| import Mathlib |
|
|
| namespace OAI |
|
|
| namespace BorsukNine |
| open scoped RealInnerProductSpace ContDiff |
| open scoped Manifold |
| open MeasureTheory Set Metric |
| open scoped ENNReal Pointwise Matrix |
| section SphereMeasure |
| variable (V : Type*) [NormedAddCommGroup V] [InnerProductSpace ℝ V] |
| [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] [Nontrivial V] |
|
|
| abbrev UnitSphere := Metric.sphere (0 : V) 1 |
|
|
| noncomputable def sphereProbability : Measure (UnitSphere V) := |
| ((volume : Measure V).toSphere Set.univ)⁻¹ • volume.toSphere |
| |
| instance sphereMeasure_neZero : NeZero ((volume : Measure V).toSphere) := by |
| constructor |
| exact (Measure.toSphere_eq_zero_iff_finrank volume).not.mpr |
| (ne_of_gt (Module.finrank_pos (R := ℝ) (M := V))) |
| |
| instance sphereProbability_isProbability : IsProbabilityMeasure (sphereProbability V) := by |
| unfold sphereProbability |
| infer_instance |
| |
| instance sphereProbability_openPos : (sphereProbability V).IsOpenPosMeasure := by |
| unfold sphereProbability |
| exact Measure.isOpenPosMeasure_smul volume.toSphere (ENNReal.inv_ne_zero.mpr (measure_ne_top _ _)) |
| |
| variable {V} |
| |
| noncomputable def sphereRotation (e : V ≃ₗᵢ[ℝ] V) : UnitSphere V ≃ₜ UnitSphere V := |
| e.toHomeomorph.subtype (fun x => by simp only [mem_sphere_zero_iff_norm, LinearIsometryEquiv.coe_toHomeomorph, e.norm_map]) |
| |
| omit [Nontrivial V] in |
| lemma sphereRotation_preserves_toSphere (e : V ≃ₗᵢ[ℝ] V) : |
| MeasurePreserving (sphereRotation e) volume.toSphere volume.toSphere := by |
| refine ⟨(sphereRotation e).measurable, ?_⟩ |
| apply Measure.ext |
| intro s hs |
| rw [Measure.map_apply (sphereRotation e).measurable hs, |
| Measure.toSphere_apply' _ ((sphereRotation e).measurable hs), Measure.toSphere_apply' _ hs] |
| congr 1 |
| have hset : Ioo (0 : ℝ) 1 • (Subtype.val '' (sphereRotation e ⁻¹' s)) = |
| e ⁻¹' (Ioo (0 : ℝ) 1 • (Subtype.val '' s)) := by |
| ext x |
| constructor |
| · rintro ⟨r, hr, v, ⟨u, hu, rfl⟩, rfl⟩ |
| exact ⟨r, hr, e u, ⟨sphereRotation e u, hu, rfl⟩, by simp⟩ |
| · rintro ⟨r, hr, v, ⟨u, hu, rfl⟩, he⟩ |
| refine ⟨r, hr, e.symm u, ⟨(sphereRotation e).symm u, ?_, rfl⟩, ?_⟩ |
| · simpa using hu |
| · apply e.injective |
| simpa using he |
| rw [hset] |
| have he := e.measurePreserving.map_eq |
| have hm := e.toMeasurableEquiv.measurableEmbedding.map_apply volume |
| (Ioo (0 : ℝ) 1 • (Subtype.val '' s)) |
| simp only [LinearIsometryEquiv.coe_toMeasurableEquiv] at hm |
| rw [he] at hm |
| exact hm.symm |
| |
| omit [Nontrivial V] in |
| lemma sphereRotation_preserves_probability (e : V ≃ₗᵢ[ℝ] V) : |
| MeasurePreserving (sphereRotation e) (sphereProbability V) (sphereProbability V) := by |
| exact (sphereRotation_preserves_toSphere e).smul_measure _ |
| |
| omit [Nontrivial V] in |
| lemma integral_sphere_rotation (e : V ≃ₗᵢ[ℝ] V) (g : UnitSphere V → ℝ) : |
| (∫ x, g (sphereRotation e x) ∂sphereProbability V) = |
| ∫ x, g x ∂sphereProbability V := |
| (sphereRotation_preserves_probability e).integral_comp |
| (sphereRotation e).measurableEmbedding g |
| |
| end SphereMeasure |
| |
| end BorsukNine |
| |
| end OAI |
| |
