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f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 | 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
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