`BookProof.ChapterShannonSampling.integral_haar_eq_real` (g : AddCircle T → ℝ) : T * ∫ z : AddCircle T, g z ∂haarAddCircle = ∫ ξ in (-(T / 2))..(-(T / 2) + T), g ξ

Proved

BookProof.ChapterShannonSampling.integral_haar_eq_real

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterShannonSampling.

BookProof.ChapterShannonSampling.integral_haar_eq_real (g : AddCircle T → ℝ) : T * ∫ z : AddCircle T, g z ∂haarAddCircle = ∫ ξ in (-(T / 2))..(-(T / 2) + T), g ξ

Formalization note: Lean 4 identifier BookProof.ChapterShannonSampling.integral_haar_eq_real.

Preamble

-- Generated from ChapterShannonSampling.lean — theorem BookProof.ChapterShannonSampling.integral_haar_eq_real
import Mathlib
import Definitions.Def_ChapterShannonSampling
open BookProof.ChapterShannonSampling



open MeasureTheory Complex AddCircle intervalIntegral Set
open scoped Real ComplexConjugate

variable {T : ℝ} [hT : Fact (0 < T)]

Formal statement

theorem BookProof.ChapterShannonSampling.integral_haar_eq_real (g : AddCircle T → ℝ) :
    T * ∫ z : AddCircle T, g z ∂haarAddCircle
      = ∫ ξ in (-(T / 2))..(-(T / 2) + T), g ξ := by sorry

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