`BookProof.ChapterU.bornMeasure_isProbability` (Ψ : X → ℂ) (μ : Measure X) (hΨ : MemLp Ψ 2 μ) (hnorm : ∫ x, ‖Ψ x‖ ^ 2 ∂μ = 1) : IsProbabilityMeasure (bornMeasure Ψ μ)

Proved

BookProof.ChapterU.bornMeasure_isProbability

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterU.

BookProof.ChapterU.bornMeasure_isProbability (Ψ : X → ℂ) (μ : Measure X) (hΨ : MemLp Ψ 2 μ) (hnorm : ∫ x, ‖Ψ x‖ ^ 2 ∂μ = 1) : IsProbabilityMeasure (bornMeasure Ψ μ)

Formalization note: Lean 4 identifier BookProof.ChapterU.bornMeasure_isProbability.

Preamble

-- Generated from ChapterU.lean — theorem BookProof.ChapterU.bornMeasure_isProbability
import Mathlib
import Definitions.Def_ChapterU
import Definitions.Def_ChapterBornMeasure
open BookProof.ChapterBornMeasure
open BookProof.ChapterU


open MeasureTheory
open scoped ENNReal ProbabilityTheory TensorProduct

variable {X : Type*} [MeasurableSpace X]

Formal statement

theorem BookProof.ChapterU.bornMeasure_isProbability (Ψ : X → ℂ) (μ : Measure X)
    (hΨ : MemLp Ψ 2 μ) (hnorm : ∫ x, ‖Ψ x‖ ^ 2 ∂μ = 1) :
    IsProbabilityMeasure (bornMeasure Ψ μ) := by sorry

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