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67aee1d | 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 53 54 55 56 57 58 59 60 61 62 | import OAI.Analysis.FiniteFactor.OrbitSampling.Measures
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter
open scoped Classical ENNReal
noncomputable section
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)]
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]
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]
/-- Change of variables for an actual weighted Haar law. The inverse identity
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
end
end FiniteFactor
end OAI
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