File size: 3,839 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 | 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
/-- Finite positive untrimmed measures averaged over the constructed powers.
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
/-- The weighted source law already orders every coordinate in its prefix. -/
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
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