| import Mathlib.Analysis.SpecialFunctions.PolarCoord |
| import Mathlib.Analysis.SpecialFunctions.Integrability.Basic |
| import Mathlib.MeasureTheory.Measure.Prod |
| import Mathlib.Tactic.FieldSimp |
| import Mathlib.Tactic.Linarith |
| import Mathlib.Tactic.Ring |
|
|
| namespace OAI |
|
|
| open MeasureTheory Set |
| open scoped ENNReal |
|
|
| namespace Brennan.Sector |
|
|
| theorem lintegral_inv_Ioo_eq_top (R : ℝ) (hR : 0 < R) : |
| (∫⁻ r in Ioo (0 : ℝ) R, ENNReal.ofReal r⁻¹) = ⊤ := by |
| by_contra h |
| have hn : 0 ≤ᵐ[volume.restrict (Ioo (0 : ℝ) R)] (fun r : ℝ => r⁻¹) := by |
| filter_upwards [ae_restrict_mem measurableSet_Ioo] with r hr |
| exact inv_nonneg.mpr hr.1.le |
| have hf : HasFiniteIntegral (fun r : ℝ => r⁻¹) (volume.restrict (Ioo (0 : ℝ) R)) := |
| (hasFiniteIntegral_iff_ofReal hn).2 (lt_top_iff_ne_top.mpr h) |
| have hi : IntegrableOn (fun r : ℝ => r⁻¹) (Ioo (0 : ℝ) R) := |
| ⟨measurable_inv.aestronglyMeasurable, hf⟩ |
| have hp : IntegrableOn (fun r : ℝ => r ^ (-1 : ℝ)) (Ioo (0 : ℝ) R) := by |
| simpa only [Real.rpow_neg_one] using hi |
| exact (lt_irrefl (-1 : ℝ)) ((intervalIntegral.integrableOn_Ioo_rpow_iff hR).mp hp) |
|
|
| theorem lintegral_eq_top_of_sector_lower_bound (f : ℂ → ℝ≥0∞) |
| (R a b c : ℝ) (hR : 0 < R) (hab : a < b) (hc : 0 < c) |
| (ha : -Real.pi ≤ a) (hb : b ≤ Real.pi) |
| (hlower : ∀ r ∈ Ioo (0 : ℝ) R, ∀ θ ∈ Ioo a b, |
| ENNReal.ofReal (c / r ^ 2) ≤ f (Complex.polarCoord.symm (r, θ))) : |
| (∫⁻ z, f z) = ⊤ := by |
| let box : Set (ℝ × ℝ) := Ioo (0 : ℝ) R ×ˢ Ioo a b |
| let radial : ℝ × ℝ → ℝ≥0∞ := fun p => ENNReal.ofReal c * ENNReal.ofReal p.1⁻¹ |
| have hradial : Measurable radial := |
| measurable_const.mul (ENNReal.measurable_ofReal.comp measurable_fst.inv) |
| have hcinv : (∫⁻ r in Ioo (0 : ℝ) R, ENNReal.ofReal c * ENNReal.ofReal r⁻¹) = ⊤ := by |
| have hmeas : Measurable (fun r : ℝ => ENNReal.ofReal r⁻¹) := |
| ENNReal.measurable_ofReal.comp measurable_inv |
| rw [lintegral_const_mul (ENNReal.ofReal c) hmeas, |
| lintegral_inv_Ioo_eq_top R hR, |
| ENNReal.mul_top (ne_of_gt (ENNReal.ofReal_pos.mpr hc))] |
| have hbox : (∫⁻ p in box, radial p) = ⊤ := by |
| change (∫⁻ p in Ioo (0 : ℝ) R ×ˢ Ioo a b, radial p ∂volume.prod volume) = ⊤ |
| rw [setLIntegral_prod_symm _ hradial.aemeasurable] |
| simp only [radial, hcinv] |
| rw [lintegral_const, Measure.restrict_apply_univ, Real.volume_Ioo, |
| ENNReal.top_mul (ne_of_gt (ENNReal.ofReal_pos.mpr (sub_pos.mpr hab)))] |
| have hsub : box ⊆ polarCoord.target := by |
| rw [polarCoord_target] |
| intro p hp |
| exact ⟨hp.1.1, lt_of_le_of_lt ha hp.2.1, lt_of_lt_of_le hp.2.2 hb⟩ |
| have hlower' (p : ℝ × ℝ) (hp : p ∈ box) : |
| radial p ≤ ENNReal.ofReal p.1 • f (Complex.polarCoord.symm p) := by |
| have hr : 0 < p.1 := hp.1.1 |
| have hid : ENNReal.ofReal c * ENNReal.ofReal p.1⁻¹ = |
| ENNReal.ofReal p.1 * ENNReal.ofReal (c / p.1 ^ 2) := by |
| rw [← ENNReal.ofReal_mul hc.le, ← ENNReal.ofReal_mul hr.le] |
| congr 1 |
| field_simp [ne_of_gt hr] |
| change ENNReal.ofReal c * ENNReal.ofReal p.1⁻¹ ≤ |
| ENNReal.ofReal p.1 * f (Complex.polarCoord.symm p) |
| rw [hid] |
| exact mul_le_mul_right (hlower p.1 hp.1 p.2 hp.2) _ |
| apply top_unique |
| calc |
| ⊤ = ∫⁻ p in box, radial p := hbox.symm |
| _ ≤ ∫⁻ p in box, ENNReal.ofReal p.1 • f (Complex.polarCoord.symm p) := |
| setLIntegral_mono' (measurableSet_Ioo.prod measurableSet_Ioo) hlower' |
| _ ≤ ∫⁻ p in polarCoord.target, ENNReal.ofReal p.1 • f (Complex.polarCoord.symm p) := |
| lintegral_mono_set hsub |
| _ = ∫⁻ z, f z := Complex.lintegral_comp_polarCoord_symm f |
| |
| end Brennan.Sector |
| |
| end OAI |
| |
