| 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 |
| |
