| 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 |
|
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| end OAI |
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|
