| import OAI.Probability.ProjectionMoments.Asymptotics |
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| namespace OAI |
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| section |
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| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal RealInnerProductSpace |
| noncomputable section |
| namespace ProjectionMoments |
| open NoiselessRegression |
| variable {d : ℕ} |
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| def extendSphereWeight (f : UnitSphere d → ℝ≥0∞) : Vector d → ℝ≥0∞ := |
| Function.extend Subtype.val f (fun _ => 0) |
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| @[simp] lemma extendSphereWeight_val (f : UnitSphere d → ℝ≥0∞) (s : UnitSphere d) : |
| extendSphereWeight f s.val = f s := |
| Subtype.val_injective.extend_apply f (fun _ => 0) s |
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| lemma measurable_extendSphereWeight {f : UnitSphere d → ℝ≥0∞} (hf : Measurable f) : |
| Measurable (extendSphereWeight f) := |
| (MeasurableEmbedding.subtype_coe Metric.isClosed_sphere.measurableSet).measurable_extend hf measurable_const |
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| lemma extendSphereWeight_le_one {f : UnitSphere d → ℝ≥0∞} (hf : ∀ s, f s ≤ 1) : |
| ∀ x, extendSphereWeight f x ≤ 1 := by |
| intro x |
| by_cases hx : ∃ s : UnitSphere d, s.val = x |
| · obtain ⟨s,rfl⟩ := hx |
| simpa using hf s |
| · rw [extendSphereWeight,Function.extend_apply' _ _ _ hx] |
| exact zero_le_one |
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| lemma sphereWeight_measure (f : UnitSphere d → ℝ≥0∞) (hf : Measurable f) : |
| ambientSphereLaw.withDensity (extendSphereWeight f) = |
| ((uniformSphere d).withDensity f).map Subtype.val := by |
| ext S hS |
| rw [withDensity_apply _ hS,Measure.map_apply measurable_subtype_coe hS, |
| withDensity_apply _ (measurable_subtype_coe hS)] |
| change (∫⁻ x in S, extendSphereWeight f x ∂(uniformSphere d).map Subtype.val) = _ |
| rw [← lintegral_indicator hS,lintegral_map ((measurable_extendSphereWeight hf).indicator hS) |
| measurable_subtype_coe,← lintegral_indicator (measurable_subtype_coe hS)] |
| apply lintegral_congr |
| intro s |
| by_cases hs : s.val ∈ S <;> simp [hs,extendSphereWeight_val] |
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| lemma sphereWeight_potential (α : ℝ) (f : UnitSphere d → ℝ≥0∞) (hf : Measurable f) : |
| rieszPotential α (extendSphereWeight f) = |
| ⨆ z : Vector d, ∫⁻ s, f s * (ENNReal.ofReal ‖s.val-z‖)^(-α) ∂uniformSphere d := by |
| unfold rieszPotential |
| congr 1 |
| funext z |
| change (∫⁻ x, extendSphereWeight f x*rieszKernel α z x ∂(uniformSphere d).map Subtype.val) = _ |
| rw [lintegral_map (show Measurable (fun x => extendSphereWeight f x*rieszKernel α z x) from |
| (measurable_extendSphereWeight hf).mul (measurable_rieszKernel α z)) measurable_subtype_coe] |
| apply lintegral_congr |
| intro s |
| simp only [extendSphereWeight_val,rieszKernel] |
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| lemma sphereWeight_restriction (f : UnitSphere d → ℝ≥0∞) (hf : Measurable f) |
| (H : Set (UnitSphere d)) (hH : MeasurableSet H) : |
| sphereWeightedRestriction (extendSphereWeight f) (Subtype.val '' H) = |
| (((uniformSphere d).withDensity f).restrict H).map Subtype.val := by |
| have he := MeasurableEmbedding.subtype_coe (Metric.isClosed_sphere.measurableSet : |
| MeasurableSet (Metric.sphere (0 : Vector d) 1)) |
| rw [sphereWeightedRestriction,sphereWeight_measure f hf,Measure.restrict_map measurable_subtype_coe |
| (he.measurableSet_image.mpr hH)] |
| congr 1 |
| congr 1 |
| exact Set.preimage_image_eq _ Subtype.val_injective |
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| lemma sphere_weight_representation (f : UnitSphere d → ℝ≥0∞) (hf : Measurable f) |
| (hf1 : ∀ s, f s ≤ 1) : |
| ∃ F : Vector d → ℝ≥0∞, Measurable F ∧ (∀ x, F x ≤ 1) ∧ |
| (∀ s : UnitSphere d, F s.val = f s) ∧ |
| ambientSphereLaw.withDensity F = ((uniformSphere d).withDensity f).map Subtype.val := |
| ⟨extendSphereWeight f,measurable_extendSphereWeight hf,extendSphereWeight_le_one hf1, |
| extendSphereWeight_val f,sphereWeight_measure f hf⟩ |
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| end ProjectionMoments |
| end |
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| end |
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| end OAI |
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