| import OAI.Analysis.FiniteFactor.Coupling.FiniteLaw |
| import OAI.Analysis.FiniteFactor.ProductDensityLimit |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Filter Finset |
| open scoped Topology ENNReal |
| noncomputable section |
|
|
| /-- Finite-coordinate factorization under the actual infinite product measure. -/ |
| theorem integral_infinitePi_finset_prod {ι : Type*} {X : ι → Type*} |
| [∀ i, MeasurableSpace (X i)] (μ : ∀ i, Measure (X i)) [∀ i, IsProbabilityMeasure (μ i)] |
| (I : Finset ι) (f : ∀ i, X i → ℝ) (hf : ∀ i, Measurable (f i)) : |
| (∫ x, ∏ i ∈ I, f i (x i) ∂Measure.infinitePi μ) = ∏ i ∈ I, ∫ t, f i t ∂μ i := by |
| classical |
| have he (x : ∀ i, X i) : (∏ i ∈ I, f i (x i)) = ∏ i : I, f i (x i) := |
| (Finset.prod_coe_sort I (fun i => f i (x i))).symm |
| simp_rw [he] |
| change (∫ x, (∏ i : I, f i (I.restrict x i)) ∂Measure.infinitePi μ) = _ |
| rw [integral_restrict_infinitePi μ |
| (f := fun x : ∀ i : I, X i => ∏ i : I, f i (x i)) |
| ((Finset.measurable_prod _ fun (i : I) _ => (hf i).comp (measurable_pi_apply i)).aestronglyMeasurable)] |
| rw [integral_fintype_prod_eq_prod] |
| exact Finset.prod_coe_sort I (fun i => ∫ t, f i t ∂μ i) |
|
|
| namespace CouplingParameters |
| variable {ι : Type*} |
|
|
| def prefixDensity (p : ι → CouplingParameters) (right : ι → Bool) |
| (I : Finset ι) (x : ι → ℝ) : ℝ := ∏ i ∈ I, (p i).endpointDensity (right i) (x i) |
|
|
| theorem prefixDensity_measurable (p : ι → CouplingParameters) (right : ι → Bool) (I : Finset ι) : |
| Measurable (prefixDensity p right I) := |
| Finset.measurable_prod _ fun i _ => ((p i).endpointDensity_measurable (right i)).comp (measurable_pi_apply i) |
|
|
| theorem prefixDensity_nonneg (p : ι → CouplingParameters) (right : ι → Bool) |
| (I : Finset ι) (x : ι → ℝ) : 0 ≤ prefixDensity p right I x := |
| Finset.prod_nonneg fun i _ => ((p i).diagonalMass_pos.trans_le ((p i).endpointDensity_lower (right i) (x i))).le |
|
|
| theorem prefixDensity_memLp (p : ι → CouplingParameters) (right : ι → Bool) (I : Finset ι) : |
| MemLp (prefixDensity p right I) 2 (Measure.infinitePi (fun _ : ι => unitIntervalMeasure)) := by |
| apply MemLp.of_bound (prefixDensity_measurable p right I).aestronglyMeasurable (10 ^ I.card) |
| apply Eventually.of_forall |
| intro x |
| rw [Real.norm_of_nonneg (prefixDensity_nonneg p right I x)] |
| calc |
| _ ≤ ∏ i ∈ I, (10 : ℝ) := Finset.prod_le_prod₀ |
| (fun i _ => ((p i).diagonalMass_pos.trans_le ((p i).endpointDensity_lower (right i) (x i))).le) |
| (fun i _ => ((p i).endpointDensity_lt_ten (right i) (x i)).le) |
| _ = _ := by simp |
|
|
| theorem prefixDensity_integral (p : ι → CouplingParameters) (right : ι → Bool) (I : Finset ι) : |
| (∫ x, prefixDensity p right I x ∂Measure.infinitePi (fun _ : ι => unitIntervalMeasure)) = 1 := by |
| unfold prefixDensity |
| rw [integral_infinitePi_finset_prod _ _ _ (fun i => (p i).endpointDensity_measurable (right i))] |
| simp [endpointDensity_integral] |
|
|
| theorem prefixDensity_sq_integral (p : ι → CouplingParameters) (right : ι → Bool) (I : Finset ι) : |
| (∫ x, prefixDensity p right I x ^ 2 ∂Measure.infinitePi (fun _ : ι => unitIntervalMeasure)) = |
| ∏ i ∈ I, ∫ t, (p i).endpointDensity (right i) t ^ 2 ∂unitIntervalMeasure := by |
| simp only [prefixDensity, ← Finset.prod_pow] |
| exact integral_infinitePi_finset_prod _ _ _ (fun i => ((p i).endpointDensity_measurable (right i)).pow_const 2) |
|
|
| theorem prefixDensity_energy (p : ι → CouplingParameters) (right : ι → Bool) (I : Finset ι) : |
| (∫ x, prefixDensity p right I x ^ 2 ∂Measure.infinitePi (fun _ : ι => unitIntervalMeasure)) ≤ |
| Real.exp (∑ i ∈ I, 20*(2*(p i).kappa^2/(p i).m + (p i).tagMass)) := by |
| classical |
| rw [prefixDensity_sq_integral, ← Finset.prod_coe_sort I, |
| ← Finset.sum_coe_sort I] |
| exact product_energy_le_exp _ _ (fun i => integral_nonneg (fun _ => sq_nonneg _)) |
| (fun i => (p i).endpointDensity_source_energy (right i)) |
|
|
| /-- Exact orthogonality identity for nested finite product densities. -/ |
| theorem prefixDensity_inner (p : ι → CouplingParameters) (right : ι → Bool) |
| {I J : Finset ι} (hIJ : I ⊆ J) : |
| (∫ x, prefixDensity p right I x * prefixDensity p right J x |
| ∂Measure.infinitePi (fun _ : ι => unitIntervalMeasure)) = |
| ∫ x, prefixDensity p right I x ^ 2 ∂Measure.infinitePi (fun _ : ι => unitIntervalMeasure) := by |
| classical |
| have he (x : ι → ℝ) : prefixDensity p right I x * prefixDensity p right J x = |
| ∏ i ∈ J, if i ∈ I then (p i).endpointDensity (right i) (x i)^2 |
| else (p i).endpointDensity (right i) (x i) := by |
| have hi : prefixDensity p right I x = ∏ i ∈ J, |
| if i ∈ I then (p i).endpointDensity (right i) (x i) else 1 := by |
| rw [← Finset.prod_subset hIJ] |
| · apply Finset.prod_congr rfl |
| intro i hi |
| simp [hi] |
| · intro i _ hi |
| simp [hi] |
| rw [hi, prefixDensity, ← Finset.prod_mul_distrib] |
| apply Finset.prod_congr rfl |
| intro i _ |
| split <;> simp_all [sq] |
| simp_rw [he] |
| rw [integral_infinitePi_finset_prod (fun _ : ι => unitIntervalMeasure) J |
| (fun i t => if i ∈ I then (p i).endpointDensity (right i) t ^ 2 |
| else (p i).endpointDensity (right i) t) (fun i => by |
| split |
| · exact ((p i).endpointDensity_measurable (right i)).pow_const 2 |
| · exact (p i).endpointDensity_measurable (right i)), prefixDensity_sq_integral] |
| calc |
| _ = ∏ i ∈ I, ∫ t, (if i ∈ I then (p i).endpointDensity (right i) t ^ 2 |
| else (p i).endpointDensity (right i) t) ∂unitIntervalMeasure := by |
| symm |
| apply Finset.prod_subset hIJ |
| intro i _ hi |
| simp [hi, endpointDensity_integral] |
| _ = _ := by |
| apply Finset.prod_congr rfl |
| intro i hi |
| simp [hi] |
|
|
| /-- The lifted prefix density has exactly the finite endpoint law on every |
| Borel cylinder at that prefix. -/ |
| theorem prefixDensity_cylinder (p : ι → CouplingParameters) (right : ι → Bool) |
| (I : Finset ι) {S : Set (I → ℝ)} (hS : MeasurableSet S) : |
| ENNReal.ofReal (∫ x in (I.restrict (π := fun _ : ι => ℝ)) ⁻¹' S, prefixDensity p right I x |
| ∂Measure.infinitePi (fun _ : ι => unitIntervalMeasure)) = |
| Measure.pi (fun i : I => (p i).selectedLaw true (right i)) S := by |
| classical |
| rw [finiteProductLaw_density, withDensity_apply _ hS] |
| rw [← Measure.infinitePi_map_restrict (fun _ : ι => unitIntervalMeasure), |
| setLIntegral_map hS (finiteProductDensity_measurable |
| (fun i : I => p i) (fun _ => true) (fun i => right i)).ennreal_ofReal (measurable_restrict I)] |
| have he (x : ι → ℝ) : finiteProductDensity (fun i : I => p i) |
| (fun _ => true) (fun i => right i) ((I.restrict (π := fun _ : ι => ℝ)) x) = prefixDensity p right I x := by |
| exact Finset.prod_coe_sort I (fun i => (p i).endpointDensity (right i) (x i)) |
| simp_rw [he] |
| exact ofReal_integral_eq_lintegral_ofReal |
| ((prefixDensity_memLp p right I).integrable (by norm_num)).restrict |
| (Eventually.of_forall (prefixDensity_nonneg p right I)) |
|
|
| /-- Enlarging the prefix does not change any earlier cylinder law. -/ |
| theorem prefixDensity_cylinder_of_subset (p : ι → CouplingParameters) (right : ι → Bool) |
| {J I : Finset ι} (hJI : J ⊆ I) {S : Set (J → ℝ)} (hS : MeasurableSet S) : |
| ENNReal.ofReal (∫ x in (J.restrict (π := fun _ : ι => ℝ)) ⁻¹' S, prefixDensity p right I x |
| ∂Measure.infinitePi (fun _ : ι => unitIntervalMeasure)) = |
| Measure.pi (fun i : J => (p i).selectedLaw true (right i)) S := by |
| have he : (J.restrict (π := fun _ : ι => ℝ)) ⁻¹' S = (I.restrict (π := fun _ : ι => ℝ)) ⁻¹' ((Finset.restrict₂ (π := fun _ : ι => ℝ) hJI : (I → ℝ) → (J → ℝ)) ⁻¹' S) := by |
| rw [← Set.preimage_comp, Finset.restrict₂_comp_restrict hJI] |
| rw [he, prefixDensity_cylinder p right I (hS.preimage (measurable_restrict₂ hJI))] |
| have hh := congrArg (fun ν : Measure (J → ℝ) => ν S) |
| (isProjectiveMeasureFamily_pi (fun i => (p i).selectedLaw true (right i)) I J hJI) |
| rw [Measure.map_apply (show Measurable (Finset.restrict₂ (π := fun _ : ι => ℝ) hJI) from measurable_restrict₂ hJI) hS] at hh |
| exact hh.symm |
|
|
| end CouplingParameters |
| end |
| end FiniteFactor |
|
|
| end OAI |
|
|
