File: Mathlib/Dynamics/Ergodic/Function.lean

full_name: QuasiErgodic.ae_eq_const_of_ae_eq_comp_of_ae_range₀
code: /-- Let `f : α → α` be a (quasi)ergodic map. Let `g : α → X` is a null-measurable function
from `α` to a nonempty space with a countable family of measurable sets
separating points of a set `s` such that `f x ∈ s` for a.e. `x`.
If `g` that is a.e.-invariant under `f`, then `g` is a.e. constant. -/
theorem QuasiErgodic.ae_eq_const_of_ae_eq_comp_of_ae_range₀ [Nonempty X] [MeasurableSpace X]
    {s : Set X} [MeasurableSpace.CountablySeparated s] {f : α → α} {g : α → X}
    (h : QuasiErgodic f μ) (hs : ∀ᵐ x ∂μ, g x ∈ s) (hgm : NullMeasurable g μ)
    (hg_eq : g ∘ f =ᵐ[μ] g) :
    ∃ c, g =ᵐ[μ] const α c 
goal: ∃ c, g =ᵐ[μ] Function.const α c

full_name: PreErgodic.ae_eq_const_of_ae_eq_comp
code: /-- Let `f : α → α` be a (pre)ergodic map.
Let `g : α → X` be a measurable function from `α` to a nonempty measurable space
with a countable family of measurable sets separating the points of `X`.
If `g` is invariant under `f`, then `g` is a.e. constant. -/
theorem PreErgodic.ae_eq_const_of_ae_eq_comp (h : PreErgodic f μ) (hgm : Measurable g)
    (hg_eq : g ∘ f = g) : ∃ c, g =ᵐ[μ] const α c 
goal: ∃ c, g =ᵐ[μ] Function.const α c

full_name: QuasiErgodic.ae_eq_const_of_ae_eq_comp₀
code: /-- Let `f : α → α` be a quasi ergodic map.
Let `g : α → X` be a null-measurable function from `α` to a nonempty measurable space
with a countable family of measurable sets separating the points of `X`.
If `g` is a.e.-invariant under `f`, then `g` is a.e. constant. -/
theorem QuasiErgodic.ae_eq_const_of_ae_eq_comp₀ (h : QuasiErgodic f μ) (hgm : NullMeasurable g μ)
    (hg_eq : g ∘ f =ᵐ[μ] g) : ∃ c, g =ᵐ[μ] const α c 
goal: ∃ c, g =ᵐ[μ] Function.const α c

full_name: Ergodic.ae_eq_const_of_ae_eq_comp₀
code: /-- Let `f : α → α` be an ergodic map.
Let `g : α → X` be a null-measurable function from `α` to a nonempty measurable space
with a countable family of measurable sets separating the points of `X`.
If `g` is a.e.-invariant under `f`, then `g` is a.e. constant. -/
theorem Ergodic.ae_eq_const_of_ae_eq_comp₀ (h : Ergodic f μ) (hgm : NullMeasurable g μ)
    (hg_eq : g ∘ f =ᵐ[μ] g) : ∃ c, g =ᵐ[μ] const α c 
goal: ∃ c, g =ᵐ[μ] Function.const α c

full_name: QuasiErgodic.ae_eq_const_of_ae_eq_comp_ae
code: /-- Let `f : α → α` be a quasi ergodic map.
Let `g : α → X` be an a.e. strongly measurable function
from `α` to a nonempty metrizable topological space.
If `g` is a.e.-invariant under `f`, then `g` is a.e. constant. -/
theorem ae_eq_const_of_ae_eq_comp_ae {g : α → X} (h : QuasiErgodic f μ)
    (hgm : AEStronglyMeasurable g μ) (hg_eq : g ∘ f =ᵐ[μ] g) : ∃ c, g =ᵐ[μ] const α c 
goal: ∃ c, g =ᵐ[μ] Function.const α c

full_name: QuasiErgodic.eq_const_of_compQuasiMeasurePreserving_eq
code: theorem eq_const_of_compQuasiMeasurePreserving_eq (h : QuasiErgodic f μ) {g : α →ₘ[μ] X}
    (hg_eq : g.compQuasiMeasurePreserving f h.1 = g) : ∃ c, g = .const α c 
goal: ∃ c, g = MeasureTheory.AEEqFun.const α c

full_name: Ergodic.ae_eq_const_of_ae_eq_comp_ae
code: /-- Let `f : α → α` be an ergodic map.
Let `g : α → X` be an a.e. strongly measurable function
from `α` to a nonempty metrizable topological space.
If `g` is a.e.-invariant under `f`, then `g` is a.e. constant. -/
theorem ae_eq_const_of_ae_eq_comp_ae {g : α → X} (h : Ergodic f μ) (hgm : AEStronglyMeasurable g μ)
    (hg_eq : g ∘ f =ᵐ[μ] g) : ∃ c, g =ᵐ[μ] const α c 
goal: ∃ c, g =ᵐ[μ] Function.const α c

full_name: Ergodic.eq_const_of_compMeasurePreserving_eq
code: theorem eq_const_of_compMeasurePreserving_eq (h : Ergodic f μ) {g : α →ₘ[μ] X}
    (hg_eq : g.compMeasurePreserving f h.1 = g) : ∃ c, g = .const α c 
goal: ∃ c, g = MeasureTheory.AEEqFun.const α c