Theorems · Theorem · dynamical systems

MeasureTheory.Measure.QuasiMeasurePreserving.birkhoffAverage_ae_eq_of_ae_eq

∀ {α : Type u_1} {M : Type u_2} [inst : MeasurableSpace α] [inst_1 : AddCommMonoid M] {f : α → α}
  {μ : MeasureTheory.Measure α} {φ ψ : α → M} (R : Type u_3) [inst_2 : DivisionSemiring R] [inst_3 : Module R M],
  MeasureTheory.Measure.QuasiMeasurePreserving f μ μ →
    φ =ᵐ[μ] ψ → ∀ (n : ℕ), birkhoffAverage R f φ n =ᵐ[μ] birkhoffAverage R f ψ n

If observables φ and ψ are μ-a.e. equal then the corresponding birkhoffAverage are μ-a.e. equal.

Defined in

Mathlib.Dynamics.BirkhoffSum.QuasiMeasurePreserving

Cited by

0 results in Mathlib

Foundations

Depth 205 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceAddCommMonoidDivisionSemiringModule