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Nevanlinna/meromorphicOn_integrability.lean

@@ -6,6 +6,8 @@ import Nevanlinna.meromorphicOn_divisor
 import Nevanlinna.stronglyMeromorphicOn
 import Nevanlinna.mathlibAddOn
 import Mathlib.MeasureTheory.Integral.CircleIntegral
+import Mathlib.MeasureTheory.Integral.IntervalIntegral
+
 
 
 open scoped Interval Topology
@@ -43,27 +45,85 @@ lemma c
   exact TopologicalSpace.SecondCountableTopology.to_separableSpace
 
 
-theorem integrability_congr_changeDiscrete
+theorem d
+  {U S : Set ℂ}
+  {r : ℝ}
+  (hU : Metric.sphere 0 |r| ⊆ U)
+  (hS : S ∈ Filter.codiscreteWithin U) :
+  Countable ((circleMap 0 r)⁻¹' (S ∪ Uᶜ)ᶜ) := by
+
+  have : (circleMap 0 r)⁻¹' U = ⊤ := by
+    simpa
+  simp [this]
+
+
+
+  sorry
+
+
+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
 
-  apply IntervalIntegrable.congr 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} ∩ U)
+    use (circleMap 0 r)⁻¹' ({z | f₁ z = f₂ z} ∪ Uᶜ)
     constructor
     · apply Set.Countable.measure_zero
-    have : (circleMap 0 r ⁻¹' ({z | f₁ z = f₂ z} ∩ U))ᶜ = (circleMap 0 r ⁻¹' ({z | f₁ z = f₂ z} ∩ U)ᶜ) := by
+      have : (circleMap 0 r ⁻¹' ({z | f₁ z = f₂ z} ∪ Uᶜ))ᶜ = (circleMap 0 r ⁻¹' ({z | f₁ z = f₂ z} ∪ Uᶜ)ᶜ) := by
         exact rfl
       rw [this]
       apply Set.Countable.preimage_circleMap
-    apply c
-    sorry
-    sorry
+      let A := c {z | f₁ z = f₂ z} U hf
+      simp at A
+      simpa
+      exact hr
     · intro x hx
       simp at hx
       simp
-    exact hx.1
+      rcases hx with h|h
+      · assumption
+      · let A := hU (circleMap_mem_sphere' 0 r x)
+        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}
+  (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} ∪ Uᶜ)
+
+
+  sorry