| import Mathlib | |
| import OAI.Analysis.BiholderTransport.Geodesics.ChartLipschitz | |
| import OAI.Analysis.BiholderTransport.Volume.EuclideanVolumeFactorNe | |
| import OAI.Analysis.BiholderTransport.Coordinates.ChartAlexandrov | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Filter Metric Manifold Bundle MeasureTheory | |
| open scoped Topology ContDiff ENNReal | |
| namespace WeakMTWTransport | |
| section OriginalNullSet | |
| variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M] [Nonempty M] | |
| [MeasurableSpace M] [BorelSpace M] | |
| [ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M] | |
| [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n) | |
| (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsRiemannianManifold 𝓘(ℝ,Model n) M] | |
| omit [Nonempty M] in | |
| lemma original_ae_chart_null {P:M → Prop} (hP:∀ᵐ x ∂metricVolume n,P x) (a:M) : | |
| ∃r>0,∃N:Set (Model n),volume N=0 ∧ | |
| ball (extChartAt 𝓘(ℝ,Model n) a a) r ⊆ (extChartAt 𝓘(ℝ,Model n) a).target ∧ | |
| ∀z∈ball (extChartAt 𝓘(ℝ,Model n) a a) r,z∉N → | |
| P ((extChartAt 𝓘(ℝ,Model n) a).symm z) := by | |
| let χ:=extChartAt 𝓘(ℝ,Model n) a | |
| obtain ⟨C,R,hR,hS,hC⟩:=exists_lipschitzOn_chart (n:=n) a | |
| have ht:χ a∈χ.target:=mem_extChartAt_target (I:=𝓘(ℝ,Model n)) a | |
| have hi:ContinuousAt χ.symm (χ a):=continuousAt_extChartAt_symm'' ht | |
| have he:χ.symm (χ a)=a:=χ.left_inv (mem_extChartAt_source a) | |
| obtain ⟨r,hr,hsub⟩:=Metric.mem_nhds_iff.mp (inter_mem | |
| ((isOpen_extChartAt_target a).mem_nhds ht) | |
| (hi.preimage_mem_nhds (he.symm ▸ ball_mem_nhds a hR))) | |
| let B:Set M:=ball a R ∩ {x | ¬P x} | |
| let N:Set (Model n):=χ '' B | |
| have hB:metricVolume n B=0:=measure_mono_null inter_subset_right (ae_iff.mp hP) | |
| have hN:volume N=0:=by | |
| apply (Measure.absolutelyContinuous_isAddHaarMeasure volume (metricVolume (M:=Model n) n)) | |
| apply le_antisymm _ zero_le | |
| have H:=lipschitzOn_metricVolume_image_le (hC.mono (show B⊆ball a R from inter_subset_left)) n | |
| rw [hB,mul_zero] at H | |
| exact H | |
| refine ⟨r,hr,N,hN,fun z hz=>(hsub hz).1,?_⟩ | |
| intro z hz hn | |
| by_contra hp | |
| apply hn | |
| exact ⟨χ.symm z,⟨by | |
| have hb:= (hsub hz).2 | |
| change dist (χ.symm z) (χ.symm (χ a)) < R at hb | |
| rw [he] at hb | |
| exact hb,hp⟩,χ.right_inv (hsub hz).1⟩ | |
| omit [Nonempty M] in | |
| lemma original_chart_good_null {P:M → Prop} (hP:∀ᵐ x ∂metricVolume n,P x) | |
| {u v:M → ℝ} (hu:Continuous u) (hdual:IsCostDualPair u v) (a:M) : | |
| ∃r>0,∃N:Set (Model n),volume N=0 ∧ | |
| ball (extChartAt 𝓘(ℝ,Model n) a a) r ⊆ (extChartAt 𝓘(ℝ,Model n) a).target ∧ | |
| ∀z∈ball (extChartAt 𝓘(ℝ,Model n) a a) r,z∉N → | |
| P ((extChartAt 𝓘(ℝ,Model n) a).symm z) ∧ | |
| ∃p:Model n,∃A:Model n →L[ℝ] Model n, | |
| (∀d e,inner ℝ (A d) e=inner ℝ d (A e)) ∧ | |
| HasQuadraticExpansion (fun h=>v ((extChartAt 𝓘(ℝ,Model n) a).symm (z+h))) p A := by | |
| obtain ⟨r,hr,N,hN,ht,hg⟩:=original_ae_chart_null (n:=n) hP a | |
| obtain ⟨s,hs,Q,hQ,hst,hq⟩:=original_chart_alexandrov_null (n:=n) hu hdual a | |
| refine ⟨min r s,lt_min hr hs,N∪Q,measure_union_null hN hQ,?_,?_⟩ | |
| · exact (ball_subset_ball (min_le_left _ _)).trans ht | |
| · intro z hz hn | |
| have hNr:z∉N:=fun hm=>hn (Or.inl hm) | |
| have hQr:z∉Q:=fun hm=>hn (Or.inr hm) | |
| exact ⟨hg z (ball_subset_ball (min_le_left _ _) hz) hNr, | |
| hq z (ball_subset_ball (min_le_right _ _) hz) hQr⟩ | |
| end OriginalNullSet | |
| end WeakMTWTransport | |
| end | |
| end | |
| end OAI | |