`BookProof.WeylCauchyRiemann.weak_cauchyRiemann_iff_ae_eq_analytic` {f : ℂ → ℂ} {U : Set ℂ} (hU : IsOpen U) (hf : LocallyIntegrableOn f U) : WeakCauchyRiemannOn f U ↔ ∃ g : ℂ → ℂ,

Proved

BookProof.WeylCauchyRiemann.weak_cauchyRiemann_iff_ae_eq_analytic

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterWeylCauchyRiemann.

BookProof.WeylCauchyRiemann.weak_cauchyRiemann_iff_ae_eq_analytic {f : ℂ → ℂ} {U : Set ℂ} (hU : IsOpen U) (hf : LocallyIntegrableOn f U) : WeakCauchyRiemannOn f U ↔ ∃ g : ℂ → ℂ, AnalyticOn ℂ g U ∧ ∀ᵐ z : ℂ, z ∈ U → f z = g z

Formalization note: Lean 4 identifier BookProof.WeylCauchyRiemann.weak_cauchyRiemann_iff_ae_eq_analytic.

Preamble

-- Generated from ChapterWeylCauchyRiemann.lean — theorem BookProof.WeylCauchyRiemann.weak_cauchyRiemann_iff_ae_eq_analytic
import Definitions.Def_ChapterRadialMollifier
import Mathlib
import Definitions.Def_ChapterWeylCauchyRiemann
open BookProof.WeylCauchyRiemann



open MeasureTheory Metric Set Filter Complex BookProof.RadialMollifier
open scoped Convolution Topology ContDiff

noncomputable section

Formal statement

theorem BookProof.WeylCauchyRiemann.weak_cauchyRiemann_iff_ae_eq_analytic {f : ℂ → ℂ} {U : Set ℂ} (hU : IsOpen U)
    (hf : LocallyIntegrableOn f U) :
    WeakCauchyRiemannOn f U ↔ ∃ g : ℂ → ℂ, AnalyticOn ℂ g U ∧ ∀ᵐ z : ℂ, z ∈ U → f z = g z := by sorry

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