| import OAI.NumberTheory.DukePrimeDegree.Dynamics.HopfPairs |
| import Mathlib.Probability.Kernel.Disintegration.StandardBorel |
|
|
| namespace OAI |
|
|
| /-! |
| # Pairs sampled from a conditional measure |
|
|
| Resampling the second coordinate from its conditional distribution |
| preserves both coordinate marginals. These pairs let the Hopf principle |
| apply inside a foliation chart, with both ambient marginals dominated |
| by the original finite invariant measure. |
| -/ |
|
|
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped ENNReal Topology Uniformity |
|
|
| namespace DukePrimeDegree |
|
|
| variable {S B : Type*} [MeasurableSpace S] [MeasurableSpace B] |
| [StandardBorelSpace B] [Nonempty B] |
|
|
| / |
| coordinate using its conditional law. -/ |
| noncomputable def conditionalPairMeasure (ρ : Measure (S × B)) [IsFiniteMeasure ρ] : |
| Measure ((S × B) × B) := |
| ρ ⊗ₘ ρ.condKernel.comap Prod.fst measurable_fst |
| |
| instance (ρ : Measure (S × B)) [IsFiniteMeasure ρ] : |
| IsFiniteMeasure (conditionalPairMeasure ρ) := by |
| unfold conditionalPairMeasure |
| infer_instance |
| |
| lemma conditionalPairMeasure_left (ρ : Measure (S × B)) [IsFiniteMeasure ρ] : |
| (conditionalPairMeasure ρ).map Prod.fst = ρ := by |
| exact Measure.fst_compProd ρ _ |
| |
| lemma conditionalPairMeasure_right (ρ : Measure (S × B)) [IsFiniteMeasure ρ] : |
| (conditionalPairMeasure ρ).map (fun z => (z.1.1, z.2)) = ρ := by |
| ext A hA |
| have hm : Measurable (fun z : (S × B) × B => (z.1.1, z.2)) := by fun_prop |
| rw [Measure.map_apply hm hA, conditionalPairMeasure, |
| Measure.compProd_apply (hA.preimage hm)] |
| change (∫⁻ z : S × B, ρ.condKernel z.1 (Prod.mk z.1 ⁻¹' A) ∂ρ) = ρ A |
| calc |
| _ = ∫⁻ s, ρ.condKernel s (Prod.mk s ⁻¹' A) ∂ρ.fst := |
| (lintegral_map (Kernel.measurable_kernel_prodMk_left hA) measurable_fst).symm |
| _ = (ρ.fst ⊗ₘ ρ.condKernel) A := (Measure.compProd_apply hA).symm |
| _ = ρ A := by rw [Measure.disintegrate] |
| |
| lemma map_conditionalPair_left {α : Type*} [MeasurableSpace α] |
| (ρ : Measure (S × B)) [IsFiniteMeasure ρ] |
| (R : S × B → α) (hR : Measurable R) : |
| (conditionalPairMeasure ρ).map (fun z => R z.1) = ρ.map R := by |
| change (conditionalPairMeasure ρ).map (R ∘ Prod.fst) = ρ.map R |
| rw [← Measure.map_map hR measurable_fst, conditionalPairMeasure_left] |
| |
| lemma map_conditionalPair_right {α : Type*} [MeasurableSpace α] |
| (ρ : Measure (S × B)) [IsFiniteMeasure ρ] |
| (R : S × B → α) (hR : Measurable R) : |
| (conditionalPairMeasure ρ).map (fun z => R (z.1.1, z.2)) = ρ.map R := by |
| let r : (S × B) × B → S × B := fun z => (z.1.1, z.2) |
| have hr : Measurable r := by fun_prop |
| change (conditionalPairMeasure ρ).map (R ∘ r) = ρ.map R |
| rw [← Measure.map_map hR hr] |
| exact congrArg (Measure.map R) (conditionalPairMeasure_right ρ) |
| |
| / |
| lemma ae_conditionalPair_of_ambient (ρ : Measure (S × B)) [IsFiniteMeasure ρ] |
| {P : S × B → Prop} (hP : ∀ᵐ z ∂ρ, P z) : |
| ∀ᵐ z ∂conditionalPairMeasure ρ, P z.1 ∧ P (z.1.1, z.2) := by |
| have hleft : ∀ᵐ z ∂conditionalPairMeasure ρ, P z.1 := by |
| apply ae_of_ae_map measurable_fst.aemeasurable |
| rwa [conditionalPairMeasure_left] |
| have hright : ∀ᵐ z ∂conditionalPairMeasure ρ, P (z.1.1, z.2) := by |
| apply ae_of_ae_map (by fun_prop : |
| Measurable (fun z : (S × B) × B => (z.1.1, z.2))).aemeasurable |
| rwa [conditionalPairMeasure_right] |
| exact hleft.and hright |
| |
| / |
| constant conditional values for every integrable invariant observable. |
| The conclusion is with respect to the actual conditional kernel. -/ |
| theorem ae_conditionalPair_eq_of_asymptotic_orbits |
| {α : Type*} [UniformSpace α] [T2Space α] [NormalSpace α] |
| [WeaklyLocallyCompactSpace α] [MeasurableSpace α] [BorelSpace α] |
| (μ : Measure α) [μ.Regular] |
| (ρ : Measure (S × B)) [IsFiniteMeasure ρ] |
| (R : S × B → α) (hR : Measurable R) (hRμ : ρ.map R ≤ μ) |
| (T : α → α) (hT : MeasurePreserving T μ μ) |
| (hasymptotic : ∀ᵐ z ∂conditionalPairMeasure ρ, |
| Tendsto (fun n => (T^[n] (R z.1), T^[n] (R (z.1.1, z.2)))) atTop (𝓤 α)) |
| (f : α → ℝ) (hf : Integrable f μ) (hfinv : (fun x => f (T x)) =ᵐ[μ] f) : |
| ∀ᵐ z ∂conditionalPairMeasure ρ, f (R z.1) = f (R (z.1.1, z.2)) := by |
| have hl : (conditionalPairMeasure ρ).map (fun z => R z.1) ≤ μ := by |
| rw [map_conditionalPair_left ρ R hR] |
| exact hRμ |
| have hr : (conditionalPairMeasure ρ).map (fun z => R (z.1.1, z.2)) ≤ μ := by |
| rw [map_conditionalPair_right ρ R hR] |
| exact hRμ |
| have hh := ae_eq_of_asymptotic_pair_orbits μ (conditionalPairMeasure ρ) |
| (fun z => R z.1) (fun z => R (z.1.1, z.2)) |
| (hR.comp measurable_fst) (by fun_prop) hl hr T hT hasymptotic f hf hfinv |
| exact hh |
| |
| |
| theorem ae_conditional_eq_of_asymptotic_orbits |
| {α : Type*} [UniformSpace α] [T2Space α] [NormalSpace α] |
| [WeaklyLocallyCompactSpace α] [MeasurableSpace α] [BorelSpace α] |
| (μ : Measure α) [μ.Regular] |
| (ρ : Measure (S × B)) [IsFiniteMeasure ρ] |
| (R : S × B → α) (hR : Measurable R) (hRμ : ρ.map R ≤ μ) |
| (T : α → α) (hT : MeasurePreserving T μ μ) |
| (hasymptotic : ∀ᵐ z ∂conditionalPairMeasure ρ, |
| Tendsto (fun n => (T^[n] (R z.1), T^[n] (R (z.1.1, z.2)))) atTop (𝓤 α)) |
| (f : α → ℝ) (hf : Integrable f μ) (hfinv : (fun x => f (T x)) =ᵐ[μ] f) : |
| ∀ᵐ z ∂ρ, ∀ᵐ t ∂ρ.condKernel z.1, f (R z) = f (R (z.1, t)) := by |
| have hh := ae_conditionalPair_eq_of_asymptotic_orbits μ ρ R hR hRμ T hT |
| hasymptotic f hf hfinv |
| exact Measure.ae_ae_of_ae_compProd |
| (μ := ρ) (κ := ρ.condKernel.comap Prod.fst measurable_fst) |
| (p := fun z : (S × B) × B => f (R z.1) = f (R (z.1.1, z.2))) hh |
| |
| end DukePrimeDegree |
| |
| end OAI |
| |
