`BookProof.HalfLineLimitCircle.inner_eq_integral` {u v : HL} {a b : ℝ → ℂ} (hu : (u : ℝ → ℂ) =ᵐ[hlMeasure] a) (hv : (v : ℝ → ℂ) =ᵐ[hlMeasure] b) : (inner ℂ u v : ℂ) =...

Proved

BookProof.HalfLineLimitCircle.inner_eq_integral

by leonardopedro · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterHalfLineLimitCircle.

BookProof.HalfLineLimitCircle.inner_eq_integral {u v : HL} {a b : ℝ → ℂ} (hu : (u : ℝ → ℂ) =ᵐ[hlMeasure] a) (hv : (v : ℝ → ℂ) =ᵐ[hlMeasure] b) : (inner ℂ u v : ℂ) = ∫ x in Ioi (0:ℝ), (starRingEnd ℂ) (a x) * b x

Formalization note: Lean 4 identifier BookProof.HalfLineLimitCircle.inner_eq_integral.

Preamble

-- Generated from ChapterHalfLineLimitCircle.lean — theorem BookProof.HalfLineLimitCircle.inner_eq_integral
import Definitions.Def_ChapterFarisLavine
import Mathlib
import Definitions.Def_ChapterHalfLineLimitCircle
open BookProof.HalfLineLimitCircle



open MeasureTheory Set BookProof.FarisLavine

noncomputable section

local notation "smoothTop" => ((⊤ : ℕ∞) : WithTop ℕ∞)

Formal statement

theorem BookProof.HalfLineLimitCircle.inner_eq_integral {u v : HL} {a b : ℝ → ℂ}
    (hu : (u : ℝ → ℂ) =ᵐ[hlMeasure] a) (hv : (v : ℝ → ℂ) =ᵐ[hlMeasure] b) :
    (inner ℂ u v : ℂ) = ∫ x in Ioi (0:ℝ), (starRingEnd ℂ) (a x) * b x := by sorry

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