34 lines
1.1 KiB
Plaintext
34 lines
1.1 KiB
Plaintext
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import Mathlib.Analysis.Analytic.Meromorphic
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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.mathlibAddOn
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import Mathlib.MeasureTheory.Integral.CircleIntegral
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open scoped Interval Topology
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open Real Filter MeasureTheory intervalIntegral
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theorem integrability_congr_changeDiscrete
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{f₁ f₂ : ℂ → ℂ}
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{r : ℝ}
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(hf : f₁ =ᶠ[Filter.codiscreteWithin ⊤] 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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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
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have : (circleMap 0 r ⁻¹' {z | f₁ z = f₂ z})ᶜ = (circleMap 0 r ⁻¹' {z | f₁ z = f₂ z}ᶜ) := by
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exact rfl
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rw [this]
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apply Set.Countable.preimage_circleMap
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sorry
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sorry
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· sorry
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