| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Slicing.ProfileIntegral |
| import OAI.Geometry.CAT0Fillings.Mass.Duality |
|
|
| namespace OAI |
|
|
| section |
|
|
| open Set Filter MeasureTheory |
| open scoped Topology NNReal ENNReal BigOperators |
|
|
| namespace CAT0Fillings |
| namespace BorelCoefficients |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [CompactSpace X] |
|
|
| abbrev CoefficientList (N : ℕ) := |
| {b : Fin N → C(X,ℝ) // (∀ i, BoundedLip (b i)) ∧ ∀ x, ∑ i, |b i x| ≤ 1} |
| |
| instance coefficientList_nonempty (N : ℕ) : Nonempty (CoefficientList (X := X) N) := by |
| refine ⟨⟨fun _ => 0,?_,?_⟩⟩ |
| · intro i |
| exact ⟨⟨0,LipschitzWith.const 0⟩,0,by simp⟩ |
| · simp |
| |
| lemma coefficientList_eval_continuous {k N : ℕ} {T : Functional X k} (hT : IsMetricCurrent T) |
| (π : Fin N → Fin k → X → ℝ) (hπ : ∀ j i, LipschitzWith 1 (π j i)) : |
| Continuous (fun b : CoefficientList (X := X) N => ∑ i, T (b.val i) (π i)) := by |
| apply continuous_finsetSum |
| intro i hi |
| apply LipschitzWith.continuous (K := ⟨mass T,mass_nonneg _⟩) |
| apply LipschitzWith.of_dist_le_mul |
| intro b c |
| have hh := Foundations.IsMetricCurrent.first_difference_bound hT (b.property.1 i) (c.property.1 i) |
| (π i) 1 (hπ i) (nndist b c) (fun x => |
| (ContinuousMap.dist_apply_le_dist x).trans (dist_le_pi_dist b.val c.val i)) |
| change |T (b.val i) (π i)-T (c.val i) (π i)| ≤ mass T*dist b c |
| simpa only [Real.dist_eq,NNReal.coe_one,one_pow,one_mul,coe_nndist,mul_comm] using hh |
| |
| lemma finite_list_le_mass {k N : ℕ} {T : Functional X k} (hT : IsMetricCurrent T) |
| (b : Fin N → X → ℝ) (hb : ∀ i, BoundedLip (b i)) |
| (hbudget : ∀ x, ∑ i, |b i x| ≤ 1) |
| (π : Fin N → Fin k → X → ℝ) (hπ : ∀ j i, LipschitzWith 1 (π j i)) : |
| (∑ i, T (b i) (π i)) ≤ mass T := by |
| let μ := MassMeasure.currentMassMeasure hT |
| have hi i : Integrable (fun x => |b i x|) μ := (integrable_boundedLip μ (hb i)).abs |
| calc (∑ i, T (b i) (π i)) ≤ ∑ i, |T (b i) (π i)| := |
| Finset.sum_le_sum fun i _ => le_abs_self _ |
| _ ≤ ∑ i, ∫ x, |b i x| ∂μ := Finset.sum_le_sum fun i _ => |
| MassMeasure.currentMassMeasure_controls hT _ _ (hb i) (hπ i) |
| _ = ∫ x, ∑ i, |b i x| ∂μ := (integral_finsetSum _ (fun i _ => hi i)).symm |
| _ ≤ ∫ x, (1:ℝ) ∂μ := integral_mono (integrable_finsetSum _ fun i _ => hi i) |
| (integrable_const _) hbudget |
| _ = mass T := by simpa using MassMeasure.currentMassMeasure_total hT |
| |
| theorem exists_countable_mass_lists (k : ℕ) : |
| ∃ (π : ℕ → Fin k → X → ℝ) (b : (N : ℕ) → ℕ → Fin (N+1) → X → ℝ), |
| (∀ j i, LipschitzWith 1 (π j i)) ∧ |
| (∀ N j i, BoundedLip (b N j i)) ∧ |
| (∀ N j x, ∑ i, |b N j i x| ≤ 1) ∧ |
| ∀ {T : Functional X k}, IsMetricCurrent T → ∀ C : ℝ, |
| mass T ≤ C ↔ ∀ N j, (∑ i : Fin (N+1), T (b N j i) (π i)) ≤ C := by |
| classical |
| obtain ⟨π,hπ,hspec⟩ := exists_mass_spectrum (X := X) k |
| choose bs hbs using fun N : ℕ => TopologicalSpace.exists_dense_seq (CoefficientList (X := X) (N+1)) |
| refine ⟨π,fun N j i => (bs N j).val i,hπ,fun N j => (bs N j).property.1, |
| fun N j => (bs N j).property.2,?_⟩ |
| intro T hT C |
| constructor |
| · intro h N j |
| exact (finite_list_le_mass hT _ (bs N j).property.1 (bs N j).property.2 |
| (fun i => π i) (fun i => hπ i)).trans h |
| · intro h |
| obtain ⟨ρ,hρm,hρ1,hρ,hmass⟩ := hspec hT |
| apply mass_le_of_finite_tests hT π hπ ρ hρm hρ1 hρ hmass |
| intro N a ha hbudget |
| let q : CoefficientList (X := X) (N+1) := ⟨fun i => ⟨a i,(ha i).continuous⟩,ha,hbudget⟩ |
| change (∑ i, T (q.val i) (π i)) ≤ C |
| apply (hbs N).induction_on (p := fun r : CoefficientList (X := X) (N+1) => |
| (∑ i, T (r.val i) (π i)) ≤ C) q |
| · exact isClosed_le (coefficientList_eval_continuous hT _ (fun i => hπ i)) continuous_const |
| · exact h N |
| |
| theorem ae_mass_le_of_listwise {A : Type*} [MeasurableSpace A] (ν : Measure A) |
| {k : ℕ} {T : A → Functional X k} {C : A → ℝ} |
| (hT : ∀ᵐ a ∂ν, IsMetricCurrent (T a)) |
| (h : ∀ N (b : Fin (N+1) → X → ℝ) (π : Fin (N+1) → Fin k → X → ℝ), |
| (∀ i, BoundedLip (b i)) → (∀ x, ∑ i, |b i x| ≤ 1) → |
| (∀ j i, LipschitzWith 1 (π j i)) → |
| ∀ᵐ a ∂ν, (∑ i, T a (b i) (π i)) ≤ C a) : |
| ∀ᵐ a ∂ν, mass (T a) ≤ C a := by |
| obtain ⟨π,b,hπ,hb,hbudget,hmass⟩ := exists_countable_mass_lists (X := X) k |
| have hh : ∀ᵐ a ∂ν, ∀ N j, (∑ i : Fin (N+1), T a (b N j i) (π i)) ≤ C a := |
| ae_all_iff.mpr fun N => ae_all_iff.mpr fun j => |
| h N (b N j) (fun i => π i) (hb N j) (hbudget N j) (fun i => hπ i) |
| filter_upwards [hT,hh] with a hta ha |
| exact (hmass hta (C a)).mpr ha |
| |
| end BorelCoefficients |
| end CAT0Fillings |
| end |
| |
| section |
| |
| open Set Filter MeasureTheory Metric |
| open scoped Topology NNReal ENNReal |
| |
| namespace CAT0Fillings.SmoothCutoff |
| |
| lemma dprofile_integrable_measure (μ : Measure ℝ) [IsFiniteMeasure μ] (a : ℝ≥0) (t : ℝ) : |
| Integrable (dprofile a t) μ := by |
| obtain ⟨C,hC⟩ := exists_profile_bounds |
| exact Integrable.of_bound (hC a t).2.1.continuous.aestronglyMeasurable ((C*a:ℝ≥0):ℝ) |
| (Eventually.of_forall fun x => by simpa only [Real.norm_eq_abs] using (hC a t).2.2 x) |
| |
| theorem ae_tendsto_dprofile_measure (μ : Measure ℝ) [IsFiniteMeasure μ] : |
| ∀ᵐ t : ℝ, Tendsto (fun n : ℕ => ∫ x : ℝ, |
| dprofile ((n:ℝ≥0)+1) t x ∂μ) atTop (𝓝 ((μ.rnDeriv volume t).toReal)) := by |
| obtain ⟨C,hC⟩ := exists_profile_bounds |
| let σ := μ.singularPart volume |
| let f : ℝ → ℝ := fun x => (μ.rnDeriv volume x).toReal |
| have hf : Integrable f := μ.integrable_toReal_rnDeriv |
| let a : ℕ → ℝ≥0 := fun n => (n:ℝ≥0)+1 |
| let r : ℕ → ℝ := fun n => ((a n:ℝ))⁻¹ |
| have hap n : 0 < a n := by dsimp [a]; positivity |
| have harp n : 0 < (a n:ℝ) := hap n |
| have hrp n : 0 < r n := inv_pos.mpr (harp n) |
| have hrt : Tendsto r atTop (𝓝[>] 0) := by |
| apply tendsto_nhdsWithin_iff.mpr |
| refine ⟨?_,Eventually.of_forall fun n => hrp n⟩ |
| exact tendsto_inv_atTop_zero.comp |
| (show Tendsto (fun n : ℕ => (a n:ℝ)) atTop atTop from by |
| simpa only [a,NNReal.coe_add,NNReal.coe_natCast,NNReal.coe_one] using |
| tendsto_atTop_add_const_right atTop 1 (tendsto_natCast_atTop_atTop (R := ℝ))) |
| filter_upwards [ae_tendsto_dprofile_integral hf,Besicovitch.ae_tendsto_rnDeriv σ volume, |
| μ.rnDeriv_singularPart volume] with t ht hσ hz |
| have hz' : σ.rnDeriv volume t = 0 := hz |
| have hs : Tendsto (fun n => ∫ x : ℝ, dprofile (a n) t x ∂σ) atTop (𝓝 0) := by |
| have hratio := (ENNReal.continuousAt_toReal (by simpa only [hz'] using ENNReal.zero_ne_top)).tendsto.comp |
| (hσ.comp hrt) |
| have hbound n : ‖∫ x : ℝ, dprofile (a n) t x ∂σ‖ ≤ |
| (2*(C:ℝ)) * (σ (closedBall t (r n))/volume (closedBall t (r n))).toReal := by |
| have hnz (x : ℝ) (hx : x ∉ closedBall t (r n)) : dprofile (a n) t x = 0 := by |
| apply dprofile_zero (hap n) |
| intro hi |
| apply hx |
| rw [Real.closedBall_eq_Icc] |
| change t-r n ≤ x ∧ x ≤ t+r n |
| exact ⟨by linarith [hi.1,hrp n],hi.2⟩ |
| calc |
| _ ≤ ∫ x : ℝ, ‖dprofile (a n) t x‖ ∂σ := norm_integral_le_integral_norm _ |
| _ = ∫ x in closedBall t (r n), ‖dprofile (a n) t x‖ ∂σ := |
| (setIntegral_eq_integral_of_forall_compl_eq_zero (fun x hx => by rw [hnz x hx,norm_zero])).symm |
| _ ≤ ∫ _x in closedBall t (r n), (C:ℝ)*(a n:ℝ) ∂σ := by |
| apply integral_mono (dprofile_integrable_measure σ _ _).norm.integrableOn (integrable_const _) |
| intro x |
| simpa only [Real.norm_eq_abs,NNReal.coe_mul] using (hC (a n) t).2.2 x |
| _ = _ := by |
| rw [integral_const,measureReal_def,Measure.restrict_apply_univ,smul_eq_mul,ENNReal.toReal_div] |
| change σ.real (closedBall t (r n)) * ((C:ℝ)*(a n:ℝ)) = |
| (2*(C:ℝ))*(σ.real (closedBall t (r n))/volume.real (closedBall t (r n))) |
| rw [Real.volume_real_closedBall (hrp n).le] |
| dsimp only [r] |
| field_simp |
| apply squeeze_zero_norm hbound |
| simpa only [hz',ENNReal.toReal_zero,mul_zero,Function.comp_apply] using hratio.const_mul (2*(C:ℝ)) |
| have heq n : (∫ x : ℝ, dprofile (a n) t x ∂μ) = |
| (∫ x : ℝ, dprofile (a n) t x ∂σ) + |
| ∫ x : ℝ, dprofile (a n) t x*f x := by |
| calc |
| _ = (∫ x : ℝ, dprofile (a n) t x ∂σ) + |
| ∫ x : ℝ, dprofile (a n) t x ∂volume.withDensity (μ.rnDeriv volume) := by |
| rw [←integral_add_measure (dprofile_integrable_measure σ _ _) |
| (dprofile_integrable_measure (volume.withDensity (μ.rnDeriv volume)) _ _)] |
| rw [show σ + volume.withDensity (μ.rnDeriv volume) = μ from μ.singularPart_add_rnDeriv volume] |
| _ = _ := by |
| congr 1 |
| rw [integral_withDensity_eq_integral_toReal_smul (μ.measurable_rnDeriv volume) |
| (μ.rnDeriv_lt_top volume)] |
| apply integral_congr_ae |
| exact Eventually.of_forall fun x => mul_comm _ _ |
| change Tendsto (fun n => ∫ x : ℝ, dprofile (a n) t x ∂μ) atTop (𝓝 (f t)) |
| simp_rw [heq] |
| simpa only [zero_add] using hs.add ht |
| |
| end CAT0Fillings.SmoothCutoff |
| end |
| |
| end OAI |
| |
