File size: 4,388 Bytes
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 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 | import OAI.Analysis.FiniteFactor.OrbitSampling.UniformEvent
import OAI.Analysis.FiniteFactor.OrbitSampling.RotationTransport
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter Finset
open scoped Classical ENNReal
noncomputable section
/-- The actual second marginal of one orbit summand, obtained by weighted
Haar transport and not supplied as an assumption. -/
theorem sourceSamplingSummand_second_density (I : Finset SourceCoordinate) (stage : ℕ)
(θ : SourceCoordinate → ℝ) (n : ℤ) :
(sourceSamplingSummand I stage θ n).map Prod.snd =
sourceSamplingHaar.withDensity (fun y => ENNReal.ofReal
(sourceSamplingWeight I stage θ n (coordinateRotate θ (-n) y))) := by
unfold sourceSamplingSummand
have hp : Measurable (fun x : SourceCoordinate → ℝ => (x,coordinateRotate θ n x)) :=
measurable_id.prodMk (coordinateRotate_preserving θ n).measurable
rw [Measure.map_map measurable_snd hp]
exact sourceSamplingHaar_rotate_withDensity θ n _
(sourceSamplingWeight_measurable I stage θ n).ennreal_ofReal
/-- The actual second marginal is the second-oriented empirical density. -/
theorem sourceSamplingUntrimmed_second_density (I : Finset SourceCoordinate) (stage : ℕ)
(θ : SourceCoordinate → ℝ) :
(sourceSamplingUntrimmed I stage θ).map Prod.snd =
sourceSamplingHaar.withDensity (fun x => ENNReal.ofReal
(sourceSamplingEmpiricalDensity stage true (fun i => decide (i ∈ I))
(fun _ => Set.Icc 0 1) (fun _ => Set.Icc 0 1) θ x)) := by
let S := sourceSamplingPowers stage
let c : ℝ≥0∞ := ENNReal.ofReal (1/((sourceSamplingPowers stage).card : ℝ))
let w (n : ℤ) := fun x => sourceSamplingWeight I stage θ n (coordinateRotate θ (-n) x)
have hrep : (fun x => ENNReal.ofReal (sourceSamplingEmpiricalDensity stage true (fun i => decide (i ∈ I))
(fun _ => Set.Icc 0 1) (fun _ => Set.Icc 0 1) θ x)) =ᵐ[sourceSamplingHaar]
(fun x => c*(∑ n ∈ S, ENNReal.ofReal (w n x))) := by
filter_upwards [sourceSamplingHaar_cube] with x hx
have hw (n : ℤ) : sourceSamplingProduct stage true (fun i => decide (i ∈ I))
(fun _ => Set.Icc 0 1) (fun _ => Set.Icc 0 1) n θ x = w n x := by
apply Finset.prod_congr rfl
intro i _
have hx' : x i ∈ Set.Icc (0 : ℝ) 1 := ⟨(hx i).1,(hx i).2.le⟩
have hz : Int.fract (x i-Int.fract ((n : ℝ)*θ i)) ∈ Set.Icc (0 : ℝ) 1 :=
⟨Int.fract_nonneg _,(Int.fract_lt_one _).le⟩
simp only [samplingOrientedWeight,ite_true,hz,hx',and_self]
rw [coordinateRotate_neg_phase]
simp only [sourceSamplingEmpiricalDensity,hw]
rw [div_eq_mul_inv,mul_comm,← one_div,
ENNReal.ofReal_mul (by positivity),ENNReal.ofReal_sum_of_nonneg (fun n _ => (sourceSamplingWeight_bounds I stage θ n (coordinateRotate θ (-n) x)).1)]
rw [withDensity_congr_ae hrep]
ext A hA
rw [Measure.map_apply measurable_snd hA]
unfold sourceSamplingUntrimmed
rw [Measure.smul_apply,Measure.finsetSum_apply,withDensity_apply _ hA]
have hterm (n : ℤ) : sourceSamplingSummand I stage θ n (Prod.snd ⁻¹' A) =
∫⁻ x in A, ENNReal.ofReal (w n x) ∂sourceSamplingHaar := by
rw [← Measure.map_apply measurable_snd hA,sourceSamplingSummand_second_density]
exact withDensity_apply _ hA
simp only [hterm,smul_eq_mul]
rw [lintegral_const_mul' c _ ENNReal.ofReal_ne_top]
congr 1
exact (lintegral_finsetSum S (fun n _ =>
((sourceSamplingWeight_measurable I stage θ n).comp (coordinateRotate_preserving θ (-n)).measurable).ennreal_ofReal)).symm
/-- Uniform L2 densities for the actual untrimmed second marginals. -/
theorem sourceSamplingUntrimmed_second_L2 (I : Finset SourceCoordinate) :
∀ᵐ θ ∂sourceAngleLaw, ∃ C : ℝ, ∀ stage,
∃ f : (SourceCoordinate → ℝ) → ℝ,
(sourceSamplingUntrimmed I stage θ).map Prod.snd = sourceSamplingHaar.withDensity (fun x => ENNReal.ofReal (f x)) ∧
MemLp f 2 sourceSamplingHaar ∧ (∫ x, f x^2 ∂sourceSamplingHaar) ≤ C := by
filter_upwards [sourceSamplingEmpiricalDensity_uniform true (fun i => decide (i ∈ I))
(fun _ => 0) (fun _ => 1) (fun _ => 0) (fun _ => 1)] with θ hθ
obtain ⟨C,hC⟩ := hθ
refine ⟨C,fun stage => ⟨_,sourceSamplingUntrimmed_second_density I stage θ,(hC stage).1,(hC stage).2⟩⟩
end
end FiniteFactor
end OAI
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