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b317718 | 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 | import Mathlib
import OAI.Probability.GaussianInformation.Kernel.Mixtures
namespace OAI
open MeasureTheory ProbabilityTheory Metric Set
open scoped ENNReal
namespace KernelPotential
open Filter
open scoped Topology NNReal
namespace Projection
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal Topology
namespace Support
lemma lintegral_pi_product {α : Type*} [MeasurableSpace α] (μ : Measure α) [SigmaFinite μ]
(f : α → ℝ≥0∞) (hf : Measurable f) (m : ℕ) :
(∫⁻ z : Fin m → α, ∏ i, f (z i) ∂Measure.pi (fun _ => μ)) = (∫⁻ z, f z ∂μ)^m := by
induction m with
| zero => simp
| succ m ih =>
rw [((measurePreserving_piFinSuccAbove (fun _ : Fin (m+1) => μ) 0).symm).lintegral_map_equiv]
simp only [MeasurableEquiv.piFinSuccAbove_symm_apply, Fin.insertNthEquiv,
Fin.insertNth_zero, Fin.prod_univ_succ, Fin.zero_succAbove]
change (∫⁻ z : α × (Fin m → α), f z.1 * ∏ i, f (z.2 i)
∂μ.prod (Measure.pi (fun _ => μ))) = _
rw [lintegral_prod_mul (μ := μ) (ν := Measure.pi (fun _ : Fin m => μ))
(g := fun z : Fin m → α => ∏ i, f (z i)) hf.aemeasurable (by
exact (Finset.measurable_prod _ fun i _ => hf.comp (measurable_pi_apply i)).aemeasurable),
ih, pow_succ']
noncomputable def closedBox {k : ℕ} (c : Fin k → ℝ) (η : ℝ) : Set (Fin k → ℝ) :=
{u | ∀ i, |u i-c i| ≤ η}
lemma measurableSet_closedBox {k : ℕ} (c : Fin k → ℝ) (η : ℝ) :
MeasurableSet (closedBox c η) := by
simp only [closedBox, ofPred_forall]
exact MeasurableSet.iInter fun i =>
measurableSet_le ((measurable_pi_apply i).sub measurable_const).abs measurable_const
lemma volume_closedBox {k : ℕ} (c : Fin k → ℝ) (η : ℝ) (hη : 0 ≤ η) :
volume (closedBox c η) = ENNReal.ofReal ((2*η)^k) := by
have heq : closedBox c η = univ.pi (fun i => Icc (c i-η) (c i+η)) := by
ext u
simp only [closedBox, mem_ofPred_eq, Set.mem_pi, mem_univ, forall_true_left,
mem_Icc, abs_le]
congr! 1
constructor <;> intro h <;> constructor <;> linarith [h.1, h.2]
rw [heq, volume_pi, Measure.pi_pi]
simp only [Real.volume_Icc, show ∀ i, c i+η-(c i-η)=2*η from fun i => by ring,
Finset.prod_const, Finset.card_univ, Fintype.card_fin]
rw [ENNReal.ofReal_pow (by positivity)]
noncomputable def boxScale (k : ℕ) (η : ℝ) : ℝ≥0∞ := (ENNReal.ofReal ((2*η)^k))⁻¹
noncomputable def boxKernel {k : ℕ} (η : ℝ) (c u : Fin k → ℝ) : ℝ≥0∞ :=
(closedBox c η).indicator (fun _ => boxScale k η) u
lemma measurable_boxKernel {k : ℕ} (η : ℝ) :
Measurable (fun p : (Fin k → ℝ) × (Fin k → ℝ) => boxKernel η p.1 p.2) := by
have hs : MeasurableSet {p : (Fin k → ℝ) × (Fin k → ℝ) | ∀ i, |p.2 i-p.1 i| ≤ η} := by
simp only [Set.ofPred_forall]
exact MeasurableSet.iInter fun i => measurableSet_le
(((measurable_pi_apply i).comp measurable_snd).sub
((measurable_pi_apply i).comp measurable_fst)).abs measurable_const
exact measurable_const.indicator hs
lemma boxKernel_le {k : ℕ} (η : ℝ) (c u : Fin k → ℝ) :
boxKernel η c u ≤ boxScale k η := Set.indicator_le_self (closedBox c η) (fun _ => boxScale k η) u
lemma boxKernel_ne_zero_mem {k : ℕ} {η : ℝ} {c u : Fin k → ℝ}
(h : boxKernel η c u ≠ 0) : u ∈ closedBox c η := by
by_contra hu
exact h (Set.indicator_of_notMem hu _)
lemma lintegral_boxKernel {k : ℕ} (η : ℝ) (hη : 0 < η) (c : Fin k → ℝ) :
(∫⁻ u, boxKernel η c u) = 1 := by
change (∫⁻ u, (closedBox c η).indicator (fun _ => boxScale k η) u) = 1
rw [lintegral_indicator (measurableSet_closedBox c η), lintegral_const,
Measure.restrict_apply_univ, volume_closedBox c η hη.le]
exact ENNReal.inv_mul_cancel (by positivity) ENNReal.ofReal_ne_top
lemma lintegral_boxKernel_product_le {k q : ℕ} (η : ℝ) (hη : 0 < η)
(c : Fin (q+1) → Fin k → ℝ) :
(∫⁻ u, ∏ i, boxKernel η (c i) u) ≤
if ∀ (i : Fin q) r, |c i.succ r-c 0 r| ≤ 2*η then boxScale k η ^ q else 0 := by
classical
by_cases hd : ∀ (i : Fin q) r, |c i.succ r-c 0 r| ≤ 2*η
· rw [ite_eq_left hd]
calc
_ ≤ ∫⁻ u, boxScale k η ^ q * boxKernel η (c 0) u := by
apply lintegral_mono
intro u
dsimp only
rw [Fin.prod_univ_succ, mul_comm]
apply mul_le_mul_left
simpa using Finset.prod_le_prod (s := (Finset.univ : Finset (Fin q)))
(fun i _ => boxKernel_le η (c i.succ) u)
_ = boxScale k η ^ q * ∫⁻ u, boxKernel η (c 0) u :=
lintegral_const_mul _ ((measurable_boxKernel η).comp (measurable_const.prodMk measurable_id))
_ = boxScale k η ^ q := by rw [lintegral_boxKernel η hη, mul_one]
· rw [ite_eq_right hd]
have hz : ∀ u, (∏ i, boxKernel η (c i) u) = 0 := by
intro u
by_contra hp
have hall : ∀ i, u ∈ closedBox (c i) η := by
intro i
apply boxKernel_ne_zero_mem
exact (Finset.prod_ne_zero_iff.mp hp) i (Finset.mem_univ _)
apply hd
intro i r
have h0 := hall 0 r
have hi := hall i.succ r
have he := abs_sub_le (c i.succ r) (u r) (c 0 r)
rw [abs_sub_comm (c i.succ r) (u r)] at he
exact he.trans (by linarith)
simp only [hz, lintegral_zero, le_refl]
end Support
open InnerProductSpace Matrix MeasureTheory Set
open scoped RealInnerProductSpace ENNReal
end Projection
end KernelPotential
end OAI
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