| 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 |
|
|
