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8e4d533 | 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 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 | 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
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