| import OAI.Analysis.FiniteFactor.OrbitSampling.Weights |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Finset Filter |
| open scoped ENNReal |
| noncomputable section |
|
|
| abbrev sourceSamplingHaar : Measure (SourceCoordinate → ℝ) := |
| Measure.infinitePi (fun _ : SourceCoordinate => unitIntervalMeasure) |
|
|
| def sourceSamplingWeightedMeasure (I : Finset SourceCoordinate) (s : ℕ) |
| (θ : SourceCoordinate → ℝ) (n : ℤ) : Measure (SourceCoordinate → ℝ) := |
| sourceSamplingHaar.withDensity (fun x => ENNReal.ofReal (sourceSamplingWeight I s θ n x)) |
| |
| instance sourceSamplingWeightedMeasure_finite (I : Finset SourceCoordinate) (s : ℕ) |
| (θ : SourceCoordinate → ℝ) (n : ℤ) : IsFiniteMeasure (sourceSamplingWeightedMeasure I s θ n) := by |
| apply isFiniteMeasure_withDensity |
| let C := ((sourceBlockSize s : ℝ)^(13/5 : ℝ))^(sourceSamplingPrefix s).card |
| have hh : (∫⁻ x, ENNReal.ofReal (sourceSamplingWeight I s θ n x) ∂sourceSamplingHaar) ≤ ENNReal.ofReal C := by |
| calc |
| _ ≤ ∫⁻ _x, ENNReal.ofReal C ∂sourceSamplingHaar := |
| lintegral_mono (fun x => ENNReal.ofReal_le_ofReal (sourceSamplingWeight_bounds I s θ n x).2) |
| _ = _ := by simp |
| exact ne_top_of_le_ne_top ENNReal.ofReal_ne_top hh |
| |
| def sourceSamplingSummand (I : Finset SourceCoordinate) (s : ℕ) |
| (θ : SourceCoordinate → ℝ) (n : ℤ) : |
| Measure ((SourceCoordinate → ℝ) × (SourceCoordinate → ℝ)) := |
| (sourceSamplingWeightedMeasure I s θ n).map (fun x => (x,coordinateRotate θ n x)) |
| |
| instance sourceSamplingSummand_finite (I : Finset SourceCoordinate) (s : ℕ) |
| (θ : SourceCoordinate → ℝ) (n : ℤ) : IsFiniteMeasure (sourceSamplingSummand I s θ n) := by |
| unfold sourceSamplingSummand |
| infer_instance |
| |
| theorem sourceSamplingSummand_first_marginal (I : Finset SourceCoordinate) (s : ℕ) |
| (θ : SourceCoordinate → ℝ) (n : ℤ) : |
| (sourceSamplingSummand I s θ n).map Prod.fst = sourceSamplingWeightedMeasure I s θ n := 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_fst hp] |
| exact Measure.map_id |
| |
| / |
| No weak limit or uniform marginal bound is asserted by this definition. -/ |
| def sourceSamplingUntrimmed (I : Finset SourceCoordinate) (s : ℕ) (θ : SourceCoordinate → ℝ) : |
| Measure ((SourceCoordinate → ℝ) × (SourceCoordinate → ℝ)) := |
| ENNReal.ofReal (1/((sourceSamplingPowers s).card : ℝ)) • |
| ∑ n ∈ sourceSamplingPowers s, sourceSamplingSummand I s θ n |
| |
| theorem sourceSamplingHaar_cube : ∀ᵐ x ∂sourceSamplingHaar, |
| ∀ i : SourceCoordinate, x i ∈ Set.Ico (0 : ℝ) 1 := by |
| rw [ae_all_iff] |
| intro i |
| have hh : ∀ᵐ t ∂sourceSamplingHaar.map (fun x => x i), t ∈ Set.Ico (0 : ℝ) 1 := by |
| rw [Measure.infinitePi_map_eval] |
| exact ae_restrict_mem measurableSet_Ico |
| exact (ae_map_iff (measurable_pi_apply i).aemeasurable measurableSet_Ico).mp hh |
| |
| / |
| theorem sourceSamplingWeightedMeasure_orders (I : Finset SourceCoordinate) (s : ℕ) |
| (θ : SourceCoordinate → ℝ) (n : ℤ) : |
| ∀ᵐ x ∂sourceSamplingWeightedMeasure I s θ n, |
| ∀ i ∈ sourceSamplingPrefix s, x i < coordinateRotate θ n x i := by |
| rw [sourceSamplingWeightedMeasure,ae_withDensity_iff |
| ((sourceSamplingWeight_measurable I s θ n).ennreal_ofReal)] |
| filter_upwards [sourceSamplingHaar_cube] with x hx hw |
| have hw' : sourceSamplingWeight I s θ n x ≠ 0 := by |
| intro he |
| exact hw (by rw [he]; simp) |
| exact sourceSamplingWeight_orders_prefix I s θ n x hx hw' |
| |
| end |
| end FiniteFactor |
| |
| end OAI |
| |
