| import OAI.Analysis.FiniteFactor.Sampling.LayerCake |
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| namespace OAI |
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| namespace FiniteFactor |
| open MeasureTheory Filter |
| open scoped Classical ENNReal |
| noncomputable section |
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| theorem sampling_measureReal_le_one {α : Type*} [MeasurableSpace α] (ρ : Measure α) [IsProbabilityMeasure ρ] |
| (S : Set α) : |(ρ S).toReal| ≤ 1 := by |
| rw [abs_of_nonneg ENNReal.toReal_nonneg] |
| calc |
| _ ≤ (ρ Set.univ).toReal := ENNReal.toReal_mono (measure_ne_top _ _) (measure_mono (Set.subset_univ S)) |
| _ = 1 := by simp |
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| /-- Joint superlevel probabilities are measurable in both thresholds. -/ |
| theorem sampling_levelMass_measurable {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsFiniteMeasure μ] (f g : Ω → ℝ) |
| (hf : Measurable f) (hg : Measurable g) : |
| Measurable (fun p : ℝ × ℝ => (μ {x | p.1 ≤ f x ∧ p.2 ≤ g x}).toReal) := by |
| let K : (ℝ × ℝ) × Ω → ℝ := fun p => |
| if p.1.1 ≤ f p.2 ∧ p.1.2 ≤ g p.2 then 1 else 0 |
| have hK : Measurable K := measurable_const.ite |
| ((measurableSet_le measurable_fst.fst (hf.comp measurable_snd)).inter |
| (measurableSet_le measurable_fst.snd (hg.comp measurable_snd))) measurable_const |
| have he (p : ℝ × ℝ) : (∫ x, K (p,x) ∂μ) = |
| (μ {x | p.1 ≤ f x ∧ p.2 ≤ g x}).toReal := by |
| have hs : MeasurableSet {x | p.1 ≤ f x ∧ p.2 ≤ g x} := |
| (measurableSet_le measurable_const hf).inter (measurableSet_le measurable_const hg) |
| have hk : (fun x => K (p,x)) = |
| {x | p.1 ≤ f x ∧ p.2 ≤ g x}.indicator (fun _ => (1 : ℝ)) := by |
| funext x; simp [K,Set.indicator] |
| rw [hk,integral_indicator hs] |
| simp [measureReal_def] |
| have hKm : StronglyMeasurable (Function.uncurry (fun p x => K (p,x))) := hK.stronglyMeasurable |
| simpa only [he] using (hKm.integral_prod_right (ν := μ)).measurable |
|
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| /-- Uniform superlevel discrepancy controls weighted pair integrals. The |
| threshold square has area C^2; no density for the joint law is needed. -/ |
| theorem sampling_layer_discrepancy {Ω Ψ : Type*} [MeasurableSpace Ω] [MeasurableSpace Ψ] |
| (μ : Measure Ω) (ν : Measure Ψ) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| (f g : Ω → ℝ) (f₀ g₀ : Ψ → ℝ) |
| (hf : Measurable f) (hg : Measurable g) (hf₀ : Measurable f₀) (hg₀ : Measurable g₀) |
| (C δ : ℝ) (hC : 0 ≤ C) |
| (hfb : ∀ x, 0 ≤ f x ∧ f x ≤ C) (hgb : ∀ x, 0 ≤ g x ∧ g x ≤ C) |
| (hf₀b : ∀ x, 0 ≤ f₀ x ∧ f₀ x ≤ C) (hg₀b : ∀ x, 0 ≤ g₀ x ∧ g₀ x ≤ C) |
| (hlevel : ∀ r t : ℝ, 0 < r → r ≤ C → 0 < t → t ≤ C → |
| |(μ {x | r ≤ f x ∧ t ≤ g x}).toReal - |
| (ν {x | r ≤ f₀ x ∧ t ≤ g₀ x}).toReal| ≤ δ) : |
| |(∫ x, f x*g x ∂μ)-(∫ x, f₀ x*g₀ x ∂ν)| ≤ δ*C^2 := by |
| let τ := volume.restrict (Set.Ioc 0 C) |
| let F : ℝ × ℝ → ℝ := fun p => (μ {x | p.1 ≤ f x ∧ p.2 ≤ g x}).toReal |
| let G : ℝ × ℝ → ℝ := fun p => (ν {x | p.1 ≤ f₀ x ∧ p.2 ≤ g₀ x}).toReal |
| have hFm : Measurable F := sampling_levelMass_measurable μ f g hf hg |
| have hGm : Measurable G := sampling_levelMass_measurable ν f₀ g₀ hf₀ hg₀ |
| have hFi := sampling_integrable_bounded (τ.prod τ) F hFm 1 (fun p => by dsimp [F]; exact sampling_measureReal_le_one μ _) |
| have hGi := sampling_integrable_bounded (τ.prod τ) G hGm 1 (fun p => by dsimp [G]; exact sampling_measureReal_le_one ν _) |
| rw [sampling_product_layerCake μ f g hf hg C hC hfb hgb, |
| sampling_product_layerCake ν f₀ g₀ hf₀ hg₀ C hC hf₀b hg₀b] |
| change |(∫ r, ∫ t, F (r,t) ∂τ ∂τ)-(∫ r, ∫ t, G (r,t) ∂τ ∂τ)| ≤ _ |
| rw [← integral_prod F hFi,← integral_prod G hGi,← integral_sub hFi hGi] |
| have hb : ∀ᵐ p ∂τ.prod τ, ‖F p-G p‖ ≤ δ := by |
| apply (Measure.ae_prod_iff_ae_ae (show MeasurableSet {p : ℝ × ℝ | ‖F p-G p‖ ≤ δ} from |
| measurableSet_le (hFm.sub hGm).norm measurable_const)).mpr |
| filter_upwards [self_mem_ae_restrict measurableSet_Ioc] with r hr |
| filter_upwards [self_mem_ae_restrict measurableSet_Ioc] with t ht |
| simpa [Real.norm_eq_abs,F,G] using hlevel r t hr.1 hr.2 ht.1 ht.2 |
| have hi := norm_integral_le_of_norm_le_const hb |
| have hm : (τ.prod τ).real Set.univ = C^2 := by |
| rw [measureReal_def,← Set.univ_prod_univ,Measure.prod_prod,ENNReal.toReal_mul] |
| simp [τ,Real.volume_Ioc,ENNReal.toReal_ofReal hC,pow_two] |
| simpa only [Real.norm_eq_abs,hm,mul_comm] using hi |
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| end |
| end FiniteFactor |
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| end OAI |
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