`BookProof.QuantumGravityHalfDensity.halfDensityUnitary_apply` (g : Lp ℂ 2 (volume.restrict (Set.Ioi (0 : ℝ)))) : (halfDensityUnitary g : ℝ → ℂ) =ᵐ[qgSrcMeasure] fun y => (g : ℝ →

Proved

BookProof.QuantumGravityHalfDensity.halfDensityUnitary_apply

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterQuantumGravityHalfDensity.

BookProof.QuantumGravityHalfDensity.halfDensityUnitary_apply (g : Lp ℂ 2 (volume.restrict (Set.Ioi (0 : ℝ)))) : (halfDensityUnitary g : ℝ → ℂ) =ᵐ[qgSrcMeasure] fun y => (g : ℝ → ℂ) (y ^ 2)

Formalization note: Lean 4 identifier BookProof.QuantumGravityHalfDensity.halfDensityUnitary_apply.

Preamble

-- Generated from ChapterQuantumGravityHalfDensity.lean — theorem BookProof.QuantumGravityHalfDensity.halfDensityUnitary_apply
import Mathlib
import Definitions.Def_ChapterQuantumGravityHalfDensity
open BookProof.QuantumGravityHalfDensity



open MeasureTheory Set Filter
open scoped ENNReal

Formal statement

theorem BookProof.QuantumGravityHalfDensity.halfDensityUnitary_apply (g : Lp ℂ 2 (volume.restrict (Set.Ioi (0 : ℝ)))) :
    (halfDensityUnitary g : ℝ → ℂ) =ᵐ[qgSrcMeasure] fun y => (g : ℝ → ℂ) (y ^ 2) := by sorry

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