`BookProof.ChapterBornMeasure.bornMeasure_apply` (psi : Lp ℂ 2 μ) {s : Set α} (hs : MeasurableSet s) : bornMeasure psi s = ∫⁻ x in s, ‖(psi : α → ℂ) x‖ₑ ^ 2 ∂μ

Proved

BookProof.ChapterBornMeasure.bornMeasure_apply

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterBornMeasure.

BookProof.ChapterBornMeasure.bornMeasure_apply (psi : Lp ℂ 2 μ) {s : Set α} (hs : MeasurableSet s) : bornMeasure psi s = ∫⁻ x in s, ‖(psi : α → ℂ) x‖ₑ ^ 2 ∂μ

Formalization note: Lean 4 identifier BookProof.ChapterBornMeasure.bornMeasure_apply.

Preamble

-- Generated from ChapterBornMeasure.lean — theorem BookProof.ChapterBornMeasure.bornMeasure_apply
import Mathlib
import Definitions.Def_ChapterBornMeasure
open BookProof.ChapterBornMeasure


open MeasureTheory
open scoped ENNReal


variable {α : Type*} [MeasurableSpace α] {μ : Measure α}

Formal statement

theorem BookProof.ChapterBornMeasure.bornMeasure_apply (psi : Lp ℂ 2 μ) {s : Set α} (hs : MeasurableSet s) :
    bornMeasure psi s = ∫⁻ x in s, ‖(psi : α → ℂ) x‖ₑ ^ 2 ∂μ := by sorry

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