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ddc84c5 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 | 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
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