Datasets:

roseking
/

openAI-math-dataset

import OAI.Analysis.DiskMaximal.Telescoping

namespace OAI

open MeasureTheory Set
open scoped ENNReal NNReal Topology ContDiff
open scoped RealInnerProductSpace
open MeasureTheory Set Filter
open scoped Topology ENNReal
open scoped Real RealInnerProductSpace
open MeasureTheory Set Filter
open scoped Topology RealInnerProductSpace ContDiff
open MeasureTheory Set Filter
open scoped Topology Interval NNReal
open MeasureTheory Set Filter
open scoped Topology NNReal Real
open MeasureTheory Set Metric
open scoped ENNReal NNReal Topology

noncomputable section

namespace DiskMaximalTools

lemma volume_dilate_five (x : ℂ) (r : ℝ) :
    volume (ball x (5*r)) = 25 * volume (ball x r) := by
  rw [Complex.volume_ball, Complex.volume_ball, ENNReal.ofReal_mul (by norm_num : (0:ℝ) ≤ 5)]
  norm_num [mul_pow]
  ring

/-- The actual Vitali weak-(1,1) estimate, with no assumed maximal inequality. -/
theorem weak_bound_bounded_family {ι : Type*} (t : Set ι) (x : ι → ℂ) (r : ι → ℝ)
    (ν : Measure ℂ) {R a : ℝ} (ha : 0 < a)
    (hr : ∀ i ∈ t, 0 < r i) (hR : ∀ i ∈ t, r i ≤ R)
    (hav : ∀ i ∈ t, ENNReal.ofReal a * volume (ball (x i) (r i)) ≤ ν (ball (x i) (r i))) :
    volume (⋃ i ∈ t, ball (x i) (r i)) ≤ (25 / ENNReal.ofReal a) * ν univ := by
  obtain ⟨u,hut,hdisj,hcover⟩ := Vitali.exists_disjoint_subfamily_covering_enlargement_ball
    t x r R hR 5 (by norm_num)
  have huc : u.Countable := hdisj.countable_of_isOpen (fun _ _ => isOpen_ball)
    (fun i hi => ⟨x i,mem_ball_self (hr i (hut hi))⟩)
  have : Countable u := huc.to_subtype
  have hsub : (⋃ i ∈ t, ball (x i) (r i)) ⊆ ⋃ i : u, ball (x i) (5*r i) := by
    intro y hy
    obtain ⟨i,hi,hy⟩ := mem_iUnion₂.mp hy
    obtain ⟨j,hj,hij⟩ := hcover i hi
    exact mem_iUnion.mpr ⟨⟨j,hj⟩,hij hy⟩
  have hmass : (∑' i : u, ν (ball (x i) (r i))) ≤ ν univ := by
    rw [← measure_iUnion (fun (i j : u) hij => hdisj i.property j.property (fun h => hij (Subtype.ext h)))
      (fun _ => measurableSet_ball)]
    exact measure_mono (subset_univ _)
  calc
    _ ≤ volume (⋃ i : u, ball (x i) (5*r i)) := measure_mono hsub
    _ ≤ ∑' i : u, volume (ball (x i) (5*r i)) := measure_iUnion_le _
    _ = ∑' i : u, 25 * volume (ball (x i) (r i)) := by simp_rw [volume_dilate_five]
    _ ≤ ∑' i : u, (25 / ENNReal.ofReal a) * ν (ball (x i) (r i)) := by
      apply ENNReal.tsum_le_tsum
      intro i
      have hb := hav i (hut i.property)
      calc
        _ = (25 / ENNReal.ofReal a) * (ENNReal.ofReal a * volume (ball (x i) (r i))) := by
          rw [← mul_assoc, ENNReal.div_mul_cancel (ne_of_gt (ENNReal.ofReal_pos.mpr ha)) ENNReal.ofReal_ne_top]
        _ ≤ _ := by gcongr
    _ = (25 / ENNReal.ofReal a) * ∑' i : u, ν (ball (x i) (r i)) := by rw [ENNReal.tsum_mul_left]
    _ ≤ _ := by gcongr
lemma volume_disk_real (x : ℂ) {r : ℝ} (hr : 0 ≤ r) :
    volume (ball x r) = ENNReal.ofReal (Real.pi*r^2) := by
  rw [Complex.volume_ball,ENNReal.ofReal_mul Real.pi_nonneg,ENNReal.ofReal_pow hr]
  rw [← (ENNReal.ofReal_coe_nnreal : ENNReal.ofReal (NNReal.pi : ℝ) = (NNReal.pi : ℝ≥0∞))]
  exact mul_comm _ _
/-- All high-average balls of a finite measure have a common finite radius
bound; the bounded-family Vitali argument therefore controls every radius. -/
theorem weak_bound {ν : Measure ℂ} [IsFiniteMeasure ν] {a : ℝ} (ha : 0 < a) :
    volume {x : ℂ | ∃ r : ℝ, 0 < r ∧
      ENNReal.ofReal a * volume (ball x r) ≤ ν (ball x r)} ≤
      (25 / ENNReal.ofReal a) * ν univ := by
  let t : Set (ℂ × ℝ) := {p | 0 < p.2 ∧
    ENNReal.ofReal a * volume (ball p.1 p.2) ≤ ν (ball p.1 p.2)}
  let B : ℝ := (ν univ).toReal/(a*Real.pi)
  have hB : 0 ≤ B := div_nonneg ENNReal.toReal_nonneg (mul_pos ha Real.pi_pos).le
  have hR : ∀ p ∈ t, p.2 ≤ B+1 := by
    intro p hp
    have hh := hp.2.trans (measure_mono (subset_univ _))
    rw [volume_disk_real _ hp.1.le,← ENNReal.ofReal_mul ha.le] at hh
    have hh' : a*(Real.pi*p.2^2) ≤ (ν univ).toReal := by
      exact (ENNReal.ofReal_le_iff_le_toReal (measure_ne_top _ _)).mp hh
    have hs : p.2^2 ≤ B := (le_div_iff₀ (mul_pos ha Real.pi_pos)).mpr (by nlinarith)
    nlinarith [sq_nonneg (p.2-1)]
  have hweak := weak_bound_bounded_family t Prod.fst Prod.snd ν ha
    (fun _ hp => hp.1) hR (fun _ hp => hp.2)
  apply le_trans (measure_mono ?_) hweak
  intro x hx
  obtain ⟨r,hr,havg⟩ := hx
  exact mem_iUnion₂.mpr ⟨(x,r),⟨hr,havg⟩,mem_ball_self hr⟩

/-- A countable measurable envelope, sufficiently rich for every actual disk. -/
def rationalMaximal (ν : Measure ℂ) (x : ℂ) : ℝ≥0∞ :=
  ⨆ (q : ℚ) (_ : 0 < q), ν (ball x q) / volume (ball x q)

lemma measurable_ball_mass (ν : Measure ℂ) [SFinite ν] (r : ℝ) :
    Measurable (fun x : ℂ => ν (ball x r)) := by
  have hm : MeasurableSet {p : ℂ × ℂ | dist p.2 p.1 < r} :=
    isOpen_lt (continuous_snd.dist continuous_fst) continuous_const |>.measurableSet
  exact measurable_measure_prodMk_left hm

lemma measurable_rationalMaximal (ν : Measure ℂ) [SFinite ν] :
    Measurable (rationalMaximal ν) := by
  unfold rationalMaximal
  apply Measurable.iSup
  intro q
  apply Measurable.iSup
  intro _
  have hd : Measurable (fun x : ℂ => volume (ball x (q:ℝ))) := by
    simp only [Complex.volume_ball]
    exact measurable_const
  exact (measurable_ball_mass ν (q:ℝ)).div hd

lemma weak_rationalMaximal {ν : Measure ℂ} [IsFiniteMeasure ν] {a : ℝ} (ha : 0 < a) :
    volume {x | ENNReal.ofReal a < rationalMaximal ν x} ≤
      (25 / ENNReal.ofReal a) * ν univ := by
  apply le_trans (measure_mono ?_) (weak_bound ha)
  intro x hx
  change ENNReal.ofReal a < ⨆ (q : ℚ) (_ : 0 < q), ν (ball x q) / volume (ball x q) at hx
  obtain ⟨q,hh⟩ := lt_iSup_iff.mp hx
  obtain ⟨hq,hh⟩ := lt_iSup_iff.mp hh
  have hq' : (0:ℝ) < q := by exact_mod_cast hq
  refine ⟨q,hq',?_⟩
  have hv0 : volume (ball x (q:ℝ)) ≠ 0 := by
    rw [volume_disk_real _ hq'.le]
    exact ne_of_gt (ENNReal.ofReal_pos.mpr (mul_pos Real.pi_pos (sq_pos_of_pos hq')))
  exact ((ENNReal.lt_div_iff_mul_lt (Or.inl hv0) (Or.inl measure_ball_lt_top.ne)).mp hh).le

lemma disk_average_le_rationalMaximal (ν : Measure ℂ) (x : ℂ) {r : ℝ} (hr : 0 < r) :
    ν (ball x r) / volume (ball x r) ≤ 4*rationalMaximal ν x := by
  obtain ⟨q,hqr,hrq⟩ := exists_rat_btwn (show r < 2*r by linarith)
  have hq : (0 : ℝ) < q := hr.trans hqr
  have hqRat : (0:ℚ) < q := by exact_mod_cast hq
  have hd : volume (ball x (q:ℝ)) ≤ 4*volume (ball x r) := by
    rw [volume_disk_real _ hq.le,volume_disk_real _ hr.le,
      ← show ENNReal.ofReal (4:ℝ) = 4 by norm_num,← ENNReal.ofReal_mul (by norm_num : (0:ℝ) ≤ 4)]
    apply ENNReal.ofReal_le_ofReal
    have hs : (q:ℝ)^2 ≤ 4*r^2 := by nlinarith
    nlinarith [mul_le_mul_of_nonneg_left hs Real.pi_nonneg]
  have hv0 : volume (ball x r) ≠ 0 := by
    rw [volume_disk_real _ hr.le]
    exact ne_of_gt (ENNReal.ofReal_pos.mpr (mul_pos Real.pi_pos (sq_pos_of_pos hr)))
  have hvq0 : volume (ball x (q:ℝ)) ≠ 0 := by
    rw [volume_disk_real _ hq.le]
    exact ne_of_gt (ENNReal.ofReal_pos.mpr (mul_pos Real.pi_pos (sq_pos_of_pos hq)))
  have hav : ν (ball x (q:ℝ)) / volume (ball x (q:ℝ)) ≤ rationalMaximal ν x :=
    le_iSup_of_le q (le_iSup_of_le hqRat le_rfl)
  have hnum : ν (ball x r) ≤ rationalMaximal ν x * (4*volume (ball x r)) := by
    calc
      _ ≤ ν (ball x (q:ℝ)) := measure_mono (ball_subset_ball hqr.le)
      _ ≤ rationalMaximal ν x * volume (ball x (q:ℝ)) :=
        (ENNReal.div_le_iff hvq0 measure_ball_lt_top.ne).mp hav
      _ ≤ _ := by gcongr
  apply (ENNReal.div_le_iff hv0 measure_ball_lt_top.ne).mpr
  convert hnum using 1
  ring

/-- Truncating an input below half the testing height sharpens weak type by
charging only its large values. This is the interpolation mechanism. -/
lemma weak_split {ν η : Measure ℂ} [IsFiniteMeasure η] {a : ℝ} (ha : 0 < a)
    (hsplit : ∀ x r, 0 < r → ν (ball x r) ≤ η (ball x r)+
      ENNReal.ofReal (a/2)*volume (ball x r)) :
    volume {x | ENNReal.ofReal a < rationalMaximal ν x} ≤
      (25 / ENNReal.ofReal (a/2)) * η univ := by
  apply le_trans (measure_mono ?_) (weak_bound (by linarith : 0 < a/2))
  intro x hx
  change ENNReal.ofReal a < ⨆ (q : ℚ) (_ : 0 < q), ν (ball x q) / volume (ball x q) at hx
  obtain ⟨q,hh⟩ := lt_iSup_iff.mp hx
  obtain ⟨hq,hh⟩ := lt_iSup_iff.mp hh
  have hq' : (0:ℝ) < q := by exact_mod_cast hq
  have hv0 : volume (ball x (q:ℝ)) ≠ 0 := by
    rw [volume_disk_real _ hq'.le]
    exact ne_of_gt (ENNReal.ofReal_pos.mpr (mul_pos Real.pi_pos (sq_pos_of_pos hq')))
  have hh' := (ENNReal.lt_div_iff_mul_lt (Or.inl hv0) (Or.inl measure_ball_lt_top.ne)).mp hh
  refine ⟨q,hq',?_⟩
  by_contra h
  have hη : η (ball x (q:ℝ)) ≤ ENNReal.ofReal (a/2)*volume (ball x (q:ℝ)) := (lt_of_not_ge h).le
  have hfull := (hsplit x q hq').trans (add_le_add hη le_rfl)
  have hsum : ENNReal.ofReal (a/2)+ENNReal.ofReal (a/2) = ENNReal.ofReal a := by
    rw [← ENNReal.ofReal_add (by linarith : 0 ≤ a/2) (by linarith : 0 ≤ a/2)]
    congr 1
    ring
  rw [← add_mul,hsum] at hfull
  exact (not_lt_of_ge hfull) hh'
lemma weak_truncated {f : ℂ → ℝ≥0∞} (hf : Measurable f)
    [IsFiniteMeasure (volume.withDensity f)] {a : ℝ} (ha : 0 < a) :
    volume {x | ENNReal.ofReal a < rationalMaximal (volume.withDensity f) x} ≤
      (25 / ENNReal.ofReal (a/2)) *
        ∫⁻ x, {x | ENNReal.ofReal (a/2) < f x}.indicator f x := by
  let H := {x | ENNReal.ofReal (a/2) < f x}
  let η := volume.withDensity (H.indicator f)
  have hle : η ≤ volume.withDensity f := withDensity_mono (ae_of_all _ (indicator_le_self _ _))
  have : IsFiniteMeasure η := isFiniteMeasure_of_le _ hle
  have hweak := weak_split (ν := volume.withDensity f) (η := η) ha ?_
  · simpa only [η,withDensity_apply _ MeasurableSet.univ,setLIntegral_univ] using hweak
  · intro x r _
    rw [withDensity_apply _ measurableSet_ball,show η = volume.withDensity (H.indicator f) from rfl,
      withDensity_apply _ measurableSet_ball]
    calc
      _ ≤ ∫⁻ y in ball x r, H.indicator f y+ENNReal.ofReal (a/2) := by
        apply lintegral_mono
        intro y
        dsimp only
        by_cases hy : y ∈ H
        · rw [indicator_of_mem hy]
          exact le_add_right le_rfl
        · rw [indicator_of_notMem hy,zero_add]
          exact le_of_not_gt hy
      _ = _ := by
        rw [lintegral_add_left (hf.indicator (measurableSet_lt measurable_const hf))]
        simp only [lintegral_const,Measure.restrict_apply_univ]
        rfl
end DiskMaximalTools
open MeasureTheory Set
open scoped ENNReal NNReal
noncomputable section
namespace DiskMaximalTools
lemma lintegral_rpow_Ioo {b s : ℝ} (hb : 0 ≤ b) (hs : -1 < s) :
    (∫⁻ t in Ioo 0 b, ENNReal.ofReal (t^s)) = ENNReal.ofReal (b^(s+1)/(s+1)) := by
  rcases eq_or_lt_of_le hb with rfl | hb
  · simp [Real.zero_rpow (by linarith : s+1 ≠ 0)]
  · have hi : IntegrableOn (fun t : ℝ => t^s) (Ioo 0 b) :=
      (intervalIntegral.integrableOn_Ioo_rpow_iff hb).mpr hs
    rw [← ofReal_integral_eq_lintegral_ofReal hi]
    · rw [← integral_Ioc_eq_integral_Ioo,← intervalIntegral.integral_of_le hb.le,
        integral_rpow (Or.inl hs)]
      simp [Real.zero_rpow (by linarith : s+1 ≠ 0)]
    · filter_upwards [self_mem_ae_restrict measurableSet_Ioo] with t ht
      exact Real.rpow_nonneg ht.1.le _
lemma truncation_layer_single {b p : ℝ} (hb : 0 ≤ b) (hp : 1 < p) :
    (∫⁻ t in Ioi 0, (if t < 2*b then ENNReal.ofReal b else 0)*ENNReal.ofReal (t^(p-2))) =
      ENNReal.ofReal (2^(p-1)/(p-1))*ENNReal.ofReal (b^p) := by
  have heq : (fun t : ℝ => (if t < 2*b then ENNReal.ofReal b else 0)*ENNReal.ofReal (t^(p-2))) =
      (Iio (2*b)).indicator (fun t => ENNReal.ofReal b * ENNReal.ofReal (t^(p-2))) := by
    funext t
    by_cases ht : t < 2*b <;> simp [ht]
  rw [heq,lintegral_indicator measurableSet_Iio,Measure.restrict_restrict measurableSet_Iio]
  have hi : Iio (2*b) ∩ Ioi (0:ℝ) = Ioo 0 (2*b) := by ext t; simp [and_comm]
  rw [hi,lintegral_const_mul'' _ (by fun_prop)]
  rw [lintegral_rpow_Ioo (by positivity : 0 ≤ 2*b) (by linarith : -1 < p-2)]
  have hpp : p-2+1 = p-1 := by ring
  rw [hpp]
  have h2 : 0 ≤ (2:ℝ)^(p-1)/(p-1) := div_nonneg (Real.rpow_nonneg (by norm_num) _) (by linarith)
  rw [← ENNReal.ofReal_mul hb,← ENNReal.ofReal_mul h2]
  congr 1
  rw [Real.mul_rpow (by norm_num : (0:ℝ) ≤ 2) hb]
  have hp0 : 0 < p-1 := by linarith
  have hpow : b*b^(p-1) = b^p := by
    rcases eq_or_lt_of_le hb with rfl | hb
    · simp [Real.zero_rpow hp0.ne',Real.zero_rpow (by linarith : p ≠ 0)]
    · calc
        b*b^(p-1) = b^(1:ℝ)*b^(p-1) := by rw [Real.rpow_one]
        _ = b^(1+(p-1)) := (Real.rpow_add hb _ _).symm
        _ = b^p := by congr 1; ring
  calc
    b * (2^(p-1)*b^(p-1)/(p-1)) = (2^(p-1)/(p-1))*(b*b^(p-1)) := by ring
    _ = _ := by rw [hpow]

/-- Exact Tonelli computation behind strong maximal interpolation, for every
p > 1. The singularity at testing height zero is actually integrated. -/
theorem truncation_layer_integral {α : Type*} [MeasurableSpace α] {μ : Measure α}
    [SFinite μ] {f : α → ℝ} (hf : Measurable f) (hf0 : ∀ x, 0 ≤ f x) {p : ℝ} (hp : 1 < p) :
    (∫⁻ t in Ioi 0,
      (∫⁻ x, if t < 2*f x then ENNReal.ofReal (f x) else 0 ∂μ)*ENNReal.ofReal (t^(p-2))) =
      ENNReal.ofReal (2^(p-1)/(p-1))*(∫⁻ x, ENNReal.ofReal ((f x)^p) ∂μ) := by
  have hm : Measurable (fun z : ℝ × α =>
      (if z.1 < 2*f z.2 then ENNReal.ofReal (f z.2) else 0)*ENNReal.ofReal (z.1^(p-2))) := by
    apply Measurable.mul
    · exact ((hf.comp measurable_snd).ennreal_ofReal).ite
        (measurableSet_lt measurable_fst ((hf.comp measurable_snd).const_mul 2)) measurable_const
    · exact (measurable_fst.pow_const _).ennreal_ofReal
  calc
    _ = ∫⁻ t in Ioi 0, ∫⁻ x,
        (if t < 2*f x then ENNReal.ofReal (f x) else 0)*ENNReal.ofReal (t^(p-2)) ∂μ := by
      apply lintegral_congr
      intro t
      rw [lintegral_mul_const' _ _ ENNReal.ofReal_ne_top]
    _ = ∫⁻ x, (∫⁻ t in Ioi 0,
        (if t < 2*f x then ENNReal.ofReal (f x) else 0)*ENNReal.ofReal (t^(p-2))) ∂μ :=
      lintegral_lintegral_swap hm.aemeasurable
    _ = ∫⁻ x, ENNReal.ofReal (2^(p-1)/(p-1))*ENNReal.ofReal ((f x)^p) ∂μ := by
      apply lintegral_congr
      intro x
      exact truncation_layer_single (hf0 x) hp
    _ = _ := lintegral_const_mul'' _ ((hf.pow_const _).ennreal_ofReal.aemeasurable)
end DiskMaximalTools
namespace DiskMaximalTools
lemma maximal_layer_coefficient {t p : ℝ} (ht : 0 < t) :
    (25 / ENNReal.ofReal (t/2))*ENNReal.ofReal (t^(p-1)) =
      ENNReal.ofReal 50*ENNReal.ofReal (t^(p-2)) := by
  rw [← show ENNReal.ofReal (25:ℝ) = 25 by norm_num,
    ← ENNReal.ofReal_div_of_pos (by linarith : 0 < t/2),
    ← ENNReal.ofReal_mul (by positivity : 0 ≤ (25:ℝ)/(t/2)),
    ← ENNReal.ofReal_mul (by norm_num : (0:ℝ) ≤ 50)]
  congr 1
  have hpow : t^(p-1) = t*t^(p-2) := by
    calc
      _ = t^(1+(p-2)) := by congr 1; ring
      _ = t^(1:ℝ)*t^(p-2) := Real.rpow_add ht _ _
      _ = _ := by rw [Real.rpow_one]
  rw [hpow]
  field_simp
  ring
/-- A full strong p estimate from Vitali, truncation and Tonelli. No maximal
inequality is supplied as an assumption. The real representative is controlled
even before a separate almost-everywhere finiteness argument. -/
theorem strong_rationalMaximal {f : ℂ → ℝ} (hf : Measurable f)
    (hf0 : ∀ x, 0 ≤ f x) [IsFiniteMeasure (volume.withDensity (fun x => ENNReal.ofReal (f x)))]
    {p : ℝ} (hp : 1 < p) :
    (∫⁻ x, ENNReal.ofReal ((rationalMaximal (volume.withDensity (fun x => ENNReal.ofReal (f x))) x).toReal^p)) ≤
      ENNReal.ofReal (50*p*2^(p-1)/(p-1)) * (∫⁻ x, ENNReal.ofReal ((f x)^p)) := by
  let M := rationalMaximal (volume.withDensity (fun x => ENNReal.ofReal (f x)))
  have hM : Measurable M := measurable_rationalMaximal _
  have htail (t : ℝ) (ht : 0 < t) : volume {x | t < (M x).toReal} ≤
      (25/ENNReal.ofReal (t/2)) * ∫⁻ x, if t < 2*f x then ENNReal.ofReal (f x) else 0 := by
    have hsub : {x | t < (M x).toReal} ⊆ {x | ENNReal.ofReal t < M x} := by
      intro x hx
      exact (ENNReal.ofReal_lt_ofReal_iff_of_nonneg ht.le).mpr hx |>.trans_le ENNReal.ofReal_toReal_le
    apply le_trans (measure_mono hsub)
    have hweak := weak_truncated hf.ennreal_ofReal ht
    apply hweak.trans_eq
    congr 1
    apply lintegral_congr
    intro x
    have heq : ENNReal.ofReal (t/2) < ENNReal.ofReal (f x) ↔ t < 2*f x := by
      rw [ENNReal.ofReal_lt_ofReal_iff_of_nonneg (by linarith : 0 ≤ t/2)]
      constructor <;> intro h <;> linarith
    simp only [Set.indicator,Set.mem_ofPred_eq,heq]
  rw [lintegral_rpow_eq_lintegral_meas_lt_mul volume (ae_of_all _ (fun _ => ENNReal.toReal_nonneg))
    hM.ennreal_toReal.aemeasurable (by linarith : 0 < p)]
  have hlayer : (∫⁻ t in Ioi 0, volume {x | t < (M x).toReal}*ENNReal.ofReal (t^(p-1))) ≤
      ENNReal.ofReal 50 * (∫⁻ t in Ioi 0,
        (∫⁻ x, if t < 2*f x then ENNReal.ofReal (f x) else 0)*ENNReal.ofReal (t^(p-2))) := by
    calc
      _ ≤ ∫⁻ t in Ioi 0, ENNReal.ofReal 50 *
          ((∫⁻ x, if t < 2*f x then ENNReal.ofReal (f x) else 0)*ENNReal.ofReal (t^(p-2))) := by
        apply lintegral_mono_ae
        filter_upwards [self_mem_ae_restrict measurableSet_Ioi] with t ht
        calc
          _ ≤ ((25/ENNReal.ofReal (t/2)) * ∫⁻ x, if t < 2*f x then ENNReal.ofReal (f x) else 0)*
              ENNReal.ofReal (t^(p-1)) := by gcongr; exact htail t ht
          _ = _ := by
            rw [mul_right_comm, maximal_layer_coefficient ht]
            ring
      _ = _ := lintegral_const_mul' _ _ ENNReal.ofReal_ne_top
  calc
    _ ≤ ENNReal.ofReal p*(ENNReal.ofReal 50 * (∫⁻ t in Ioi 0,
        (∫⁻ x, if t < 2*f x then ENNReal.ofReal (f x) else 0)*ENNReal.ofReal (t^(p-2)))) := by
      gcongr
    _ = _ := by
      rw [truncation_layer_integral hf hf0 hp]
      have hc : ENNReal.ofReal p*(ENNReal.ofReal 50 * ENNReal.ofReal (2^(p-1)/(p-1))) =
          ENNReal.ofReal (50*p*2^(p-1)/(p-1)) := by
        rw [← ENNReal.ofReal_mul (by norm_num : (0:ℝ) ≤ 50),
          ← ENNReal.ofReal_mul (by linarith : 0 ≤ p)]
        congr 1
        ring
      simpa only [mul_assoc] using congrArg
        (fun c => c*(∫⁻ x, ENNReal.ofReal ((f x)^p))) hc

end DiskMaximalTools

namespace DiskMaximalTools

lemma disk_average_le_rationalMaximal_exact (ν : Measure ℂ) (x : ℂ) {r : ℝ} (hr : 0 < r) :
    ν (ball x r) / volume (ball x r) ≤ rationalMaximal ν x := by
  let I := {q : ℚ // 0 < q ∧ (q:ℝ) < r}
  have hdir : Directed (· ⊆ ·) (fun q : I => ball x (q:ℚ)) := by
    intro i j
    refine ⟨⟨max i.val j.val,?_,?_⟩,?_,?_⟩
    · exact lt_max_of_lt_left i.property.1
    · exact_mod_cast max_lt i.property.2 j.property.2
    · exact ball_subset_ball (by exact_mod_cast le_max_left i.val j.val)
    · exact ball_subset_ball (by exact_mod_cast le_max_right i.val j.val)
  have heq : (⋃ q : I, ball x (q:ℚ)) = ball x r := by
    ext y
    constructor
    · intro hy
      obtain ⟨q,hq⟩ := mem_iUnion.mp hy
      exact (ball_subset_ball q.property.2.le) hq
    · intro hy
      have hd : dist y x < r := hy
      obtain ⟨q,hq0,hqr⟩ := exists_rat_btwn (show max 0 (dist y x) < r by exact max_lt hr hd)
      have hq : 0 < (q:ℝ) := lt_of_le_of_lt (le_max_left _ _) hq0
      have hqRat : (0:ℚ) < q := by exact_mod_cast hq
      refine mem_iUnion.mpr ⟨⟨q,hqRat,hqr⟩,?_⟩
      exact lt_of_le_of_lt (le_max_right _ _) hq0
  have hv0 : volume (ball x r) ≠ 0 := by
    rw [volume_disk_real _ hr.le]
    exact ne_of_gt (ENNReal.ofReal_pos.mpr (mul_pos Real.pi_pos (sq_pos_of_pos hr)))
  apply (ENNReal.div_le_iff hv0 measure_ball_lt_top.ne).mpr
  rw [← heq,hdir.measure_iUnion]
  apply iSup_le
  intro q
  have hq0 : 0 < (q.val:ℝ) := by exact_mod_cast q.property.1
  have hqv : volume (ball x (q.val:ℝ)) ≠ 0 := by
    rw [volume_disk_real _ hq0.le]
    exact ne_of_gt (ENNReal.ofReal_pos.mpr (mul_pos Real.pi_pos (sq_pos_of_pos hq0)))
  have hav : ν (ball x (q.val:ℝ)) / volume (ball x (q.val:ℝ)) ≤ rationalMaximal ν x :=
    le_iSup_of_le q.val (le_iSup_of_le q.property.1 le_rfl)
  calc
    _ ≤ rationalMaximal ν x * volume (ball x (q.val:ℝ)) :=
      (ENNReal.div_le_iff hqv measure_ball_lt_top.ne).mp hav
    _ ≤ _ := by
      gcongr
      exact subset_iUnion (fun q : I => ball x (q:ℚ)) q

/-- Countability loses neither radii nor high values: the rational expression is
exactly the centered disk maximal function of the measure. -/
theorem all_radii_eq_rational (ν : Measure ℂ) (x : ℂ) :
    (⨆ (r : ℝ) (_ : 0 < r), ν (ball x r)/volume (ball x r)) = rationalMaximal ν x := by
  apply le_antisymm
  · apply iSup_le
    intro r
    apply iSup_le
    intro hr
    exact disk_average_le_rationalMaximal_exact ν x hr
  · unfold rationalMaximal
    apply iSup_le
    intro q
    apply iSup_le
    intro hq
    have hq' : 0 < (q:ℝ) := by exact_mod_cast hq
    exact le_iSup_of_le (q:ℝ) (le_iSup_of_le hq' le_rfl)

lemma rationalMaximal_le_of_density_le {f : ℂ → ℝ≥0∞} {B : ℝ≥0∞} (hf : ∀ x, f x ≤ B) (x : ℂ) :
    rationalMaximal (volume.withDensity f) x ≤ B := by
  unfold rationalMaximal
  apply iSup_le
  intro q
  apply iSup_le
  intro hq
  have hq' : 0 < (q:ℝ) := by exact_mod_cast hq
  have hv0 : volume (ball x (q:ℝ)) ≠ 0 := by
    rw [volume_disk_real _ hq'.le]
    exact ne_of_gt (ENNReal.ofReal_pos.mpr (mul_pos Real.pi_pos (sq_pos_of_pos hq')))
  apply (ENNReal.div_le_iff hv0 measure_ball_lt_top.ne).mpr
  rw [withDensity_apply _ measurableSet_ball]
  calc
    _ ≤ ∫⁻ y in ball x (q:ℝ), B := lintegral_mono hf
    _ = _ := by simp
end DiskMaximalTools
open MeasureTheory Set
open scoped ENNReal NNReal
noncomputable section
end
end
end
end OAI