Prove2Me

`BookProof.HalfLineLimitCircle.setIntegral_eq_integral_of_testSpace` {f : ℝ → ℂ} (hf : f ∈ testSpace) (w : ℝ → ℂ) : ∫ x in Ioi (0 : ℝ), (starRingEnd ℂ) (f x) * w x = ∫ x,...

Proved
BookProof.HalfLineLimitCircle.setIntegral_eq_integral_of_testSpace

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterHalfLineLimitCircle.

BookProof.HalfLineLimitCircle.setIntegral_eq_integral_of_testSpace {f : ℝ → ℂ} (hf : f ∈ testSpace) (w : ℝ → ℂ) : ∫ x in Ioi (0 : ℝ), (starRingEnd ℂ) (f x) * w x = ∫ x, (starRingEnd ℂ) (f x) * w x

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

Preamble
-- Generated from ChapterHalfLineLimitCircle.lean — theorem BookProof.HalfLineLimitCircle.setIntegral_eq_integral_of_testSpace
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.setIntegral_eq_integral_of_testSpace {f : ℝ → ℂ} (hf : f ∈ testSpace) (w : ℝ → ℂ) :
    ∫ x in Ioi (0 : ℝ), (starRingEnd ℂ) (f x) * w x = ∫ x, (starRingEnd ℂ) (f x) * w x := by sorry

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