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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 | import OAI.Analysis.FiniteFactor.OrbitSampling.WeakLimit
import Mathlib.MeasureTheory.Constructions.Polish.Basic
import Mathlib.Topology.Instances.AddCircle.Real
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter Finset
open scoped Classical ENNReal Topology
noncomputable section
abbrev SourceSamplingTorus := (SourceCoordinate → UnitAddCircle) × (SourceCoordinate → UnitAddCircle)
instance sourceSamplingTorus_compact : CompactSpace SourceSamplingTorus := inferInstance
instance sourceSamplingTorus_borel : BorelSpace SourceSamplingTorus := Prod.borelSpace
def sourceSamplingTorusMap (p : SourceSamplingPair) : SourceSamplingTorus :=
(fun i => (p.1 i : UnitAddCircle),fun i => (p.2 i : UnitAddCircle))
theorem sourceSamplingTorusMap_continuous : Continuous sourceSamplingTorusMap := by
exact (continuous_pi (fun i => (AddCircle.continuous_mk' (1 : ℝ)).comp
((continuous_apply i).comp continuous_fst))).prodMk
(continuous_pi (fun i => (AddCircle.continuous_mk' (1 : ℝ)).comp
((continuous_apply i).comp continuous_snd)))
def sourceSamplingUntrimmedTorus (I : Finset SourceCoordinate) (s : ℕ)
(θ : SourceCoordinate → ℝ) : FiniteMeasure SourceSamplingTorus :=
(sourceSamplingUntrimmedFinite I s θ).map sourceSamplingTorusMap
def sourceSamplingLambdaTorus (I : Finset SourceCoordinate) : FiniteMeasure SourceSamplingTorus :=
(sourceSamplingLambdaFinite I).map sourceSamplingTorusMap
theorem sourceSamplingUntrimmed_torus_event : ∀ᵐ θ ∂sourceAngleLaw,
∀ I : Finset SourceCoordinate,
Tendsto (fun s => sourceSamplingUntrimmedTorus I s θ) atTop
(𝓝 (sourceSamplingLambdaTorus I)) ∧
∀ f : C(SourceSamplingTorus,ℝ),
Tendsto (fun s => ∫ p, f p ∂(sourceSamplingUntrimmedTorus I s θ : Measure SourceSamplingTorus))
atTop (𝓝 (∫ p, f p ∂(sourceSamplingLambdaTorus I : Measure SourceSamplingTorus))) := by
filter_upwards [sourceSamplingUntrimmed_weak_event] with θ hθ
intro I
have h := FiniteMeasure.tendsto_map_of_tendsto_of_continuous _ _ (hθ I)
sourceSamplingTorusMap_continuous
refine ⟨h,fun f => ?_⟩
exact (FiniteMeasure.tendsto_iff_forall_integral_tendsto.mp h)
(BoundedContinuousFunction.mkOfCompact f)
end
end FiniteFactor
end OAI
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