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67aee1d | 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 | import Mathlib
import OAI.Analysis.HardSphere.Kernels.TrialKernel
namespace OAI
section
section
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal NNReal
namespace HardSphere.Comparison
variable {U X : Type*} [MeasurableSpace U] [MeasurableSpace X]
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))
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)
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)
end HardSphere.Comparison
end
end
end
end OAI
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