiniteKernel κ] (hκ : ρ = ρ.fst ⊗ₘ κ) :
∀ᵐ x ∂ρ.fst, κ x = ρ.condKernel x := by
have huniv : ∀ᵐ x ∂ρ.fst, κ x Set.univ = ρ.condKernel x Set.univ :=
eq_condKernel_of_measure_eq_compProd' κ hκ MeasurableSet.univ
suffices ∀ᵐ x ∂ρ.fst, ∀ ⦃t⦄, MeasurableSet t → κ x t = ρ.condKernel x t by
filter_upwards [this] with x hx
ext t ht; exact hx ht
apply MeasurableSpace.ae_induction_on_inter Real.borel_eq_generateFrom_Iic_rat
Real.isPiSystem_Iic_rat
· simp
· simp only [iUnion_singleton_eq_range, mem_range, forall_exists_index, forall_apply_eq_imp_iff]
exact ae_all_iff.2 fun q ↦ eq_condKernel_of_measure_eq_compProd' κ hκ measurableSet_Iic
· filter_upwards [huniv] with x hxuniv t ht heq
rw [measure_compl ht <| measure_ne_top _ _, heq, hxuniv, measure_compl ht <| measure_ne_top _ _]
· refine ae_of_all _ (fun x f hdisj hf heq ↦ ?_)
rw [measure_iUnion hdisj hf, measure_iUnion hdisj hf]
exact tsum_congr heq | lemma | Probability | [
"Mathlib.Probability.Kernel.Disintegration.Integral"
] | Mathlib/Probability/Kernel/Disintegration/Unique.lean | eq_condKernel_of_measure_eq_compProd_real | Auxiliary lemma for `eq_condKernel_of_measure_eq_compProd`.
Uniqueness of the disintegration kernel on ℝ. |
eq_condKernel_of_measure_eq_compProd (κ : Kernel α Ω) [IsFiniteKernel κ]
(hκ : ρ = ρ.fst ⊗ₘ κ) :
∀ᵐ x ∂ρ.fst, κ x = ρ.condKernel x := by
let f := embeddingReal Ω
have hf := measurableEmbedding_embeddingReal Ω
set ρ' : Measure (α × ℝ) := ρ.map (Prod.map id f) with hρ'def
have hρ' : ρ'.fst = ρ.fst := by
ext s hs
rw [hρ'def, Measure.fst_apply, Measure.fst_apply, Measure.map_apply]
exacts [rfl, Measurable.prod measurable_fst <| hf.measurable.comp measurable_snd,
measurable_fst hs, hs, hs]
have hρ'' : ∀ᵐ x ∂ρ.fst, Kernel.map κ f x = ρ'.condKernel x := by
rw [← hρ']
refine eq_condKernel_of_measure_eq_compProd_real (Kernel.map κ f) ?_
ext s hs
conv_lhs => rw [hρ'def, hκ]
rw [Measure.map_apply (measurable_id.prodMap hf.measurable) hs, hρ',
Measure.compProd_apply hs, Measure.compProd_apply (measurable_id.prodMap hf.measurable hs)]
congr with a
rw [Kernel.map_apply' _ hf.measurable]
exacts [rfl, measurable_prodMk_left hs]
suffices ∀ᵐ x ∂ρ.fst, ∀ s, MeasurableSet s → ρ'.condKernel x s = ρ.condKernel x (f ⁻¹' s) by
filter_upwards [hρ'', this] with x hx h
rw [Kernel.map_apply _ hf.measurable] at hx
ext s hs
rw [← Set.preimage_image_eq s hf.injective,
← Measure.map_apply hf.measurable <| hf.measurableSet_image.2 hs, hx,
h _ <| hf.measurableSet_image.2 hs]
suffices ρ.map (Prod.map id f) = (ρ.fst ⊗ₘ (Kernel.map ρ.condKernel f)) by
rw [← hρ'] at this
have heq := eq_condKernel_of_measure_eq_compProd_real _ this
rw [hρ'] at heq
filter_upwards [heq] with x hx s hs
rw [← hx, Kernel.map_apply _ hf.measurable, Measure.map_apply hf.measurable hs]
ext s hs
conv_lhs => rw [← ρ.disintegrate ρ.condKernel]
rw [Measure.compProd_apply hs, Measure.map_apply (measurable_id.prodMap hf.measurable) hs,
Measure.compProd_apply]
· congr with a
rw [Kernel.map_apply' _ hf.measurable]
exacts [rfl, measurable_prodMk_left hs]
· exact measurable_id.prodMap hf.measurable hs | theorem | Probability | [
"Mathlib.Probability.Kernel.Disintegration.Integral"
] | Mathlib/Probability/Kernel/Disintegration/Unique.lean | eq_condKernel_of_measure_eq_compProd | A finite kernel which satisfies the disintegration property is almost everywhere equal to the
disintegration kernel. |
condKernel_compProd (μ : Measure α) [IsFiniteMeasure μ] (κ : Kernel α Ω) [IsMarkovKernel κ] :
(μ ⊗ₘ κ).condKernel =ᵐ[μ] κ := by
suffices κ =ᵐ[(μ ⊗ₘ κ).fst] (μ ⊗ₘ κ).condKernel by symm; rwa [Measure.fst_compProd] at this
refine eq_condKernel_of_measure_eq_compProd _ ?_
rw [Measure.fst_compProd] | lemma | Probability | [
"Mathlib.Probability.Kernel.Disintegration.Integral"
] | Mathlib/Probability/Kernel/Disintegration/Unique.lean | condKernel_compProd | null |
Kernel.apply_eq_measure_condKernel_of_compProd_eq
{ρ : Kernel α (β × Ω)} [IsFiniteKernel ρ] {κ : Kernel (α × β) Ω} [IsFiniteKernel κ]
(hκ : Kernel.fst ρ ⊗ₖ κ = ρ) (a : α) :
(fun b ↦ κ (a, b)) =ᵐ[Kernel.fst ρ a] (ρ a).condKernel := by
have : ρ a = (ρ a).fst ⊗ₘ Kernel.comap κ (fun b ↦ (a, b)) measurable_prodMk_left := by
ext s hs
conv_lhs => rw [← hκ]
rw [Measure.compProd_apply hs, Kernel.compProd_apply hs]
rfl
have h := eq_condKernel_of_measure_eq_compProd _ this
rw [Kernel.fst_apply]
filter_upwards [h] with b hb
rw [← hb, Kernel.comap_apply] | lemma | Probability | [
"Mathlib.Probability.Kernel.Disintegration.Integral"
] | Mathlib/Probability/Kernel/Disintegration/Unique.lean | Kernel.apply_eq_measure_condKernel_of_compProd_eq | null |
Kernel.condKernel_apply_eq_condKernel [CountableOrCountablyGenerated α β]
(κ : Kernel α (β × Ω)) [IsFiniteKernel κ] (a : α) :
(fun b ↦ Kernel.condKernel κ (a, b)) =ᵐ[Kernel.fst κ a] (κ a).condKernel :=
Kernel.apply_eq_measure_condKernel_of_compProd_eq (κ.disintegrate _) a | lemma | Probability | [
"Mathlib.Probability.Kernel.Disintegration.Integral"
] | Mathlib/Probability/Kernel/Disintegration/Unique.lean | Kernel.condKernel_apply_eq_condKernel | For `fst κ a`-almost all `b`, the conditional kernel `Kernel.condKernel κ` applied to `(a, b)`
is equal to the conditional kernel of the measure `κ a` applied to `b`. |
condKernel_const [CountableOrCountablyGenerated α β] (ρ : Measure (β × Ω)) [IsFiniteMeasure ρ]
(a : α) :
(fun b ↦ Kernel.condKernel (Kernel.const α ρ) (a, b)) =ᵐ[ρ.fst] ρ.condKernel := by
have h := Kernel.condKernel_apply_eq_condKernel (Kernel.const α ρ) a
simp_rw [Kernel.fst_apply, Kernel.const_apply] at h
filter_upwards [h] with b hb using hb | lemma | Probability | [
"Mathlib.Probability.Kernel.Disintegration.Integral"
] | Mathlib/Probability/Kernel/Disintegration/Unique.lean | condKernel_const | null |
eq_condKernel_of_kernel_eq_compProd [CountableOrCountablyGenerated α β]
{ρ : Kernel α (β × Ω)} [IsFiniteKernel ρ] {κ : Kernel (α × β) Ω} [IsFiniteKernel κ]
(hκ : Kernel.fst ρ ⊗ₖ κ = ρ) (a : α) :
∀ᵐ x ∂(Kernel.fst ρ a), κ (a, x) = Kernel.condKernel ρ (a, x) := by
filter_upwards [Kernel.condKernel_apply_eq_condKernel ρ a,
Kernel.apply_eq_measure_condKernel_of_compProd_eq hκ a] with a h1 h2
rw [h1, h2] | theorem | Probability | [
"Mathlib.Probability.Kernel.Disintegration.Integral"
] | Mathlib/Probability/Kernel/Disintegration/Unique.lean | eq_condKernel_of_kernel_eq_compProd | A finite kernel which satisfies the disintegration property is almost everywhere equal to the
disintegration kernel. |
IocProdIoc (a b c : ι) (x : (Π i : Ioc a b, X i) × (Π i : Ioc b c, X i)) (i : Ioc a c) : X i :=
if h : i ≤ b
then x.1 ⟨i, mem_Ioc.2 ⟨(mem_Ioc.1 i.2).1, h⟩⟩
else x.2 ⟨i, mem_Ioc.2 ⟨not_le.1 h, (mem_Ioc.1 i.2).2⟩⟩
@[measurability, fun_prop] | def | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.Embedding",
"Mathlib.Order.Restriction"
] | Mathlib/Probability/Kernel/IonescuTulcea/Maps.lean | IocProdIoc | Gluing `Ioc a b` and `Ioc b c` into `Ioc a c`. |
measurable_IocProdIoc [∀ i, MeasurableSpace (X i)] {a b c : ι} :
Measurable (IocProdIoc (X := X) a b c) := by
refine measurable_pi_lambda _ (fun i ↦ ?_)
by_cases h : i ≤ b
· simpa [IocProdIoc, h] using measurable_fst.eval
· simpa [IocProdIoc, h] using measurable_snd.eval
variable [LocallyFiniteOrderBot ι] | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.Embedding",
"Mathlib.Order.Restriction"
] | Mathlib/Probability/Kernel/IonescuTulcea/Maps.lean | measurable_IocProdIoc | null |
IicProdIoc (a b : ι) (x : (Π i : Iic a, X i) × (Π i : Ioc a b, X i)) (i : Iic b) : X i :=
if h : i ≤ a
then x.1 ⟨i, mem_Iic.2 h⟩
else x.2 ⟨i, mem_Ioc.2 ⟨not_le.1 h, mem_Iic.1 i.2⟩⟩ | def | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.Embedding",
"Mathlib.Order.Restriction"
] | Mathlib/Probability/Kernel/IonescuTulcea/Maps.lean | IicProdIoc | Gluing `Iic a` and `Ioc a b` into `Iic b`. If `b < a`, this is just a projection on the first
coordinate followed by a restriction, see `IicProdIoc_le`. |
IicProdIoc_def (a b : ι) :
IicProdIoc (X := X) a b = fun x i ↦ if h : i.1 ≤ a then x.1 ⟨i, mem_Iic.2 h⟩
else x.2 ⟨i, mem_Ioc.2 ⟨not_le.1 h, mem_Iic.1 i.2⟩⟩ := rfl | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.Embedding",
"Mathlib.Order.Restriction"
] | Mathlib/Probability/Kernel/IonescuTulcea/Maps.lean | IicProdIoc_def | When `IicProdIoc` is only partially applied (i.e. `IicProdIoc a b x` but not
`IicProdIoc a b x i`) `simp [IicProdIoc]` won't unfold the definition.
This lemma allows to unfold it by writing `simp [IicProdIoc_def]`. |
frestrictLe₂_comp_IicProdIoc {a b : ι} (hab : a ≤ b) :
(frestrictLe₂ hab) ∘ (IicProdIoc (X := X) a b) = Prod.fst := by
ext x i
simp [IicProdIoc, mem_Iic.1 i.2] | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.Embedding",
"Mathlib.Order.Restriction"
] | Mathlib/Probability/Kernel/IonescuTulcea/Maps.lean | frestrictLe₂_comp_IicProdIoc | null |
restrict₂_comp_IicProdIoc (a b : ι) :
(restrict₂ Ioc_subset_Iic_self) ∘ (IicProdIoc (X := X) a b) = Prod.snd := by
ext x i
simp [IicProdIoc, not_le.2 (mem_Ioc.1 i.2).1]
@[simp] | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.Embedding",
"Mathlib.Order.Restriction"
] | Mathlib/Probability/Kernel/IonescuTulcea/Maps.lean | restrict₂_comp_IicProdIoc | null |
IicProdIoc_self (a : ι) : IicProdIoc (X := X) a a = Prod.fst := by
ext x i
simp [IicProdIoc, mem_Iic.1 i.2] | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.Embedding",
"Mathlib.Order.Restriction"
] | Mathlib/Probability/Kernel/IonescuTulcea/Maps.lean | IicProdIoc_self | null |
IicProdIoc_le {a b : ι} (hba : b ≤ a) :
IicProdIoc (X := X) a b = (frestrictLe₂ hba) ∘ Prod.fst := by
ext x i
simp [IicProdIoc, (mem_Iic.1 i.2).trans hba] | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.Embedding",
"Mathlib.Order.Restriction"
] | Mathlib/Probability/Kernel/IonescuTulcea/Maps.lean | IicProdIoc_le | null |
IicProdIoc_comp_restrict₂ {a b : ι} :
(restrict₂ Ioc_subset_Iic_self) ∘ (IicProdIoc (X := X) a b) = Prod.snd := by
ext x i
simp [IicProdIoc, not_le.2 (mem_Ioc.1 i.2).1]
variable [∀ i, MeasurableSpace (X i)]
@[measurability, fun_prop] | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.Embedding",
"Mathlib.Order.Restriction"
] | Mathlib/Probability/Kernel/IonescuTulcea/Maps.lean | IicProdIoc_comp_restrict₂ | null |
measurable_IicProdIoc {m n : ι} : Measurable (IicProdIoc (X := X) m n) := by
refine measurable_pi_lambda _ (fun i ↦ ?_)
by_cases h : i ≤ m
· simpa [IicProdIoc, h] using measurable_fst.eval
· simpa [IicProdIoc, h] using measurable_snd.eval | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.Embedding",
"Mathlib.Order.Restriction"
] | Mathlib/Probability/Kernel/IonescuTulcea/Maps.lean | measurable_IicProdIoc | null |
IicProdIoc {a b : ι} (hab : a ≤ b) :
((Π i : Iic a, X i) × (Π i : Ioc a b, X i)) ≃ᵐ Π i : Iic b, X i where
toFun x i := if h : i ≤ a then x.1 ⟨i, mem_Iic.2 h⟩
else x.2 ⟨i, mem_Ioc.2 ⟨not_le.1 h, mem_Iic.1 i.2⟩⟩
invFun x := ⟨fun i ↦ x ⟨i.1, Iic_subset_Iic.2 hab i.2⟩, fun i ↦ x ⟨i.1, Ioc_subset_Iic_self i.2⟩⟩
left_inv := fun x ↦ by
ext i
· simp [mem_Iic.1 i.2]
· simp [not_le.2 (mem_Ioc.1 i.2).1]
right_inv := fun x ↦ funext fun i ↦ by
by_cases hi : i.1 ≤ a <;> simp [hi]
measurable_toFun := by
refine measurable_pi_lambda _ (fun x ↦ ?_)
by_cases h : x ≤ a
· simpa [h] using measurable_fst.eval
· simpa [h] using measurable_snd.eval
measurable_invFun := by dsimp; fun_prop | def | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.Embedding",
"Mathlib.Order.Restriction"
] | Mathlib/Probability/Kernel/IonescuTulcea/Maps.lean | IicProdIoc | Gluing `Iic a` and `Ioc a b` into `Iic b`. This version requires `a ≤ b` to get a measurable
equivalence. |
coe_IicProdIoc {a b : ι} (hab : a ≤ b) :
⇑(IicProdIoc (X := X) hab) = _root_.IicProdIoc a b := rfl | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.Embedding",
"Mathlib.Order.Restriction"
] | Mathlib/Probability/Kernel/IonescuTulcea/Maps.lean | coe_IicProdIoc | null |
coe_IicProdIoc_symm {a b : ι} (hab : a ≤ b) :
⇑(IicProdIoc (X := X) hab).symm =
fun x ↦ (frestrictLe₂ hab x, restrict₂ Ioc_subset_Iic_self x) := rfl | lemma | Probability | [
"Mathlib.MeasureTheory.MeasurableSpace.Embedding",
"Mathlib.Order.Restriction"
] | Mathlib/Probability/Kernel/IonescuTulcea/Maps.lean | coe_IicProdIoc_symm | null |
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