99 lines
3.1 KiB
Plaintext
99 lines
3.1 KiB
Plaintext
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import Mathlib.Analysis.Analytic.Meromorphic
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import Mathlib.MeasureTheory.Integral.CircleIntegral
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import Mathlib.MeasureTheory.Integral.IntervalIntegral
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import Nevanlinna.analyticAt
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import Nevanlinna.divisor
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import Nevanlinna.meromorphicAt
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import Nevanlinna.meromorphicOn_divisor
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import Nevanlinna.stronglyMeromorphicOn
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import Nevanlinna.stronglyMeromorphicOn_eliminate
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import Nevanlinna.mathlibAddOn
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open scoped Interval Topology
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open Real Filter MeasureTheory intervalIntegral
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/- Integral and Integrability up to changes on codiscrete sets -/
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theorem d
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{U S : Set ℂ}
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{c : ℂ}
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{r : ℝ}
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(hr : r ≠ 0)
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(hU : Metric.sphere c |r| ⊆ U)
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(hS : S ∈ Filter.codiscreteWithin U) :
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Countable ((circleMap c r)⁻¹' Sᶜ) := by
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have : (circleMap c r)⁻¹' (S ∪ Uᶜ)ᶜ = (circleMap c r)⁻¹' Sᶜ := by
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simp [(by simpa : (circleMap c r)⁻¹' U = ⊤)]
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rw [← this]
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apply Set.Countable.preimage_circleMap _ c hr
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have : DiscreteTopology ((S ∪ Uᶜ)ᶜ : Set ℂ) := by
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rw [discreteTopology_subtype_iff]
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rw [mem_codiscreteWithin] at hS; simp at hS
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intro x hx
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rw [← mem_iff_inf_principal_compl, (by ext z; simp; tauto : S ∪ Uᶜ = (U \ S)ᶜ)]
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rw [Set.compl_union, compl_compl] at hx
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exact hS x hx.2
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apply TopologicalSpace.separableSpace_iff_countable.1
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exact TopologicalSpace.SecondCountableTopology.to_separableSpace
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theorem integrability_congr_changeDiscrete₀
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{f₁ f₂ : ℂ → ℝ}
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{U : Set ℂ}
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{r : ℝ}
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(hU : Metric.sphere 0 |r| ⊆ U)
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(hf : f₁ =ᶠ[Filter.codiscreteWithin U] f₂) :
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IntervalIntegrable (f₁ ∘ (circleMap 0 r)) MeasureTheory.volume 0 (2 * π) → IntervalIntegrable (f₂ ∘ (circleMap 0 r)) MeasureTheory.volume 0 (2 * π) := by
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intro hf₁
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by_cases hr : r = 0
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· unfold circleMap
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rw [hr]
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simp
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have : f₂ ∘ (fun (θ : ℝ) ↦ 0) = (fun r ↦ f₂ 0) := by
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exact rfl
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rw [this]
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simp
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· apply IntervalIntegrable.congr hf₁
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rw [Filter.eventuallyEq_iff_exists_mem]
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use (circleMap 0 r)⁻¹' {z | f₁ z = f₂ z}
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constructor
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· apply Set.Countable.measure_zero (d hr hU hf)
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· tauto
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theorem integrability_congr_changeDiscrete
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{f₁ f₂ : ℂ → ℝ}
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{U : Set ℂ}
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{r : ℝ}
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(hU : Metric.sphere (0 : ℂ) |r| ⊆ U)
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(hf : f₁ =ᶠ[Filter.codiscreteWithin U] f₂) :
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IntervalIntegrable (f₁ ∘ (circleMap 0 r)) MeasureTheory.volume 0 (2 * π) ↔ IntervalIntegrable (f₂ ∘ (circleMap 0 r)) MeasureTheory.volume 0 (2 * π) := by
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constructor
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· exact integrability_congr_changeDiscrete₀ hU hf
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· exact integrability_congr_changeDiscrete₀ hU (EventuallyEq.symm hf)
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theorem integral_congr_changeDiscrete
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{f₁ f₂ : ℂ → ℝ}
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{U : Set ℂ}
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{r : ℝ}
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(hr : r ≠ 0)
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(hU : Metric.sphere 0 |r| ⊆ U)
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(hf : f₁ =ᶠ[Filter.codiscreteWithin U] f₂) :
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∫ (x : ℝ) in (0)..(2 * π), f₁ (circleMap 0 r x) = ∫ (x : ℝ) in (0)..(2 * π), f₂ (circleMap 0 r x) := by
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apply intervalIntegral.integral_congr_ae
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rw [eventually_iff_exists_mem]
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use (circleMap 0 r)⁻¹' {z | f₁ z = f₂ z}
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constructor
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· apply Set.Countable.measure_zero (d hr hU hf)
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· tauto
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