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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 | import Mathlib
import OAI.Probability.GaussianInformation.Kernel.NoisySums
namespace OAI
open MeasureTheory ProbabilityTheory Metric Set
open scoped ENNReal
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal
open MeasureTheory ProbabilityTheory
open MeasureTheory
open scoped ENNReal
namespace KernelPotential.Continuity
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal Topology
open Coercivity Sphere EntropySupport
noncomputable def densityLaw {d : ℕ} (f : Ambient d → ℝ) : Measure (Ambient d) :=
(sphereProbability d).withDensity (fun x => ENNReal.ofReal (f x))
lemma tendsto_J {d : ℕ} (hd : 16 ≤ d) (f : ℕ → Ambient d → ℝ) (g : Ambient d → ℝ)
(hf : ∀ n,Measurable (f n)) (hg : Measurable g) (_hf0 : ∀ n x,0 ≤ f n x) {H : ℝ} (_hH : 0 ≤ H)
(hfH : ∀ n x,f n x ≤ H)
(hlim : ∀ᵐ x ∂sphereProbability d,Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (g x)))
(s : Ambient d) :
Filter.Tendsto (fun n => J (densityLaw (f n)) s) Filter.atTop (𝓝 (J (densityLaw g) s)) := by
let K := fun u : Ambient d => (ENNReal.ofReal ‖u-s‖)⁻¹^(2*(d/16))
have hK : Measurable K := ((measurable_id.sub_const s).norm.ennreal_ofReal.inv).pow_const _
have hKI : (∫⁻ u,K u ∂sphereProbability d) ≠ ∞ := by
apply ne_top_of_le_ne_top ENNReal.ofReal_ne_top
exact inverse_kernel_bound (sphereProbability d) (by positivity : 0 ≤ (9*3^d:ℝ))
(by omega) (by omega) (fun z r hr => cap_bound (by omega) z hr) s
have hl (h : Ambient d → ℝ) (hh : Measurable h) : J (densityLaw h) s =
∫⁻ u,ENNReal.ofReal (h u)*K u ∂sphereProbability d := by
unfold J densityLaw
exact lintegral_withDensity_eq_lintegral_mul _ hh.ennreal_ofReal hK
rw [hl g hg]
simp_rw [hl _ (hf _)]
apply tendsto_lintegral_of_dominated_convergence (fun u => ENNReal.ofReal H*K u)
(fun n => (hf n).ennreal_ofReal.mul hK)
(fun n => ae_of_all _ (fun u => mul_le_mul_left (ENNReal.ofReal_le_ofReal (hfH n u)) _))
· rw [lintegral_const_mul _ hK]
exact ENNReal.mul_ne_top ENNReal.ofReal_ne_top hKI
· filter_upwards [hlim,ae_lt_top hK hKI] with u hu hKu
exact ENNReal.Tendsto.mul_const (ENNReal.continuous_ofReal.continuousAt.tendsto.comp hu) (Or.inr hKu.ne)
lemma tendsto_Phi {d : ℕ} (hd : 16 ≤ d) (f : ℕ → Ambient d → ℝ) (g : Ambient d → ℝ)
(hf : ∀ n,Measurable (f n)) (hg : Measurable g)
(hf0 : ∀ n x,0 ≤ f n x) (hg0 : ∀ x,0 ≤ g x)
{H : ℝ} (hH : 0 < H) (hfH : ∀ n x,f n x ≤ H) (hgH : ∀ x,g x ≤ H)
(hfp : ∀ n,IsProbabilityMeasure (densityLaw (f n)))
(hgp : IsProbabilityMeasure (densityLaw g))
(hlim : ∀ᵐ x ∂sphereProbability d,Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (g x))) :
Filter.Tendsto (fun n => Phi (densityLaw (f n))) Filter.atTop (𝓝 (Phi (densityLaw g))) := by
have (n : ℕ) : IsProbabilityMeasure (densityLaw (f n)) := hfp n
have : IsProbabilityMeasure (densityLaw g) := hgp
have hfB (n : ℕ) : BoundedDensity (densityLaw (f n)) := ⟨H,hH,density_bounded _ _ (hfH n)⟩
have hgB : BoundedDensity (densityLaw g) := ⟨H,hH,density_bounded _ _ hgH⟩
have hformula (n : ℕ) := Phi_density_formula hd (f n) (hf n) (hf0 n) (hfH n) _ (hfB n) rfl
have hgformula := Phi_density_formula hd g hg hg0 hgH _ hgB rfl
rw [hgformula]
simp_rw [hformula]
let a : ℝ := (1/2:ℝ)^(2*(d/16))
let b : ℝ := H*(1+(9*3^d:ℝ)*2^((d-1)+2*(d/16)+1))
have ha : 0 < a := by dsimp [a]; positivity
obtain ⟨C,hC⟩ := ((isCompact_Icc (a := (0:ℝ)) (b := H)).image continuous_xlog).isBounded.exists_norm_le
have hbound (n : ℕ) : ∀ᵐ s ∂sphereProbability d,
‖xlog (f n s)-f n s*Real.log (J (densityLaw (f n)) s).toReal‖ ≤
C+H*(|Real.log a|+|Real.log b|) := by
filter_upwards [ae_unit_norm (by omega : 0 < d)] with s hs
have hj := J_real_bounds hd (densityLaw (f n)) hH (density_bounded _ _ (hfH n)) s hs
change a ≤ _ ∧ _ ≤ b at hj
have hll := Real.log_le_log ha hj.1
have hlu := Real.log_le_log (ha.trans_le hj.1) hj.2
have hab : |Real.log (J (densityLaw (f n)) s).toReal| ≤ |Real.log a|+|Real.log b| := by
apply abs_le.mpr
constructor
· have h := neg_abs_le (Real.log a); have hh := abs_nonneg (Real.log b); linarith
· have h := le_abs_self (Real.log b); have hh := abs_nonneg (Real.log a); linarith
have hfx : ‖xlog (f n s)‖ ≤ C := hC _ (mem_image_of_mem _ ⟨hf0 n s,hfH n s⟩)
calc
_ ≤ ‖xlog (f n s)‖+‖f n s*Real.log (J (densityLaw (f n)) s).toReal‖ := norm_sub_le _ _
_ ≤ C+H*(|Real.log a|+|Real.log b|) := by
rw [norm_mul,Real.norm_eq_abs (f n s),abs_of_nonneg (hf0 n s),Real.norm_eq_abs]
exact add_le_add hfx (mul_le_mul (hfH n s) hab (abs_nonneg _) hH.le)
apply tendsto_integral_of_dominated_convergence (fun _ => C+H*(|Real.log a|+|Real.log b|))
(fun n => ((continuous_xlog.measurable.comp (hf n)).sub ((hf n).mul (measurable_J _).ennreal_toReal.log)).aestronglyMeasurable)
(integrable_const _) hbound
filter_upwards [hlim,ae_unit_norm (by omega : 0 < d)] with s hs hsn
have hJt := tendsto_J hd f g hf hg hf0 hH.le hfH hlim s
have hJfin : J (densityLaw g) s ≠ ∞ := ne_top_of_le_ne_top ENNReal.ofReal_ne_top (bounded_J hd (densityLaw g) hH (density_bounded _ _ hgH) s)
have hJp : 0 < (J (densityLaw g) s).toReal := ha.trans_le (J_real_bounds hd (densityLaw g) hH (density_bounded _ _ hgH) s hsn).1
have hjr := (ENNReal.continuousAt_toReal hJfin).tendsto.comp hJt
exact (continuous_xlog.continuousAt.tendsto.comp hs).sub (hs.mul ((Real.continuousAt_log hJp.ne').tendsto.comp hjr))
end KernelPotential.Continuity
end OAI
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