lean/OAI/Analysis/IntegralMeans/ExpectedDensity.lean

import OAI.Analysis.IntegralMeans.AffineLaw

namespace OAI

noncomputable section
open Set MeasureTheory Filter Function InnerProductSpace
open scoped Topology ComplexConjugate Manifold NNReal ENNReal InnerProductSpace Classical
open MeasureTheory Function
open Set Filter
open Set MeasureTheory Filter Function
open Set MeasureTheory Filter Function InnerProductSpace
open TopologicalSpace
open scoped CompactlySupported
open scoped ENNReal
open scoped Manifold
open scoped Topology CompactlySupported ComplexConjugate
open scoped Topology ComplexConjugate Manifold NNReal ENNReal InnerProductSpace Classical
open scoped Topology ENNReal NNReal
namespace Brennan

attribute [local irreducible] classWeight
attribute [local irreducible] classFun
attribute [local irreducible] rerootClass

instance : PseudoMetrizableSpace DiskClass := inferInstanceAs (PseudoMetrizableSpace diskSchlichtSet)

lemma exists_weighted_law (hρ : 2 < transferRho) :
    ∃ P : ProbabilityMeasure DiskClass, ∀ z : halfPlane, ∀ f : DiskClass → ℝ≥0∞,
      Measurable f →
      (∫⁻ g, ENNReal.ofReal (classWeight g z)*f (rerootClass g z) ∂(P : Measure DiskClass)) =
        ENNReal.ofReal ((z.val.im)^(-transferBeta))*(∫⁻ g, f g ∂(P : Measure DiskClass)) := by
  obtain ⟨P,hP⟩ := exists_weighted_continuous_law hρ
  refine ⟨P,fun z f hf => ?_⟩
  exact weighted_lintegral_of_continuous (P : Measure DiskClass) (weightAt z)
    (fun g => (classWeight_pos g z).le) (actionAt z) ((z.val.im)^(-transferBeta))
    (Real.rpow_nonneg z.2.le _) (hP z) f hf

def halfRetract (z : ℂ) : halfPlane := if h : z ∈ halfPlane then ⟨z,h⟩ else halfOne

lemma halfRetract_of_mem {z : ℂ} (hz : z ∈ halfPlane) : halfRetract z = ⟨z,hz⟩ := by
  simp only [halfRetract,dite_eq_left hz]

lemma measurable_halfRetract : Measurable halfRetract := by
  have hm : Measurable (fun z : ℂ => (halfRetract z : ℂ)) := by
    have he : (fun z : ℂ => (halfRetract z : ℂ)) = (fun z => if z ∈ halfPlane then z else Complex.I) := by
      funext z
      by_cases hz : z ∈ halfPlane <;> simp [halfRetract,halfOne,hz]
    rw [he]
    exact measurable_id.piecewise isOpen_halfPlane.measurableSet measurable_const
  exact hm.subtype_mk

def averagedDensity (β : ℝ) (Φ : DiskClass → ℝ≥0∞) (p : DiskClass × ℂ) : ℝ≥0∞ :=
  {p : DiskClass × ℂ | p.2 ∈ halfPlane}.indicator
    (fun p => ENNReal.ofReal (p.2.im^(β-1)) * ENNReal.ofReal (classWeight p.1 (halfRetract p.2)) *
      Φ (rerootClass p.1 (halfRetract p.2))) p

lemma measurable_averagedDensity (β : ℝ) {Φ : DiskClass → ℝ≥0∞} (hΦ : Measurable Φ) :
    Measurable (averagedDensity β Φ) := by
  have hm : Measurable (fun p : DiskClass × ℂ => (p.1,halfRetract p.2)) :=
    measurable_fst.prodMk (measurable_halfRetract.comp measurable_snd)
  exact ((((Complex.measurable_im.comp measurable_snd).pow_const (β-1)).ennreal_ofReal.mul
    (continuous_classWeight.measurable.comp hm).ennreal_ofReal).mul
    (hΦ.comp (continuous_rerootClass.measurable.comp hm))).indicator
      (isOpen_halfPlane.measurableSet.preimage measurable_snd)

lemma averagedDensity_of_mem (β : ℝ) (Φ : DiskClass → ℝ≥0∞) (g : DiskClass) (z : halfPlane) :
    averagedDensity β Φ (g,z) = transportDensity g β z * Φ (rerootClass g z) := by
  rw [averagedDensity,indicator_of_mem (show (g,(z : ℂ)) ∈ {p : DiskClass × ℂ | p.2 ∈ halfPlane} from z.property)]
  simp only [halfRetract_of_mem z.property]
  rw [transportDensity,ENNReal.ofReal_mul (Real.rpow_nonneg z.property.le _)]
  simp only [classWeight]

lemma weighted_averagedDensity (P : ProbabilityMeasure DiskClass) (β : ℝ)
    (hP : ∀ z : halfPlane, ∀ f : DiskClass → ℝ≥0∞, Measurable f →
      (∫⁻ g, ENNReal.ofReal (classWeight g z)*f (rerootClass g z) ∂(P : Measure DiskClass)) =
        ENNReal.ofReal (z.val.im^(-β))*(∫⁻ g, f g ∂(P : Measure DiskClass)))
    {Φ : DiskClass → ℝ≥0∞} (hΦ : Measurable Φ) (z : halfPlane) :
    (∫⁻ g, averagedDensity β Φ (g,z) ∂(P : Measure DiskClass)) =
      ENNReal.ofReal (z.val.im⁻¹) * ∫⁻ g, Φ g ∂(P : Measure DiskClass) := by
  have hm : Measurable (fun g : DiskClass => ENNReal.ofReal (classWeight g z)*Φ (rerootClass g z)) := by
    have hp : Measurable (fun g : DiskClass => (g,z)) := measurable_id.prodMk measurable_const
    exact ((continuous_classWeight.measurable.comp hp).ennreal_ofReal).mul
      (hΦ.comp (continuous_rerootClass.measurable.comp hp))
  have he : (fun g : DiskClass => averagedDensity β Φ (g,z)) =
      fun g => ENNReal.ofReal (z.val.im^(β-1)) *
        (ENNReal.ofReal (classWeight g z)*Φ (rerootClass g z)) := by
    funext g
    rw [averagedDensity,indicator_of_mem (show (g,(z : ℂ)) ∈ {p : DiskClass × ℂ | p.2 ∈ halfPlane} from z.property)]
    simp only [halfRetract_of_mem z.property,mul_assoc]
  rw [he,lintegral_const_mul _ hm,hP z Φ hΦ,← mul_assoc,
    ← ENNReal.ofReal_mul (Real.rpow_nonneg z.property.le _),← Real.rpow_add z.property]
  congr 2
  rw [show β-1 + -β = (-1 : ℝ) from by ring,Real.rpow_neg_one]

lemma root_test_zero_of_averaging (P : ProbabilityMeasure DiskClass) (β : ℝ)
    (hP : ∀ z : halfPlane, ∀ f : DiskClass → ℝ≥0∞, Measurable f →
      (∫⁻ g, ENNReal.ofReal (classWeight g z)*f (rerootClass g z) ∂(P : Measure DiskClass)) =
        ENNReal.ofReal (z.val.im^(-β))*(∫⁻ g, f g ∂(P : Measure DiskClass)))
    {Φ : DiskClass → ℝ≥0∞} (hΦ : Measurable Φ)
    (hzero : ∀ᵐ g ∂(P : Measure DiskClass), (∫⁻ z in halfPlane, averagedDensity β Φ (g,z)) = 0) :
    (∫⁻ g, Φ g ∂(P : Measure DiskClass)) = 0 := by
  have hm := measurable_averagedDensity β hΦ
  have hz : (∫⁻ z in halfPlane, ∫⁻ g, averagedDensity β Φ (g,z) ∂(P : Measure DiskClass)) = 0 :=
    (lintegral_lintegral_swap (f := fun g z => averagedDensity β Φ (g,z)) (μ := (P : Measure DiskClass)) (ν := volume.restrict halfPlane) hm.aemeasurable).symm.trans (lintegral_eq_zero_of_ae_eq_zero hzero)
  have ha := (lintegral_eq_zero_iff hm.lintegral_prod_left').mp hz
  have : (ae (volume.restrict halfPlane)).NeBot := ae_restrict_neBot.mpr
    (ne_of_gt (isOpen_halfPlane.measure_pos volume ⟨Complex.I,by simp [halfPlane]⟩))
  obtain ⟨z,hz,hz0⟩ := ((ae_restrict_mem isOpen_halfPlane.measurableSet).and ha).exists
  have he := weighted_averagedDensity P β hP hΦ ⟨z,hz⟩
  change (∫⁻ g, averagedDensity β Φ (g,z) ∂(P : Measure DiskClass)) = 0 at hz0
  exact (mul_eq_zero.mp (he.symm.trans hz0)).resolve_left (ne_of_gt (ENNReal.ofReal_pos.mpr (inv_pos.mpr hz)))

end Brennan

end

end OAI