| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Currents.ComponentMass |
| import OAI.Geometry.CAT0Fillings.Currents.SumControls |
|
|
| namespace OAI |
|
|
| section |
| section |
| 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 MeasureTheory Filter Set Metric |
| open scoped Topology Pointwise NNReal |
| 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 scoped Topology NNReal |
| open Set MeasureTheory Filter |
| open scoped Topology ENNReal NNReal |
|
|
| namespace CAT0Fillings |
| open Set MeasureTheory Filter |
| open scoped Topology ENNReal NNReal |
|
|
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] {k : ℕ} |
| lemma IsMetricCurrent.control_bound_uniform {T : Functional X k} (hT : IsMetricCurrent T) |
| {μ : Measure X} [IsFiniteMeasure μ] (hμ : Controls T μ) |
| {b : X → ℝ} (hb : BoundedLip b) {π : Fin k → X → ℝ} |
| (K : Fin k → ℝ≥0) (hK : ∀ i, LipschitzWith (K i) (π i)) |
| {M : ℝ} (hM : ∀ x, |b x| ≤ M) : |
| |T b π| ≤ ((∏ i, (K i : ℝ))*M)*μ.real univ := by |
| have hi : Integrable (fun x => |b x|) μ := (hb.toBoundedContinuous.integrable μ).abs |
| have hbound : (∫ x, |b x| ∂μ) ≤ μ.real univ * M := by |
| simpa using integral_mono hi (integrable_const M) hM |
| calc |
| _ ≤ (∏ i, (K i : ℝ)) * ∫ x, |b x| ∂μ := hT.mass_bound hμ hb K hK |
| _ ≤ (∏ i, (K i : ℝ)) * (μ.real univ*M) := |
| mul_le_mul_of_nonneg_left hbound (Finset.prod_nonneg fun i _ => (K i).coe_nonneg) |
| _ = _ := by ring |
|
|
| lemma summable_controlled_actions {T : ℕ → Functional X k} |
| (hT : ∀ i, IsMetricCurrent (T i)) {μ : ℕ → Measure X} |
| [∀ i, IsFiniteMeasure (μ i)] (hμ : ∀ i, Controls (T i) (μ i)) |
| (hS : Summable (fun i => (μ i).real univ)) |
| (b : X → ℝ) (π : Fin k → X → ℝ) : Summable (fun i => T i b π) := by |
| by_cases h : Admissible b π |
| · choose K hK using h.2 |
| obtain ⟨M,hM⟩ := h.1.2 |
| exact (hS.mul_left ((∏ j, (K j : ℝ))*M)).of_norm_bounded |
| (fun i => by simpa only [Real.norm_eq_abs] using |
| (hT i).control_bound_uniform (hμ i) h.1 K hK hM) |
| · simp only [fun i => (hT i).offDomain b π h,summable_zero] |
|
|
| lemma isMetricCurrent_tsum_controlled {T : ℕ → Functional X k} |
| (hT : ∀ i, IsMetricCurrent (T i)) {μ : ℕ → Measure X} |
| [∀ i, IsFiniteMeasure (μ i)] (hμ : ∀ i, Controls (T i) (μ i)) |
| (hS : Summable (fun i => (μ i).real univ)) : |
| IsMetricCurrent (fun b π => ∑' i, T i b π) := by |
| have hs := summable_controlled_actions hT hμ hS |
| refine ⟨?_,?_,?_,?_,?_,?_⟩ |
| · intro b π hab |
| simp only [fun i => (hT i).offDomain b π hab,tsum_zero] |
| · intro b c π a d hb hc hπ |
| simp only [fun i => (hT i).linearFirst b c π a d hb hc hπ] |
| rw [((hs b π).mul_left a).tsum_add ((hs c π).mul_left d)] |
| simp only [tsum_mul_left] |
| · intro b π j f a c hab hf |
| simp only [fun i => (hT i).linearCoord b π j f a c hab hf] |
| rw [((hs b π).mul_left a).tsum_add ((hs b (Function.update π j f)).mul_left c)] |
| simp only [tsum_mul_left] |
| · intro b π πs hb hπs hlim |
| choose K hK using hπs |
| obtain ⟨M,hM⟩ := hb.2 |
| apply tendsto_tsum_of_dominated_convergence |
| (hS.mul_left ((∏ j, (K j : ℝ))*M)) |
| (fun i => (hT i).sequentialContinuity b π πs hb (fun j => ⟨K j,hK j⟩) hlim) |
| exact Eventually.of_forall (fun j i => |
| (hT i).control_bound_uniform (hμ i) hb K (fun l => hK l j) hM) |
| · intro b π hab hloc |
| simp only [fun i => (hT i).locality b π hab hloc,tsum_zero] |
| · let := finite_sum_measure_of_summable_total μ hS |
| exact ⟨Measure.sum μ,inferInstance,controls_sum_general hμ (fun _ _ => rfl)⟩ |
| |
| end CAT0Fillings |
| |
| end |
| end |
| |
| end OAI |
| |
