| import OAI.Analysis.CircleDomains.Probability.SimpleArcOccupation |
|
|
| namespace OAI |
|
|
| /-! Disjoint compact portions of a simple arc have a total length bounded |
| by the Hausdorff length of the whole arc, even when the map is only |
| continuous on an open part of its parameter domain. -/ |
|
|
| open Set Function MeasureTheory |
| open scoped ENNReal |
|
|
| noncomputable section |
| namespace CircleDomainRigidity |
|
|
| theorem simple_arc_Ioc_image_measurable |
| [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| (f : PathTime → X) (hf : ContinuousOn f V) (hinj : InjOn f V) |
| (hab : a ≤ b) (hsub : Icc a b ⊆ V) : |
| MeasurableSet (f '' Ioc a b) := by |
| have hcompact : IsCompact (f '' Icc a b) := |
| isCompact_Icc.image_of_continuousOn (hf.mono hsub) |
| have he : f '' Ioc a b = f '' Icc a b \ := by |
| ext x |
| constructor |
| · rintro ⟨t, ht, rfl⟩ |
| refine ⟨⟨t, Ioc_subset_Icc_self ht, rfl⟩, ?_⟩ |
| intro heq |
| exact ht.1.ne' (hinj (hsub (Ioc_subset_Icc_self ht)) |
| (hsub (left_mem_Icc.mpr hab)) heq) |
| · rintro ⟨⟨t, ht, rfl⟩, hne⟩ |
| refine ⟨t, ⟨lt_of_le_of_ne ht.1 ?_, ht.2⟩, rfl⟩ |
| intro heq |
| exact hne (congrArg f heq.symm) |
| rw [he] |
| exact hcompact.isClosed.measurableSet.diff (measurableSet_singleton _) |
|
|
| theorem simple_arc_variation_le_Ioc_image |
| [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| (f : PathTime → X) (hf : ContinuousOn f V) (hinj : InjOn f V) |
| (_hab : a ≤ b) (hsub : Icc a b ⊆ V) : |
| eVariationOn f (Icc a b) ≤ (μH[1] : Measure X) (f '' Ioc a b) := by |
| let : NullSingletonClass (μH[1] : Measure X) := |
| Measure.nullSingletonClass_hausdorff X (by norm_num) |
| have hi : f '' Icc a b ⊆ ∪ (f '' Ioc a b) := by |
| rintro _ ⟨t, ht, rfl⟩ |
| rcases ht.1.eq_or_lt with h | h |
| · exact Or.inl (congrArg f h.symm) |
| · exact Or.inr ⟨t, ⟨h, ht.2⟩, rfl⟩ |
| calc |
| _ ≤ (μH[1] : Measure X) (f '' Icc a b) := |
| simple_arc_variation_le_length f ordConnected_Icc (hf.mono hsub) (hinj.mono hsub) |
| _ ≤ (μH[1] : Measure X) ( ∪ (f '' Ioc a b)) := measure_mono hi |
| _ ≤ (μH[1] : Measure X) + (μH[1] : Measure X) (f '' Ioc a b) := measure_union_le _ _ |
| _ = _ := by rw [measure_singleton, zero_add] |
|
|
| theorem sum_disjoint_arc_variations_le [Fintype ι] |
| [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| (f : PathTime → X) (hf : ContinuousOn f V) (hinj : InjOn f V) |
| (a b : ι → PathTime) (hab : ∀ i, a i ≤ b i) |
| (hsub : ∀ i, Icc (a i) (b i) ⊆ V) |
| (hdis : Pairwise (fun i j => Disjoint (Ioc (a i) (b i)) (Ioc (a j) (b j)))) : |
| (∑ i, eVariationOn f (Icc (a i) (b i))) ≤ (μH[1] : Measure X) (f '' V) := by |
| have himdis : Pairwise (fun i j => Disjoint |
| (f '' Ioc (a i) (b i)) (f '' Ioc (a j) (b j))) := by |
| intro i j hij |
| apply Set.disjoint_left.mpr |
| rintro _ ⟨t, ht, rfl⟩ ⟨s, hs, heq⟩ |
| have hts := hinj (hsub i (Ioc_subset_Icc_self ht)) (hsub j (Ioc_subset_Icc_self hs)) heq.symm |
| exact Set.disjoint_left.mp (hdis hij) ht (hts.symm ▸ hs) |
| calc |
| _ ≤ ∑ i, (μH[1] : Measure X) (f '' Ioc (a i) (b i)) := |
| Finset.sum_le_sum (fun i _ => simple_arc_variation_le_Ioc_image f hf hinj (hab i) (hsub i)) |
| _ = (μH[1] : Measure X) (⋃ i, f '' Ioc (a i) (b i)) := by |
| rw [measure_iUnion himdis (fun i => simple_arc_Ioc_image_measurable f hf hinj (hab i) (hsub i)), |
| tsum_fintype] |
| _ ≤ _ := measure_mono (iUnion_subset fun i => image_mono (Ioc_subset_Icc_self.trans (hsub i))) |
|
|
| end CircleDomainRigidity |
|
|
| end |
|
|
| end OAI |
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|
