`BookProof.ChapterG.no_shift_invariant_probabilityMeasure` : ¬ ∃ μ : Measure ℤ, IsProbabilityMeasure μ ∧ ∀ s : Set ℤ, μ ((· + 1) ⁻¹' s) = μ s

Proved

BookProof.ChapterG.no_shift_invariant_probabilityMeasure

by leonardopedro · Oct 8, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterG.

BookProof.ChapterG.no_shift_invariant_probabilityMeasure : ¬ ∃ μ : Measure ℤ, IsProbabilityMeasure μ ∧ ∀ s : Set ℤ, μ ((· + 1) ⁻¹' s) = μ s

Formalization note: Lean 4 identifier BookProof.ChapterG.no_shift_invariant_probabilityMeasure.

Preamble

-- Generated from ChapterG.lean — theorem BookProof.ChapterG.no_shift_invariant_probabilityMeasure
import Mathlib
import Definitions.Def_ChapterG
open BookProof.ChapterG


open scoped ComplexConjugate InnerProductSpace Matrix
open MeasureTheory

Formal statement

theorem BookProof.ChapterG.no_shift_invariant_probabilityMeasure :
    ¬ ∃ μ : Measure ℤ, IsProbabilityMeasure μ ∧
      ∀ s : Set ℤ, μ ((· + 1) ⁻¹' s) = μ s := by sorry

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