`BookProof.QuantumGravityHalfDensity.halfDensityUnitary_symm_apply` (h : Lp ℂ 2 qgSrcMeasure) : (halfDensityUnitary.symm h : ℝ → ℂ) =ᵐ[volume.restrict (Set.Ioi (0 : ℝ))] fun e => (

Proved

BookProof.QuantumGravityHalfDensity.halfDensityUnitary_symm_apply

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterQuantumGravityHalfDensity.

BookProof.QuantumGravityHalfDensity.halfDensityUnitary_symm_apply (h : Lp ℂ 2 qgSrcMeasure) : (halfDensityUnitary.symm h : ℝ → ℂ) =ᵐ[volume.restrict (Set.Ioi (0 : ℝ))] fun e => (h : ℝ → ℂ) (Real.sqrt e)

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

Preamble

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



open MeasureTheory Set Filter
open scoped ENNReal

Formal statement

theorem BookProof.QuantumGravityHalfDensity.halfDensityUnitary_symm_apply (h : Lp ℂ 2 qgSrcMeasure) :
    (halfDensityUnitary.symm h : ℝ → ℂ)
      =ᵐ[volume.restrict (Set.Ioi (0 : ℝ))] fun e => (h : ℝ → ℂ) (Real.sqrt e) := by sorry

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