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f1f7953 | 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 | import Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic
import Mathlib.Probability.ProductMeasure
namespace OAI
open MeasureTheory Set
namespace Invariant233.CocycleSupport
theorem measurePreserving_circleRepresentative :
MeasurePreserving
(fun x : UnitAddCircle => ((AddCircle.equivIco (1 : ℝ) 0 x) : ℝ))
volume (volume.restrict (Ico (0 : ℝ) 1)) := by
have hmk : MeasurePreserving (fun x : ℝ => (x : UnitAddCircle))
(volume.restrict (Ico 0 (0 + 1))) volume := by
rw [restrict_Ico_eq_restrict_Ioc]
exact AddCircle.measurePreserving_mk 1 0
have hs := hmk.comp (measurePreserving_subtype_coe
(μa := (volume : Measure ℝ)) (s := Ico 0 (0 + 1)) measurableSet_Ico)
have he : MeasurePreserving (AddCircle.measurableEquivIco (1 : ℝ) 0).symm
(Measure.comap Subtype.val (volume : Measure ℝ)) volume := hs
have hi := MeasurePreserving.symm (AddCircle.measurableEquivIco (1 : ℝ) 0).symm he
convert (measurePreserving_subtype_coe
(μa := (volume : Measure ℝ)) (s := Ico 0 (0 + 1)) measurableSet_Ico).comp hi using 1
· rfl
· simp
theorem measurePreserving_productRepresentative {ι : Type*} (i : ι) :
MeasurePreserving
(fun x : ι → UnitAddCircle => ((AddCircle.equivIco (1 : ℝ) 0 (x i)) : ℝ))
(Measure.infinitePi (fun _ : ι => (volume : Measure UnitAddCircle)))
(volume.restrict (Ico (0 : ℝ) 1)) := by
let : IsProbabilityMeasure (volume : Measure UnitAddCircle) := ⟨UnitAddCircle.measure_univ⟩
exact measurePreserving_circleRepresentative.comp
(measurePreserving_eval_infinitePi (fun _ : ι => (volume : Measure UnitAddCircle)) i)
end Invariant233.CocycleSupport
end OAI
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