`BookProof.WeylCauchyRiemann.exists_analyticOn_ae_eq_ball` {f : ℂ → ℂ} {U : Set ℂ} (hf : LocallyIntegrableOn f U) (hCR : WeakCauchyRiemannOn f U) {c : ℂ} {R : ℝ} (hR : 0 < R) (hKU

Proved

BookProof.WeylCauchyRiemann.exists_analyticOn_ae_eq_ball

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterWeylCauchyRiemann.

BookProof.WeylCauchyRiemann.exists_analyticOn_ae_eq_ball {f : ℂ → ℂ} {U : Set ℂ} (hf : LocallyIntegrableOn f U) (hCR : WeakCauchyRiemannOn f U) {c : ℂ} {R : ℝ} (hR : 0 < R) (hKU : closedBall c R ⊆ U) : ∃ g : ℂ → ℂ, AnalyticOn ℂ g (ball c (R / 4)) ∧ ∀ᵐ z : ℂ, z ∈ ball c (R / 4) → f z = g z

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

Preamble

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



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

noncomputable section

Formal statement

theorem BookProof.WeylCauchyRiemann.exists_analyticOn_ae_eq_ball {f : ℂ → ℂ} {U : Set ℂ}
    (hf : LocallyIntegrableOn f U) (hCR : WeakCauchyRiemannOn f U)
    {c : ℂ} {R : ℝ} (hR : 0 < R) (hKU : closedBall c R ⊆ U) :
    ∃ g : ℂ → ℂ, AnalyticOn ℂ g (ball c (R / 4)) ∧
      ∀ᵐ z : ℂ, z ∈ ball c (R / 4) → f z = g z := by sorry

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