| import OAI.Analysis.FiniteFactor.OrbitSampling.DiagonalLimit |
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| namespace OAI |
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| namespace FiniteFactor |
| open MeasureTheory Filter Finset |
| open scoped Classical ENNReal Topology |
| noncomputable section |
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| /-- 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 |
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| /-- 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] |
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| end |
| end FiniteFactor |
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| end OAI |
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