| import OAI.Analysis.IntegralMeans.AffineLaw |
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| namespace OAI |
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| 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 |
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|
| attribute [local irreducible] classWeight |
| attribute [local irreducible] classFun |
| attribute [local irreducible] rerootClass |
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| instance : PseudoMetrizableSpace DiskClass := inferInstanceAs (PseudoMetrizableSpace diskSchlichtSet) |
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| 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 |
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| def halfRetract (z : ℂ) : halfPlane := if h : z ∈ halfPlane then ⟨z,h⟩ else halfOne |
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| lemma halfRetract_of_mem {z : ℂ} (hz : z ∈ halfPlane) : halfRetract z = ⟨z,hz⟩ := by |
| simp only [halfRetract,dite_eq_left hz] |
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| 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 |
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| 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 |
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| 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) |
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| 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] |
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| 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] |
|
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| 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 |
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| end |
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| end OAI |
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