| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory | |
| namespace Problem367.FactorTransport | |
| variable {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] | |
| theorem map_invariant_of_ae_equivariant | |
| {μ : Measure α} {a : α → α} {b : β → β} {f : α → β} | |
| (ha : MeasurePreserving a μ μ) (hb : Measurable b) (hf : Measurable f) | |
| (heq : ∀ᵐ x ∂μ, f (a x) = b (f x)) : | |
| Measure.map b (Measure.map f μ) = Measure.map f μ := by | |
| calc | |
| Measure.map b (Measure.map f μ) = Measure.map (b ∘ f) μ := | |
| Measure.map_map hb hf | |
| _ = Measure.map (f ∘ a) μ := Measure.map_congr (Filter.EventuallyEq.symm heq) | |
| _ = Measure.map f (Measure.map a μ) := (Measure.map_map hf ha.measurable).symm | |
| _ = Measure.map f μ := by rw [ha.map_eq] | |
| omit [MeasurableSpace β] in | |
| theorem ae_equivariant_congr | |
| {μ : Measure α} {a : α → α} {b : β → β} {f f' : α → β} | |
| (ha : Measure.QuasiMeasurePreserving a μ μ) (hff' : f =ᵐ[μ] f') | |
| (heq : ∀ᵐ x ∂μ, f (a x) = b (f x)) : | |
| ∀ᵐ x ∂μ, f' (a x) = b (f' x) := by | |
| filter_upwards [ha.ae hff', hff', heq] with x hx h'x heqx | |
| calc | |
| f' (a x) = f (a x) := hx.symm | |
| _ = b (f x) := heqx | |
| _ = b (f' x) := congrArg b h'x | |
| omit [MeasurableSpace β] in | |
| theorem ae_equivariant_countable | |
| {G : Type*} [Countable G] {μ : Measure α} | |
| (a : G → α → α) (b : G → β → β) (f : α → β) | |
| (heq : ∀ g, ∀ᵐ x ∂μ, f (a g x) = b g (f x)) : | |
| ∀ᵐ x ∂μ, ∀ g, f (a g x) = b g (f x) := by | |
| exact ae_all_iff.mpr heq | |
| end Problem367.FactorTransport | |
| end | |
| end OAI | |