| import Mathlib |
| import OAI.Geometry.IntegralFillings.Currents.BorelAction |
|
|
| 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 Set MeasureTheory Filter |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
| open Filter Set |
| open scoped Topology NNReal |
| open Set Filter MeasureTheory TopologicalSpace |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology Pointwise NNReal |
| open Set MeasureTheory |
| open scoped RealInnerProductSpace |
| open Matrix |
| open scoped RealInnerProductSpace MatrixOrder |
|
|
| namespace SharpIntegralFillings |
| attribute [local instance] Classical.propDecidable |
| universe u |
| namespace BorelCoefficients |
| variable {X : Type u} [MetricSpace X] [CompactSpace X] |
| variable [MeasurableSpace X] [BorelSpace X] (μ : Measure X) [IsFiniteMeasure μ] |
| variable {k : ℕ} {T : Functional X k} |
| omit [MetricSpace X] [CompactSpace X] [BorelSpace X] [IsFiniteMeasure μ] in |
|
|
| lemma toL1_mul_dist_le {f g b : X → ℝ} (hf : Integrable f μ) (hg : Integrable g μ) |
| (hb : AEStronglyMeasurable b μ) {M : ℝ} (hM : ∀ᵐ x ∂μ, ‖b x‖ ≤ M) : |
| ‖(hf.mul_bdd hb hM).toL1 (fun x => f x * b x) - |
| (hg.mul_bdd hb hM).toL1 (fun x => g x * b x)‖ ≤ |
| M * ‖hf.toL1 f - hg.toL1 g‖ := by |
| apply Lp.norm_le_mul_norm_of_ae_le_mul |
| filter_upwards [Lp.coeFn_sub ((hf.mul_bdd hb hM).toL1 (fun x => f x * b x)) |
| ((hg.mul_bdd hb hM).toL1 (fun x => g x * b x)), |
| (hf.mul_bdd hb hM).coeFn_toL1, (hg.mul_bdd hb hM).coeFn_toL1, |
| Lp.coeFn_sub (hf.toL1 f) (hg.toL1 g), hf.coeFn_toL1, hg.coeFn_toL1, hM] |
| with x hsub hfb hgb hsub' hfx hgx hMx |
| simp only [Pi.sub_apply] at hsub hsub' |
| rw [hsub, hfb, hgb, hsub', hfx, hgx, ← sub_mul, norm_mul] |
| nlinarith [norm_nonneg (f x - g x)] |
| |
| omit [MetricSpace X] [CompactSpace X] [BorelSpace X] [IsFiniteMeasure μ] in |
| lemma tendsto_toL1_mul {ι : Type*} {l : Filter ι} {fs : ι → X → ℝ} {f b : X → ℝ} |
| (hfs : ∀ j, Integrable (fs j) μ) (hf : Integrable f μ) |
| (hb : AEStronglyMeasurable b μ) {M : ℝ} (hM : ∀ᵐ x ∂μ, ‖b x‖ ≤ M) |
| (hlim : Tendsto (fun j => (hfs j).toL1 (fs j)) l (𝓝 (hf.toL1 f))) : |
| Tendsto (fun j => ((hfs j).mul_bdd hb hM).toL1 (fun x => fs j x * b x)) l |
| (𝓝 ((hf.mul_bdd hb hM).toL1 (fun x => f x * b x))) := by |
| apply tendsto_iff_norm_sub_tendsto_zero.mpr |
| apply squeeze_zero (fun j => norm_nonneg _) |
| (fun j => toL1_mul_dist_le μ (hfs j) hf hb hM) |
| simpa only [mul_zero] using |
| (tendsto_iff_norm_sub_tendsto_zero.mp hlim).const_mul M |
| |
| lemma borelAction_locality_mul (hT : IsMetricCurrent T) (hμ : Controls T μ) |
| {f b : X → ℝ} (hf : Integrable f μ) (hb : BoundedLip b) |
| (π : Fin k → X → ℝ) (hπ : ∀ i, ∃ K : ℝ≥0, LipschitzWith K (π i)) |
| (hloc : ∃ (i : Fin k) (U : Set X) (c : ℝ), IsOpen U ∧ |
| Function.support b ⊆ U ∧ ∀ x ∈ U, π i x = c) : |
| borelAction μ hT (fun x => f x * b x) π = 0 := by |
| obtain ⟨M, hM⟩ := hb.2 |
| have hbM : ∀ᵐ x ∂μ, ‖b x‖ ≤ M := Eventually.of_forall fun x => by |
| simpa only [Real.norm_eq_abs] using hM x |
| have hbmeas := hb.continuous.aestronglyMeasurable (μ := μ) |
| obtain ⟨as, has, halim⟩ := mem_closure_iff_seq_limit.mp (lipToL1_dense μ (hf.toL1 f)) |
| choose fs hfs using has |
| have hnorm : Tendsto (fun j => lipToL1 μ (fs j)) atTop (𝓝 (hf.toL1 f)) := by |
| simpa only [hfs] using halim |
| have hLp (j) : (integrable_boundedLip μ (fs j).property).toL1 (fs j).val = |
| lipToL1 μ (fs j) := by |
| apply Lp.ext |
| exact (Integrable.coeFn_toL1 _).trans |
| (ContinuousMap.coeFn_toLp (p := 1) (𝕜 := ℝ) μ (fs j).val).symm |
| have hnorm' : Tendsto (fun j => (integrable_boundedLip μ (fs j).property).toL1 (fs j).val) |
| atTop (𝓝 (hf.toL1 f)) := by simpa only [hLp] using hnorm |
| have hprod := tendsto_toL1_mul μ (fun j => integrable_boundedLip μ (fs j).property) |
| hf hbmeas hbM hnorm' |
| have hact := (l1Action μ hT π hπ).continuous.continuousAt.tendsto.comp hprod |
| have hz (j) : l1Action μ hT π hπ |
| (((integrable_boundedLip μ (fs j).property).mul_bdd hbmeas hbM).toL1 |
| (fun x => (fs j).val x * b x)) = 0 := by |
| rw [← borelAction_of_integrable μ hT _ π hπ, |
| borelAction_eq μ hT hμ ⟨(fs j).property.mul hb, hπ⟩] |
| apply hT.locality _ _ ⟨(fs j).property.mul hb, hπ⟩ |
| obtain ⟨i, U, c, hU, hbU, hc⟩ := hloc |
| refine ⟨i, U, c, hU, ?_, hc⟩ |
| intro x hx |
| apply hbU |
| change b x ≠ 0 |
| intro hbx |
| exact hx (by simp only [hbx, mul_zero]) |
| have heq : l1Action μ hT π hπ ((hf.mul_bdd hbmeas hbM).toL1 (fun x => f x * b x)) = 0 := |
| tendsto_nhds_unique hact (by simpa only [Function.comp_def, hz] using |
| (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0 : ℝ)) atTop (𝓝 0))) |
| rwa [borelAction_of_integrable μ hT (hf.mul_bdd hbmeas hbM) π hπ] |
| |
| lemma borelAction_linearCoord (hT : IsMetricCurrent T) (hμ : Controls T μ) |
| {b : X → ℝ} (hb : Integrable b μ) (π : Fin k → X → ℝ) |
| (hπ : ∀ i, ∃ K : ℝ≥0, LipschitzWith K (π i)) (i : Fin k) |
| (g : X → ℝ) (hg : ∃ K : ℝ≥0, LipschitzWith K g) (a c : ℝ) : |
| borelAction μ hT b (Function.update π i (fun x => a * π i x + c * g x)) = |
| a * borelAction μ hT b π + c * borelAction μ hT b (Function.update π i g) := by |
| have hπg : ∀ j, ∃ K : ℝ≥0, LipschitzWith K ((Function.update π i g) j) := by |
| intro j |
| by_cases hji : j = i |
| · simpa only [hji, Function.update_self] using hg |
| · simpa only [Function.update_of_ne hji] using hπ j |
| have hac : ∃ K : ℝ≥0, LipschitzWith K (fun x => a * π i x + c * g x) := by |
| obtain ⟨K, hK⟩ := hπ i |
| obtain ⟨L, hL⟩ := hg |
| exact ⟨_, (lipschitz_mul_real hK a).add (lipschitz_mul_real hL c)⟩ |
| have hπac : ∀ j, ∃ K : ℝ≥0, |
| LipschitzWith K ((Function.update π i (fun x => a * π i x + c * g x)) j) := by |
| intro j |
| by_cases hji : j = i |
| · simpa only [hji, Function.update_self] using hac |
| · simpa only [Function.update_of_ne hji] using hπ j |
| have heq : (fun f : Lp ℝ 1 μ => |
| l1Action μ hT (Function.update π i (fun x => a * π i x + c * g x)) hπac f) = |
| (fun f => a * l1Action μ hT π hπ f + |
| c * l1Action μ hT (Function.update π i g) hπg f) := by |
| apply (lipToL1_dense μ).equalizer |
| · exact (l1Action μ hT _ _).continuous |
| · exact ((l1Action μ hT π hπ).continuous.const_mul a).add |
| ((l1Action μ hT _ _).continuous.const_mul c) |
| · funext f |
| simp only [Function.comp_apply, l1Action_eq μ hT hμ] |
| exact hT.linearCoord _ _ i g a c ⟨f.property, hπ⟩ hg |
| simpa only [borelAction_of_integrable μ hT hb _ hπac, |
| borelAction_of_integrable μ hT hb _ hπ, |
| borelAction_of_integrable μ hT hb _ hπg] using congrFun heq (hb.toL1 b) |
| |
| noncomputable def weightedCurrent (hT : IsMetricCurrent T) (b : X → ℝ) : Functional X k := |
| fun f π => if Admissible f π then borelAction μ hT (fun x => b x * f x) π else 0 |
| |
| lemma weightedCurrent_isMetricCurrent (hT : IsMetricCurrent T) (hμ : Controls T μ) |
| {b : X → ℝ} (hb : Integrable b μ) {M : ℝ≥0} (hM : ∀ᵐ x ∂μ, |b x| ≤ M) : |
| IsMetricCurrent (weightedCurrent μ hT b) := by |
| have hprod {f : X → ℝ} (hf : BoundedLip f) : Integrable (fun x => b x * f x) μ := by |
| obtain ⟨N, hN⟩ := hf.2 |
| exact hb.mul_bdd hf.continuous.aestronglyMeasurable |
| (Eventually.of_forall fun x => by simpa only [Real.norm_eq_abs] using hN x) |
| have hap {f : X → ℝ} {π : Fin k → X → ℝ} (h : Admissible f π) : |
| weightedCurrent μ hT b f π = borelAction μ hT (fun x => b x * f x) π := ite_eq_left h |
| refine ⟨?_, ?_, ?_, ?_, ?_, ?_⟩ |
| · intro f π h |
| exact ite_eq_right h |
| · intro f g π a c hf hg hπ |
| rw [hap ⟨(hf.const_mul a).add (hg.const_mul c), hπ⟩, |
| hap ⟨hf, hπ⟩, hap ⟨hg, hπ⟩] |
| have heq : (fun x => b x * (a * f x + c * g x)) = |
| a • (fun x => b x * f x) + c • (fun x => b x * g x) := by funext x; simp; ring |
| rw [heq, borelAction_add μ hT ((hprod hf).smul a) ((hprod hg).smul c) π hπ, |
| borelAction_smul μ hT (hprod hf) π a hπ, |
| borelAction_smul μ hT (hprod hg) π c hπ] |
| · intro f π i g a c h hg |
| have hπg : ∀ j, ∃ K : ℝ≥0, LipschitzWith K ((Function.update π i g) j) := by |
| intro j |
| by_cases hji : j = i |
| · simpa only [hji, Function.update_self] using hg |
| · simpa only [Function.update_of_ne hji] using h.2 j |
| have hac : ∃ K : ℝ≥0, LipschitzWith K (fun x => a * π i x + c * g x) := by |
| obtain ⟨K, hK⟩ := h.2 i |
| obtain ⟨L, hL⟩ := hg |
| exact ⟨_, (lipschitz_mul_real hK a).add (lipschitz_mul_real hL c)⟩ |
| have hπac : ∀ j, ∃ K : ℝ≥0, |
| LipschitzWith K ((Function.update π i (fun x => a * π i x + c * g x)) j) := by |
| intro j |
| by_cases hji : j = i |
| · simpa only [hji, Function.update_self] using hac |
| · simpa only [Function.update_of_ne hji] using h.2 j |
| rw [hap ⟨h.1, hπac⟩, hap h, hap ⟨h.1, hπg⟩] |
| exact borelAction_linearCoord μ hT hμ (hprod h.1) π h.2 i g hg a c |
| · intro f π πs hf hLip hpt |
| choose K hK using hLip |
| have hπ (i) : LipschitzWith (K i) (π i) := by |
| apply lipschitzWith_iff_dist_le_mul.mpr |
| intro x y |
| exact le_of_tendsto ((hpt i x).dist (hpt i y)) |
| (Eventually.of_forall fun j => (hK i j).dist_le_mul x y) |
| simp only [hap ⟨hf, fun i => ⟨K i, hK i _⟩⟩, hap ⟨hf, fun i => ⟨K i, hπ i⟩⟩] |
| exact borelAction_sequentialContinuity μ hT hμ (hprod hf) π πs |
| (fun i => ⟨K i, hK i⟩) hpt |
| · intro f π h hloc |
| rw [hap h] |
| exact borelAction_locality_mul μ hT hμ hb h.1 π h.2 hloc |
| · have hfin : IsFiniteMeasure ((M : ℝ≥0∞) • μ) := by |
| constructor |
| simp only [Measure.smul_apply, smul_eq_mul] |
| exact ENNReal.mul_lt_top ENNReal.coe_lt_top (measure_lt_top μ univ) |
| refine ⟨(M : ℝ≥0∞) • μ, hfin, ?_⟩ |
| intro f π hf hπ |
| rw [hap ⟨hf, fun i => ⟨1, hπ i⟩⟩, |
| integral_smul_measure, ENNReal.coe_toReal, smul_eq_mul] |
| have hbound := borelAction_bound μ hT hμ (hprod hf) π (fun _ => 1) hπ |
| simp only [NNReal.coe_one, Finset.prod_const_one, one_mul] at hbound |
| apply hbound.trans |
| rw [← integral_const_mul] |
| apply integral_mono_ae (hprod hf).abs ((integrable_boundedLip μ hf).abs.const_mul M) |
| filter_upwards [hM] with x hx |
| rw [abs_mul] |
| exact mul_le_mul_of_nonneg_right hx (abs_nonneg _) |
| |
| end BorelCoefficients |
| end SharpIntegralFillings |
| |
| namespace SharpIntegralFillings |
| attribute [local instance] Classical.propDecidable |
| end SharpIntegralFillings |
| end |
| |
| end OAI |
| |
