| import OAI.Analysis.FiniteFactor.OrbitSampling.Measures |
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| namespace OAI |
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| namespace FiniteFactor |
| open MeasureTheory Filter |
| open scoped Classical ENNReal |
| noncomputable section |
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| theorem coordinateRotate_neg_cube (θ x : SourceCoordinate → ℝ) (n : ℤ) |
| (hx : ∀ i, x i ∈ Set.Ico (0 : ℝ) 1) : |
| coordinateRotate θ (-n) (coordinateRotate θ n x) = x := by |
| funext i |
| unfold coordinateRotate |
| rw [Int.cast_neg,neg_mul] |
| have he : Int.fract (x i+(n : ℝ)*θ i) + -((n : ℝ)*θ i) = |
| x i - (⌊x i+(n : ℝ)*θ i⌋ : ℝ) := by |
| linarith [Int.fract_add_floor (x i+(n : ℝ)*θ i)] |
| rw [he,Int.fract_sub_intCast,Int.fract_eq_self.mpr (hx i)] |
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| theorem coordinateRotate_phase (θ x : SourceCoordinate → ℝ) (n : ℤ) (i : SourceCoordinate) : |
| coordinateRotate θ n x i = Int.fract (x i+Int.fract ((n : ℝ)*θ i)) := by |
| unfold coordinateRotate |
| have he : x i+(n : ℝ)*θ i = (x i+Int.fract ((n : ℝ)*θ i)) + (⌊(n : ℝ)*θ i⌋ : ℝ) := by |
| linarith [Int.fract_add_floor ((n : ℝ)*θ i)] |
| rw [he,Int.fract_add_intCast] |
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| theorem coordinateRotate_neg_phase (θ x : SourceCoordinate → ℝ) (n : ℤ) (i : SourceCoordinate) : |
| coordinateRotate θ (-n) x i = Int.fract (x i-Int.fract ((n : ℝ)*θ i)) := by |
| unfold coordinateRotate |
| rw [Int.cast_neg,neg_mul] |
| have he : x i + -((n : ℝ)*θ i) = (x i-Int.fract ((n : ℝ)*θ i)) - (⌊(n : ℝ)*θ i⌋ : ℝ) := by |
| linarith [Int.fract_add_floor ((n : ℝ)*θ i)] |
| rw [he,Int.fract_sub_intCast] |
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| / |
| is needed only on the conull cube of representatives. -/ |
| theorem sourceSamplingHaar_rotate_withDensity (θ : SourceCoordinate → ℝ) (n : ℤ) |
| (w : (SourceCoordinate → ℝ) → ℝ≥0∞) (hw : Measurable w) : |
| (sourceSamplingHaar.withDensity w).map (coordinateRotate θ n) = |
| sourceSamplingHaar.withDensity (fun y => w (coordinateRotate θ (-n) y)) := by |
| let R := coordinateRotate θ n |
| let R' := coordinateRotate θ (-n) |
| have hR := coordinateRotate_preserving θ n |
| have hR' := coordinateRotate_preserving θ (-n) |
| ext A hA |
| rw [Measure.map_apply hR.measurable hA,withDensity_apply _ (hR.measurable hA),withDensity_apply _ hA] |
| rw [← lintegral_indicator (hR.measurable hA),← lintegral_indicator hA] |
| have hm : Measurable (A.indicator (fun y => w (R' y))) := (hw.comp hR'.measurable).indicator hA |
| calc |
| _ = ∫⁻ x, A.indicator (fun y => w (R' y)) (R x) ∂sourceSamplingHaar := by |
| apply lintegral_congr_ae |
| filter_upwards [sourceSamplingHaar_cube] with x hx |
| change (if R x ∈ A then w x else 0) = (if R x ∈ A then w (R' (R x)) else 0) |
| rw [show R' (R x) = x from coordinateRotate_neg_cube θ x n hx] |
| _ = _ := hR.lintegral_comp hm |
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| end |
| end FiniteFactor |
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| end OAI |
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