File size: 4,478 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 82 83 84 85 86 87 88 89 | import OAI.Analysis.FiniteFactor.OrbitSampling.OrderedRestriction
import OAI.Analysis.FiniteFactor.OrbitSampling.CompactTorus
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter Finset
open scoped Classical ENNReal Topology
noncomputable section
def sourceSamplingTrimmedFinite (I : Finset SourceCoordinate) (s : ℕ)
(θ : SourceCoordinate → ℝ) : FiniteMeasure SourceSamplingPair :=
⟨sourceSamplingTrimmed I s θ,inferInstance⟩
def sourceSamplingTrimmedTorus (I : Finset SourceCoordinate) (s : ℕ)
(θ : SourceCoordinate → ℝ) : FiniteMeasure SourceSamplingTorus :=
(sourceSamplingTrimmedFinite I s θ).map sourceSamplingTorusMap
theorem sourceSampling_discarded_tendsto (I : Finset SourceCoordinate)
(θ : SourceCoordinate → ℝ) (hθ : SourceAngles θ)
(hmass : Tendsto (fun s => ((sourceSamplingUntrimmed I s θ) Set.univ).toReal)
atTop (𝓝 (((sourceSamplingLambda I) Set.univ).toReal))) :
Tendsto (fun s => ((sourceSamplingUntrimmed I s θ) sourceSamplingOrderedᶜ).toReal)
atTop (𝓝 0) := by
have hlim := hmass.mul sourceSampling_tail_error_tendsto
rw [mul_zero] at hlim
apply squeeze_zero (fun s => ENNReal.toReal_nonneg) _ hlim
intro s
have h := ENNReal.toReal_mono (by finiteness)
(sourceSamplingUntrimmed_discarded_le I s θ hθ)
simpa only [ENNReal.toReal_mul,ENNReal.toReal_ofReal (by positivity :
0 ≤ 4*(sourceBlockSize s : ℝ)^(-(sourceBlockSize s : ℝ)/20))] using h
def finiteMeasureRestrict {Ω : Type*} [MeasurableSpace Ω]
(μ : FiniteMeasure Ω) (A : Set Ω) : FiniteMeasure Ω :=
⟨(μ : Measure Ω).restrict A,inferInstance⟩
/-- A restriction whose discarded mass tends to zero has the same finite
weak limit. The norm bound gives the actual error for every test function. -/
theorem finiteMeasure_restrict_tendsto {Ω : Type*} [MeasurableSpace Ω]
[TopologicalSpace Ω] [OpensMeasurableSpace Ω]
(μ : ℕ → FiniteMeasure Ω) (ν : FiniteMeasure Ω) (A : Set Ω) (hA : MeasurableSet A)
(hμ : Tendsto μ atTop (𝓝 ν))
(hloss : Tendsto (fun n => ((μ n : Measure Ω) Aᶜ).toReal) atTop (𝓝 0)) :
Tendsto (fun n => finiteMeasureRestrict (μ n) A)
atTop (𝓝 ν) := by
apply FiniteMeasure.tendsto_of_forall_integral_tendsto
intro f
have hf := FiniteMeasure.tendsto_iff_forall_integral_tendsto.mp hμ f
have he : Tendsto (fun n => ∫ x in Aᶜ, f x ∂(μ n : Measure Ω)) atTop (𝓝 0) := by
apply tendsto_zero_iff_norm_tendsto_zero.mpr
have hb := hloss.const_mul ‖f‖
rw [mul_zero] at hb
apply squeeze_zero (fun n => norm_nonneg _) _ hb
intro n
have hh := norm_integral_le_of_norm_le_const
(μ := (μ n : Measure Ω).restrict Aᶜ)
(Eventually.of_forall (fun x => f.norm_coe_le_norm x))
simpa only [measureReal_def,Measure.restrict_apply MeasurableSet.univ,Set.univ_inter] using hh
have hsub := hf.sub he
rw [sub_zero] at hsub
convert hsub using 1
funext n
exact (eq_sub_iff_add_eq.mpr (integral_add_compl (μ := (μ n : Measure Ω)) hA (f.integrable (μ := (μ n : Measure Ω)))))
/-- Actual ordered measures have the same compact-torus weak limit, for
all continuous tests, on one probability-one angle event. -/
theorem sourceSamplingTrimmed_weak_event : ∀ᵐ θ ∂sourceAngleLaw,
∀ I : Finset SourceCoordinate,
Tendsto (fun s => sourceSamplingTrimmedFinite I s θ) atTop (𝓝 (sourceSamplingLambdaFinite I)) ∧
Tendsto (fun s => sourceSamplingTrimmedTorus I s θ) atTop (𝓝 (sourceSamplingLambdaTorus I)) ∧
∀ f : C(SourceSamplingTorus,ℝ),
Tendsto (fun s => ∫ p, f p ∂(sourceSamplingTrimmedTorus I s θ : Measure SourceSamplingTorus))
atTop (𝓝 (∫ p, f p ∂(sourceSamplingLambdaTorus I : Measure SourceSamplingTorus))) := by
filter_upwards [sourceSamplingUntrimmed_weak_event,sourceSampling_mass_limit_event,sourceAngleLaw_admissible]
with θ hw hm ha
intro I
have h := finiteMeasure_restrict_tendsto (fun s => sourceSamplingUntrimmedFinite I s θ)
(sourceSamplingLambdaFinite I) sourceSamplingOrdered sourceSamplingOrdered_measurable
(hw I) (sourceSampling_discarded_tendsto I θ ha (hm I))
have ht := FiniteMeasure.tendsto_map_of_tendsto_of_continuous _ _ h sourceSamplingTorusMap_continuous
refine ⟨h,ht,fun f => ?_⟩
exact (FiniteMeasure.tendsto_iff_forall_integral_tendsto.mp ht) (BoundedContinuousFunction.mkOfCompact f)
end
end FiniteFactor
end OAI
|