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Sub-invariant functions of an ergodic system are constant

Lax606786.SubinvariantFunctions · concepts/Lax606786/SubinvariantFunctions.lean · lax-606786

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    Natural Language Statement

    Theorem

    Let σ\sigma be an ergodic measure-preserving map of a probability space (Ω,μ)(\Omega, \mu) and h:Ω→[−∞,∞]h : \Omega \to [-\infty, \infty] measurable with h(ω)≤h(σω)h(\omega) \le h(\sigma\omega) for every ω\omega. Then hh is almost everywhere equal to a constant.

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    Lean source view on GitLab

    1import Mathlib.Dynamics.Ergodic.Ergodic
    2import Mathlib.Data.EReal.Basic
    3import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
    4
    5/-!
    6---
    7title: Sub-invariant functions of an ergodic system are constant
    8type: theorem
    9---
    10Let σ\sigma be an ergodic measure-preserving map of a probability space (Ω,μ)(\Omega, \mu) and
    11h:Ω→[−∞,∞]h : \Omega \to [-\infty, \infty] measurable with h(ω)≤h(σω)h(\omega) \le h(\sigma\omega) for every
    12ω\omega. Then hh is almost everywhere equal to a constant.
    13-/
    14
    15namespace Lax606786.SubinvariantFunctions
    16
    17open MeasureTheory
    18
    19/-- A measurable `h ≤ h ∘ σ` is a.e. constant. -/
    20axiom exists_ae_eq_const_of_le_comp {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
    21 [IsProbabilityMeasure μ] {σ : Ω → Ω} (hσ : Ergodic σ μ) {h : Ω → EReal} (hm : Measurable h)
    22 (hle : ∀ ω, h ω ≤ h (σ ω)) :
    23 ∃ c : EReal, ∀ᵐ ω ∂μ, h ω = c
    24
    25end Lax606786.SubinvariantFunctions
    26
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