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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 | import OAI.Analysis.FiniteFactor.OrbitSampling.DiagonalLimit
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter Finset
open scoped Classical ENNReal Topology
noncomputable section
/-- The actual one-coordinate diagonal model converges to uniform diagonal
mass on every measurable first test and closed-interval second test. -/
theorem sourceSampling_diagonal_integral_tendsto (E : Set ℝ) (hE : MeasurableSet E) (a c : ℝ) :
Tendsto (fun stage => ∫ s,
samplingTestedFirstDensity false (sourceSamplingWidth stage) (sourceSamplingCutoff stage)
E (Set.Icc a c) s ∂unitIntervalMeasure)
atTop (𝓝 ((unitIntervalMeasure (E ∩ Set.Icc a c)).toReal)) := by
have : NullSingletonClass unitIntervalMeasure := by unfold unitIntervalMeasure; infer_instance
let f : ℝ → ℝ := (E ∩ Set.Icc a c).indicator (fun _ => 1)
have hlim : ∀ᵐ s ∂unitIntervalMeasure,
Tendsto (fun stage => samplingTestedFirstDensity false
(sourceSamplingWidth stage) (sourceSamplingCutoff stage) E (Set.Icc a c) s)
atTop (𝓝 (f s)) := by
filter_upwards [ae_restrict_mem measurableSet_Ico,unitIntervalMeasure.ae_ne 0,
unitIntervalMeasure.ae_ne a,unitIntervalMeasure.ae_ne c] with s hs hs0 hsa hsc
have hh := sourceSampling_diagonal_eventually s a c E ⟨lt_of_le_of_ne hs.1 (Ne.symm hs0),hs.2⟩ hsa hsc
have he : (fun stage => samplingTestedFirstDensity false (sourceSamplingWidth stage)
(sourceSamplingCutoff stage) E (Set.Icc a c) s) =ᶠ[atTop] (fun _ => f s) := by
filter_upwards [hh] with stage hstage
simpa only [f,Set.indicator_apply,Set.mem_inter_iff] using hstage
exact tendsto_const_nhds.congr' he.symm
have hm (stage : ℕ) : AEStronglyMeasurable
(samplingTestedFirstDensity false (sourceSamplingWidth stage) (sourceSamplingCutoff stage)
E (Set.Icc a c)) unitIntervalMeasure := (samplingTestedFirstDensity_measurable false
(sourceSamplingWidth stage) (sourceSamplingCutoff stage) E (Set.Icc a c) hE measurableSet_Icc).aestronglyMeasurable
have hb (stage : ℕ) : ∀ᵐ s ∂unitIntervalMeasure,
‖samplingTestedFirstDensity false (sourceSamplingWidth stage) (sourceSamplingCutoff stage)
E (Set.Icc a c) s‖ ≤ (1 : ℝ) := by
filter_upwards [ae_restrict_mem measurableSet_Ico] with s hs
have hp := sourceSampling_parameters stage
have h := samplingTestedFirstDensity_bounds false _ _ E (Set.Icc a c)
hp.1 hp.2.1 hp.2.2 hE measurableSet_Icc s hs
rw [Real.norm_eq_abs,abs_of_nonneg h.1]
exact h.2
have ht := tendsto_integral_of_dominated_convergence (fun _ : ℝ => (1 : ℝ)) hm (integrable_const 1) hb hlim
have hf : (∫ s, f s ∂unitIntervalMeasure) = (unitIntervalMeasure (E ∩ Set.Icc a c)).toReal := by
rw [integral_indicator (hE.inter measurableSet_Icc)]
simp [measureReal_def]
rwa [hf] at ht
/-- Triangular tested factors are independent of the sampling width and cutoff. -/
theorem samplingTestedFirstDensity_triangular_parameters (h α h' α' s : ℝ) (E G : Set ℝ) :
samplingTestedFirstDensity true h α E G s = samplingTestedFirstDensity true h' α' E G s := by
unfold samplingTestedFirstDensity samplingTestedFirstWeight samplingWeight
simp only [ite_true]
end
end FiniteFactor
end OAI
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