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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 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 | import OAI.Analysis.FiniteFactor.OrbitSampling.TestedTransport
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter Finset
open scoped Classical ENNReal
noncomputable section
def sourceSamplingEndpoint (r : Bool)
(p : (SourceCoordinate → ℝ) × (SourceCoordinate → ℝ)) : SourceCoordinate → ℝ :=
if r then p.2 else p.1
theorem sourceSamplingEndpoint_measurable (r : Bool) : Measurable (sourceSamplingEndpoint r) := by
cases r
· exact measurable_fst
· exact measurable_snd
/-- Both actual rectangle-restricted marginal measures for a single power. -/
theorem sourceSamplingSummand_tested_marginal (I : Finset SourceCoordinate) (stage : ℕ)
(r : Bool) (E G : SourceCoordinate → Set ℝ)
(hE : ∀ i, MeasurableSet (E i)) (hG : ∀ i, MeasurableSet (G i))
(houtside : ∀ i ∉ sourceSamplingPrefix stage,
Set.Ico (0 : ℝ) 1 ⊆ E i ∧ Set.Ico (0 : ℝ) 1 ⊆ G i)
(θ : SourceCoordinate → ℝ) (n : ℤ) :
((sourceSamplingSummand I stage θ n).restrict (sourceSamplingRectangle E G)).map
(sourceSamplingEndpoint r) =
sourceSamplingHaar.withDensity (fun x => ENNReal.ofReal
(sourceSamplingProduct stage r (fun i => decide (i ∈ I)) E G n θ x)) := by
rw [sourceSamplingSummand_restrict_rectangle I stage E G hE hG houtside θ n]
have hp : Measurable (fun x : SourceCoordinate → ℝ => (x,coordinateRotate θ n x)) :=
measurable_id.prodMk (coordinateRotate_preserving θ n).measurable
rw [Measure.map_map (sourceSamplingEndpoint_measurable r) hp]
cases r
· exact Measure.map_id
· change (sourceSamplingHaar.withDensity (fun x => ENNReal.ofReal
(sourceSamplingProduct stage false (fun i => decide (i ∈ I)) E G n θ x))).map
(coordinateRotate θ n) = _
have hw : Measurable (fun x => ENNReal.ofReal
(sourceSamplingProduct stage false (fun i => decide (i ∈ I)) E G n θ x)) :=
(((sourceSamplingProduct_measurable stage false (fun i => decide (i ∈ I)) E G hE hG n).comp
(measurable_const.prodMk measurable_id)).ennreal_ofReal)
rw [sourceSamplingHaar_rotate_withDensity θ n _ hw]
apply withDensity_congr_ae
filter_upwards [sourceSamplingHaar_cube] with y hy
rw [sourceSamplingProduct_second_inverse I stage E G θ y n hy]
/-- Averaging commutes with rectangle restriction and either endpoint map.
The density is the previously estimated empirical function, exactly. -/
theorem sourceSamplingUntrimmed_tested_marginal (I : Finset SourceCoordinate) (stage : ℕ)
(r : Bool) (E G : SourceCoordinate → Set ℝ)
(hE : ∀ i, MeasurableSet (E i)) (hG : ∀ i, MeasurableSet (G i))
(houtside : ∀ i ∉ sourceSamplingPrefix stage,
Set.Ico (0 : ℝ) 1 ⊆ E i ∧ Set.Ico (0 : ℝ) 1 ⊆ G i)
(θ : SourceCoordinate → ℝ) :
((sourceSamplingUntrimmed I stage θ).restrict (sourceSamplingRectangle E G)).map
(sourceSamplingEndpoint r) =
sourceSamplingHaar.withDensity (fun x => ENNReal.ofReal
(sourceSamplingEmpiricalDensity stage r (fun i => decide (i ∈ I)) E G θ x)) := by
let S := sourceSamplingPowers stage
let c : ℝ≥0∞ := ENNReal.ofReal (1/((sourceSamplingPowers stage).card : ℝ))
let w := fun n => sourceSamplingProduct stage r (fun i => decide (i ∈ I)) E G n θ
have hrep : (fun x => ENNReal.ofReal (sourceSamplingEmpiricalDensity stage r
(fun i => decide (i ∈ I)) E G θ x)) =
(fun x => c*∑ n ∈ S, ENNReal.ofReal (w n x)) := by
funext x
simp only [sourceSamplingEmpiricalDensity]
rw [div_eq_mul_inv,mul_comm,← one_div,ENNReal.ofReal_mul (by positivity),
ENNReal.ofReal_sum_of_nonneg (fun n _ => (sourceSamplingProduct_bounds stage r
(fun i => decide (i ∈ I)) E G n θ x).1)]
rw [hrep]
have hf := sourceSamplingEndpoint_measurable r
have hR := sourceSamplingRectangle_measurable E G hE hG
ext A hA
rw [Measure.map_apply hf hA,Measure.restrict_apply (hf hA)]
unfold sourceSamplingUntrimmed
rw [Measure.smul_apply,Measure.finsetSum_apply,withDensity_apply _ hA]
have hterm (n : ℤ) : sourceSamplingSummand I stage θ n
(sourceSamplingEndpoint r ⁻¹' A ∩ sourceSamplingRectangle E G) =
∫⁻ x in A, ENNReal.ofReal (w n x) ∂sourceSamplingHaar := by
rw [← Measure.restrict_apply (hf hA),← Measure.map_apply hf hA,
sourceSamplingSummand_tested_marginal I stage r E G hE hG houtside θ n]
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 _ =>
(((sourceSamplingProduct_measurable stage r (fun i => decide (i ∈ I)) E G hE hG n).comp
(measurable_const.prodMk measurable_id)).ennreal_ofReal))).symm
end
end FiniteFactor
end OAI
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