nevanlinna/Nevanlinna/meromorphicOn_integrability.lean

import Mathlib.Analysis.Analytic.Meromorphic
import Mathlib.MeasureTheory.Integral.CircleIntegral
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Nevanlinna.analyticAt
import Nevanlinna.divisor
import Nevanlinna.meromorphicAt
import Nevanlinna.meromorphicOn_divisor
import Nevanlinna.stronglyMeromorphicOn
import Nevanlinna.mathlibAddOn

open scoped Interval Topology
open Real Filter MeasureTheory intervalIntegral


/- Integral and Integrability up to changes on codiscrete sets -/

theorem d
  {U S : Set ℂ}
  {c : ℂ}
  {r : ℝ}
  (hr : r ≠ 0)
  (hU : Metric.sphere c |r| ⊆ U)
  (hS : S ∈ Filter.codiscreteWithin U) :
  Countable ((circleMap c r)⁻¹' Sᶜ) := by

  have : (circleMap c r)⁻¹' (S ∪ Uᶜ)ᶜ = (circleMap c r)⁻¹' Sᶜ := by
    simp [(by simpa : (circleMap c r)⁻¹' U = ⊤)]
  rw [← this]

  apply Set.Countable.preimage_circleMap _ c hr
  have : DiscreteTopology ((S ∪ Uᶜ)ᶜ : Set ℂ) := by
    rw [discreteTopology_subtype_iff]
    rw [mem_codiscreteWithin] at hS; simp at hS

    intro x hx
    rw [← mem_iff_inf_principal_compl, (by ext z; simp; tauto : S ∪ Uᶜ = (U \ S)ᶜ)]
    rw [Set.compl_union, compl_compl] at hx
    exact hS x hx.2

  apply TopologicalSpace.separableSpace_iff_countable.1
  exact TopologicalSpace.SecondCountableTopology.to_separableSpace


theorem integrability_congr_changeDiscrete₀
  {f₁ f₂ : ℂ → ℂ}
  {U : Set ℂ}
  {r : ℝ}
  (hU : Metric.sphere 0 |r| ⊆ U)
  (hf : f₁ =ᶠ[Filter.codiscreteWithin U] f₂) :
  IntervalIntegrable (f₁ ∘ (circleMap 0 r)) MeasureTheory.volume 0 (2 * π) → IntervalIntegrable (f₂ ∘ (circleMap 0 r)) MeasureTheory.volume 0 (2 * π) := by
  intro hf₁

  by_cases hr : r = 0
  · unfold circleMap
    rw [hr]
    simp
    have : f₂ ∘ (fun (θ : ℝ) ↦ 0) = (fun r ↦ f₂ 0) := by
      exact rfl
    rw [this]
    simp

  · apply IntervalIntegrable.congr hf₁
    rw [Filter.eventuallyEq_iff_exists_mem]
    use (circleMap 0 r)⁻¹' {z | f₁ z = f₂ z}
    constructor
    · apply Set.Countable.measure_zero (d hr hU hf)
    · tauto


theorem integrability_congr_changeDiscrete
  {f₁ f₂ : ℂ → ℂ}
  {U : Set ℂ}
  {r : ℝ}
  (hU : Metric.sphere 0 |r| ⊆ U)
  (hf : f₁ =ᶠ[Filter.codiscreteWithin U] f₂) :
  IntervalIntegrable (f₁ ∘ (circleMap 0 r)) MeasureTheory.volume 0 (2 * π) ↔ IntervalIntegrable (f₂ ∘ (circleMap 0 r)) MeasureTheory.volume 0 (2 * π) := by

  constructor
  · exact integrability_congr_changeDiscrete₀ hU hf
  · exact integrability_congr_changeDiscrete₀ hU (EventuallyEq.symm hf)


theorem integral_congr_changeDiscrete
  {f₁ f₂ : ℂ → ℂ}
  {U : Set ℂ}
  {r : ℝ}
  (hr : r ≠ 0)
  (hU : Metric.sphere 0 |r| ⊆ U)
  (hf : f₁ =ᶠ[Filter.codiscreteWithin U] f₂) :
  ∫ (x : ℝ) in (0)..(2 * π), f₁ (circleMap 0 r x) = ∫ (x : ℝ) in (0)..(2 * π), f₂ (circleMap 0 r x) := by

  apply intervalIntegral.integral_congr_ae
  rw [eventually_iff_exists_mem]
  use (circleMap 0 r)⁻¹' {z | f₁ z = f₂ z}
  constructor
  · apply Set.Countable.measure_zero (d hr hU hf)
  · tauto