| import Mathlib |
| import OAI.Analysis.HardSphere.Kernels.TrialKernel |
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| namespace OAI |
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| section |
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| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal |
| namespace HardSphere.Comparison |
| variable {U X : Type*} [MeasurableSpace U] [MeasurableSpace X] |
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|
| lemma successKernel_restrict_le (κ : Kernel U X) [IsFiniteKernel κ] (S : Set X) (hS : MeasurableSet S) |
| {u : U} {c : ℝ≥0∞} (hc : c≤κ u S) : |
| successKernel κ S hS u≤c⁻¹ • (κ u).restrict S := by |
| rw [successKernel_apply] |
| intro B |
| simp only [Measure.smul_apply,smul_eq_mul] |
| exact mul_le_mul_left (ENNReal.inv_le_inv.mpr hc) _ |
|
|
| lemma successful_marginal_le (μ ν : Measure U) [SFinite μ] [SFinite ν] |
| (κ : Kernel U X) [IsFiniteKernel κ] (S : Set X) (hS : MeasurableSet S) |
| {c A : ℝ≥0∞} (hμ : μ≤A • ν) (hc : ∀ᵐu ∂μ,c≤κ u S) : |
| (μ.compProd (successKernel κ S hS)).map Prod.snd ≤ |
| (c⁻¹*A) • (((ν.compProd κ).map Prod.snd).restrict S) := by |
| apply Measure.le_iff.mpr |
| intro B hB |
| rw [Measure.map_apply measurable_snd hB,Measure.compProd_apply (hB.preimage measurable_snd), |
| Measure.smul_apply,smul_eq_mul,Measure.restrict_apply hB, |
| Measure.map_apply measurable_snd (hB.inter hS),Measure.compProd_apply ((hB.inter hS).preimage measurable_snd)] |
| calc |
| _ ≤ ∫⁻u,c⁻¹*(κ u (B∩S)) ∂μ := by |
| apply lintegral_mono_ae |
| filter_upwards [hc] with u hu |
| change successKernel κ S hS u B≤_ |
| simpa only [preimage_preimage,preimage_id,Measure.smul_apply,smul_eq_mul,Measure.restrict_apply hB] using |
| successKernel_restrict_le κ S hS hu B |
| _ = c⁻¹*(∫⁻u,κ u (B∩S) ∂μ) := lintegral_const_mul _ (κ.measurable_coe (hB.inter hS)) |
| _ ≤ c⁻¹*(∫⁻u,κ u (B∩S) ∂A • ν) := mul_le_mul_right (lintegral_mono' hμ le_rfl) _ |
| _ = _ := by rw [lintegral_smul_measure]; simp only [smul_eq_mul]; rw [mul_assoc]; rfl |
| |
| lemma normalizer_positive_of_uniform_success (μ ν : Measure U) [IsProbabilityMeasure μ] [SFinite ν] |
| (κ : Kernel U X) [IsFiniteKernel κ] (S : Set X) (hS : MeasurableSet S) |
| {c A : ℝ≥0∞} (hc0 : 0<c) (hA : A≠⊤) (hμ : μ≤A • ν) |
| (hc : ∀ᵐu ∂μ,c≤κ u S) : 0<((ν.compProd κ).map Prod.snd) S := by |
| rw [Measure.map_apply measurable_snd hS,Measure.compProd_apply (hS.preimage measurable_snd)] |
| have h₁ : c≤∫⁻u,κ u S ∂μ := by |
| calc |
| _ = ∫⁻_u,c ∂μ := by simp |
| _ ≤ _ := lintegral_mono_ae hc |
| have h₂ : (∫⁻u,κ u S ∂μ)≤A*(∫⁻u,κ u S ∂ν) := by |
| convert lintegral_mono' hμ (f:=fun u ↦ κ u S) le_rfl using 1 |
| rw [lintegral_smul_measure,smul_eq_mul] |
| change 0<(∫⁻u,κ u S ∂ν) |
| exact (ENNReal.div_pos hc0.ne' hA).trans_le (ENNReal.div_le_of_le_mul' (h₁.trans h₂)) |
|
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| lemma restricted_le_normalized (P : Measure X) [IsProbabilityMeasure P] (S : Set X) : |
| P.restrict S≤(P S)⁻¹ • P.restrict S := by |
| intro B |
| rw [Measure.smul_apply,smul_eq_mul] |
| have hi : (1:ℝ≥0∞)≤(P S)⁻¹ := by |
| simpa using ENNReal.inv_le_inv.mpr (prob_le_one (μ:=P) (s:=S)) |
| simpa using mul_le_mul_left hi (P.restrict S B) |
|
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| lemma successful_marginal_le_reference (μ ν : Measure U) [SFinite μ] [IsProbabilityMeasure ν] |
| (κ : Kernel U X) [IsMarkovKernel κ] (S : Set X) (hS : MeasurableSet S) |
| {c A : ℝ≥0∞} (hμ : μ≤A • ν) (hc : ∀ᵐu ∂μ,c≤κ u S) : |
| (μ.compProd (successKernel κ S hS)).map Prod.snd ≤ |
| (c⁻¹*A) • ((((ν.compProd κ).map Prod.snd) S)⁻¹ • (((ν.compProd κ).map Prod.snd).restrict S)) := by |
| have := successful_marginal_le μ ν κ S hS hμ hc |
| refine this.trans ?_ |
| intro B |
| simpa only [Measure.smul_apply,smul_eq_mul] using mul_le_mul_right (restricted_le_normalized ((ν.compProd κ).map Prod.snd) S B) (c⁻¹*A) |
|
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| end HardSphere.Comparison |
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| end |
| end |
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| end |
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| end OAI |
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|
