| import Mathlib |
| import OAI.Geometry.IntegralFillings.Charts.DomainRestriction |
| import OAI.Geometry.IntegralFillings.Currents.Composition |
| import OAI.Geometry.IntegralFillings.Slicing.Profiles |
| import OAI.Geometry.IntegralFillings.Slicing.StripMass |
|
|
| namespace OAI |
|
|
| section |
| open Set MeasureTheory Measure Filter Module |
| open Set Filter MeasureTheory Measure ContinuousLinearMap |
| open scoped Topology Convolution NNReal |
| open Set Filter MeasureTheory Measure Metric |
| open scoped Topology ContDiff |
| open Set Filter Metric |
| open Filter Set |
| open Set Filter MeasureTheory TopologicalSpace |
| open scoped Topology ENNReal |
| open Set MeasureTheory |
| open scoped RealInnerProductSpace |
| open Matrix |
| open scoped RealInnerProductSpace MatrixOrder |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology Pointwise NNReal |
| open scoped Topology NNReal |
| open Set MeasureTheory Filter |
| open scoped ENNReal NNReal Topology |
|
|
| namespace SharpIntegralFillings.Slicing |
| open Set MeasureTheory Filter BorelCoefficients BorelRestriction MassMeasure SmoothCutoff |
| open scoped Topology NNReal ENNReal |
|
|
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [CompactSpace X] |
| lemma controls_from_bounded_measures {ι : Type*} {l : Filter ι} [NeBot l] |
| {k : ℕ} {Ts : ι → Functional X k} {T : Functional X k} |
| (μs : ι → Measure X) [∀ j, IsFiniteMeasure (μs j)] (M : ℝ≥0) |
| (hb : ∀ j, (μs j).real univ ≤ M) (hc : ∀ j, Controls (Ts j) (μs j)) |
| (hlim : ∀ b π, Admissible b π → Tendsto (fun j => Ts j b π) l (𝓝 (T b π))) : |
| ∃ μ : Measure X, IsFiniteMeasure μ ∧ Controls T μ := by |
| let νs : ι → FiniteMeasure X := fun j => ⟨μs j,inferInstance⟩ |
| have hb' : ∀ j, νs j ∈ {ν : FiniteMeasure X | ν.mass ≤ M} := by |
| intro j |
| apply NNReal.coe_le_coe.mp |
| exact hb j |
| obtain ⟨ν,_,hcluster⟩ := |
| (isCompact_setOfPred_finiteMeasure_le_of_compactSpace X M).exists_mapClusterPt_of_frequently |
| (l := l) (Eventually.of_forall hb').frequently |
| let l' := comap νs (𝓝 ν) ⊓ l |
| have hl' : NeBot l' := neBot_inf_comap_iff_map'.mpr hcluster |
| let := hl' |
| have hν : Tendsto νs l' (𝓝 ν) := tendsto_iff_comap.mpr inf_le_left |
| exact ⟨ν,inferInstance,controls_of_weak_limits hν hc |
| (fun b π h => (hlim b π h).mono_left inf_le_right)⟩ |
|
|
| omit [MeasurableSpace X] [BorelSpace X] [CompactSpace X] in |
| lemma profile_comp_boundedLip {u : X → ℝ} (hu : BoundedLip u) |
| (a : ℝ≥0) (t : ℝ) : BoundedLip (profile a t ∘ u) := by |
| obtain ⟨K,hK⟩ := hu.1 |
| obtain ⟨C,hC⟩ := exists_profile_bounds |
| refine ⟨⟨_,(hC a t).1.comp hK⟩,1,fun x => ?_⟩ |
| change |profile a t (u x)| ≤ 1 |
| rw [abs_of_nonneg (profile_bounds a t (u x)).1] |
| exact (profile_bounds a t (u x)).2 |
|
|
| omit [MeasurableSpace X] [BorelSpace X] [CompactSpace X] in |
| lemma dprofile_comp_boundedLip {u : X → ℝ} (hu : BoundedLip u) |
| (a : ℝ≥0) (t : ℝ) : BoundedLip (dprofile a t ∘ u) := by |
| obtain ⟨K,hK⟩ := hu.1 |
| obtain ⟨C,hC⟩ := exists_profile_bounds |
| refine ⟨⟨_,(hC a t).2.1.comp hK⟩,(C*a:ℝ≥0),fun x => ?_⟩ |
| exact (hC a t).2.2 (u x) |
|
|
| lemma profile_L1_tendsto (μ : Measure X) [IsFiniteMeasure μ] {u : X → ℝ} |
| (hu : BoundedLip u) (t : ℝ) : |
| Tendsto (fun n : ℕ => ∫ x, |profile ((n:ℝ≥0)+1) t (u x) - |
| ({x | t < u x}.indicator (fun _ => (1:ℝ)) x)| ∂μ) atTop (𝓝 0) := by |
| let E : Set X := {x | t < u x} |
| have hE : MeasurableSet E := measurableSet_lt measurable_const hu.continuous.measurable |
| have hI : Integrable (E.indicator (fun _ : X => (1:ℝ))) μ := |
| (integrable_const _).indicator hE |
| have hP (n : ℕ) : Integrable (fun x => profile ((n:ℝ≥0)+1) t (u x)) μ := |
| integrable_boundedLip μ (profile_comp_boundedLip hu _ t) |
| have hpt (x : X) : Tendsto (fun n : ℕ => profile ((n:ℝ≥0)+1) t (u x)) atTop |
| (𝓝 (E.indicator (fun _ => (1:ℝ)) x)) := by |
| simpa only [E,indicator_apply,mem_Ioi,mem_ofPred_eq] using profile_sequence_tendsto t (u x) |
| have hdom (n : ℕ) : ∀ᵐ x ∂μ, |
| ‖|profile ((n:ℝ≥0)+1) t (u x)-E.indicator (fun _ => (1:ℝ)) x|‖ ≤ (2:ℝ) := by |
| filter_upwards [] with x |
| have hp := profile_bounds ((n:ℝ≥0)+1) t (u x) |
| have hi : 0 ≤ E.indicator (fun _ => (1:ℝ)) x ∧ E.indicator (fun _ => (1:ℝ)) x ≤ 1 := by |
| by_cases hx : x ∈ E <;> simp [hx] |
| simpa only [Real.norm_eq_abs,abs_abs] using |
| (abs_le.mpr (show -2 ≤ profile ((n:ℝ≥0)+1) t (u x)-E.indicator (fun _ => (1:ℝ)) x ∧ |
| profile ((n:ℝ≥0)+1) t (u x)-E.indicator (fun _ => (1:ℝ)) x ≤ 2 from by |
| constructor <;> linarith [hp.1,hp.2,hi.1,hi.2])) |
| have h := tendsto_integral_of_dominated_convergence (μ := μ) |
| (F := fun n x => |profile ((n:ℝ≥0)+1) t (u x)-E.indicator (fun _ => (1:ℝ)) x|) |
| (f := fun _ => (0:ℝ)) (fun _ => (2:ℝ)) |
| (fun n => ((hP n).sub hI).abs.aestronglyMeasurable) (integrable_const _) hdom |
| (Eventually.of_forall fun x => by |
| simpa only [sub_self,abs_zero] using ((hpt x).sub (tendsto_const_nhds (x := E.indicator (fun _ => (1:ℝ)) x))).abs) |
| simpa only [integral_zero] using h |
|
|
| lemma profile_boundary_tendsto {k : ℕ} {T : Functional X (k+1)} |
| (hT : IsMetricCurrent T) {u : X → ℝ} (hu : BoundedLip u) (t : ℝ) |
| (b : X → ℝ) (π : Fin k → X → ℝ) (hab : Admissible b π) : |
| Tendsto (fun n : ℕ => boundarySucc (weightedCurrent (currentMassMeasure hT) hT |
| (profile ((n:ℝ≥0)+1) t ∘ u)) b π) atTop |
| (𝓝 (boundarySucc (restrictCurrent hT {x | t < u x}) b π)) := by |
| let μ := currentMassMeasure hT |
| have hE : MeasurableSet {x | t < u x} := measurableSet_lt measurable_const hu.continuous.measurable |
| have hcons : Admissible (fun _ : X => (1:ℝ)) (Matrix.vecCons b π) := |
| ⟨BoundedLip.const 1,fun i => Fin.cases hab.1.1 (fun j => hab.2 j) i⟩ |
| simp only [boundarySucc,ite_eq_left hab,weightedCurrent,ite_eq_left hcons,mul_one, |
| restrictCurrent,Function.comp_apply] |
| exact borelAction_tendsto μ hT (currentMassMeasure_controls hT) |
| ((integrable_const _).indicator hE) |
| (fun n => integrable_boundedLip μ (profile_comp_boundedLip hu _ t)) |
| (profile_L1_tendsto μ hu t) (Matrix.vecCons b π) hcons.2 |
|
|
| theorem ae_finite_boundary_restrict_cycle {k : ℕ} {T : Functional X (k+1)} |
| (hT : IsMetricCurrent T) (hrect : IntegerRectifiable T) (hz : IsCycle T) |
| {u : X → ℝ} (hu : BoundedLip u) : |
| ∀ᵐ t : ℝ, ∃ ν : Measure X, IsFiniteMeasure ν ∧ |
| Controls (boundarySucc (restrictCurrent hT {x | t < u x})) ν := by |
| let μ := currentMassMeasure hT |
| obtain ⟨K,hK⟩ := hu.1 |
| obtain ⟨C,hC⟩ := exists_profile_bounds |
| filter_upwards [ae_bounded_strip_subsequence μ hu.continuous.measurable] with t ht |
| obtain ⟨j,M,hj,hM⟩ := ht |
| let a : ℕ → ℝ≥0 := fun n => (j n:ℝ≥0)+1 |
| let E : ℕ → Set X := fun n => {x | t ≤ u x ∧ u x ≤ t+(a n:ℝ)⁻¹} |
| have hE (n) : MeasurableSet (E n) := |
| (measurableSet_le measurable_const hu.continuous.measurable).inter |
| (measurableSet_le hu.continuous.measurable measurable_const) |
| let ν : ℕ → Measure X := fun n => (K*(C*a n)) • μ.restrict (E n) |
| have hνfin : ∀ n, IsFiniteMeasure (ν n) := fun _ => inferInstance |
| let := hνfin |
| apply controls_from_bounded_measures (l := atTop) ν (K*C*M) |
| (Ts := fun n => boundarySucc (weightedCurrent μ hT (profile (a n) t ∘ u))) |
| · intro n |
| have hen : ν n univ ≤ (K*C*M:ℝ≥0) := by |
| calc |
| ν n univ = (K*C:ℝ≥0) * stripMass μ u (a n) t := by |
| simp only [ν,E,stripMass,Measure.smul_apply,Measure.restrict_apply_univ] |
| simp only [ENNReal.smul_def,smul_eq_mul,ENNReal.coe_mul] |
| ring |
| _ ≤ (K*C:ℝ≥0) * (M:ℝ≥0∞) := mul_le_mul_right (hM n) _ |
| _ = (K*C*M:ℝ≥0) := by simp only [ENNReal.coe_mul] |
| exact (ENNReal.toReal_mono ENNReal.coe_ne_top hen).trans_eq (by simp) |
| · intro n |
| exact weightedCurrent_comp_boundary_controls hT hrect hz μ (currentMassMeasure_controls hT) |
| hu hK (profile_hasDerivAt (a n) t) (profile_comp_boundedLip hu _ t) |
| (dprofile_comp_boundedLip hu _ t) (hE n) (fun x => (hC (a n) t).2.2 (u x)) |
| (fun x hx => dprofile_zero (by dsimp [a]; positivity) hx) |
| · intro b π hab |
| exact (profile_boundary_tendsto hT hu t b π hab).comp hj |
|
|
| theorem ae_normal_restrict_cycle [Nonempty X] {k : ℕ} {T : Functional X (k+1)} |
| (hT : IsMetricCurrent T) (hrect : IntegerRectifiable T) (hz : IsCycle T) |
| {u : X → ℝ} (hu : BoundedLip u) : |
| ∀ᵐ t : ℝ, IsMetricCurrent (restrictCurrent hT {x | t < u x}) ∧ |
| IntegerRectifiable (restrictCurrent hT {x | t < u x}) ∧ |
| IsMetricCurrent (boundarySucc (restrictCurrent hT {x | t < u x})) := by |
| filter_upwards [ae_finite_boundary_restrict_cycle hT hrect hz hu] with t ht |
| have hE : MeasurableSet {x | t < u x} := measurableSet_lt measurable_const hu.continuous.measurable |
| have hR := restrictCurrent_isMetricCurrent hT hE |
| have hrectR := integerRectifiable_restrict hT hrect hE |
| exact ⟨hR,hrectR,boundary_isMetricCurrent_of_finiteMass hR hrectR ht⟩ |
|
|
| end SharpIntegralFillings.Slicing |
| open Set MeasureTheory Filter |
| open scoped Topology NNReal |
|
|
| end |
|
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| end OAI |
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