| import Mathlib |
| import OAI.Geometry.IntegralFillings.Currents.CountableTests |
|
|
| namespace OAI |
|
|
| 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 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 Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology Pointwise NNReal |
|
|
| namespace SharpIntegralFillings.Foundations |
| open MeasureTheory Set Filter |
| open scoped Topology NNReal |
|
|
| variable {X : Type*} [MetricSpace X] [CompactSpace X] |
| [MeasurableSpace X] [BorelSpace X] |
| abbrev UnitTestClass (d K : ℕ) := |
| {p : TestClass (X := X) d K |
|
|
| instance unitTestClass_nonempty (d K : ℕ) : Nonempty (UnitTestClass (X := X) d K) := by |
| refine ⟨⟨⟨(0,fun _ => 0),?_,?_,?_⟩,?_⟩⟩ |
| · exact (LipschitzWith.const 0).weaken (by positivity) |
| · simp |
| · intro i |
| exact (LipschitzWith.const 0).weaken (by positivity) |
| · intro i |
| exact (LipschitzWith.const 0).weaken (by positivity) |
|
|
| lemma TestClass.continuous_integral_abs (d K : ℕ) (μ : Measure X) [IsFiniteMeasure μ] : |
| Continuous (fun p : TestClass (X := X) d K => ∫ x, |p.val.1 x| ∂μ) := by |
| apply continuous_of_dominated (bound := fun _ => (K:ℝ)) |
| · intro p |
| exact p.val.1.continuous.abs.measurable.aestronglyMeasurable |
| · intro p |
| exact Eventually.of_forall fun x => by |
| simpa only [Real.norm_eq_abs,abs_abs] using (p.val.1.norm_coe_le_norm x).trans p.property.2.1 |
| · exact integrable_const _ |
| · exact Eventually.of_forall fun x => by fun_prop |
|
|
| theorem exists_countable_mass_tests (d : ℕ) : |
| ∃ (b : ℕ → ℕ → X → ℝ) (π : ℕ → ℕ → Fin d → X → ℝ), |
| (∀ K j, BoundedLip (b K j) ∧ ∀ i, LipschitzWith 1 (π K j i)) ∧ |
| ∀ T : Functional X d, IsMetricCurrent T → |
| ∀ μ : Measure X, IsFiniteMeasure μ → |
| (∀ K j, |T (b K j) (π K j)| ≤ ∫ x, |b K j x| ∂μ) → Controls T μ := by |
| choose p hp using fun K : ℕ => TopologicalSpace.exists_dense_seq (UnitTestClass (X := X) d K) |
| refine ⟨fun K j => (p K j).val.val.1,fun K j i => (p K j).val.val.2 i,?_,?_⟩ |
| · intro K j |
| exact ⟨(p K j).val.boundedLip,(p K j).property⟩ |
| · intro T hT μ hμ he b π hb hπ |
| let := hμ |
| obtain ⟨K,q,hbq,hπq⟩ := admissible_mem_testClass (show Admissible b π from ⟨hb,fun i => ⟨1,hπ i⟩⟩) |
| let q' : UnitTestClass (X := X) d K := ⟨q,by |
| intro i |
| have hi : (q.val.2 i : X → ℝ) = π i := congrFun hπq i |
| simpa only [hi] using hπ i⟩ |
| have hh : |TestClass.eval T q'.val| ≤ ∫ x, |q'.val.val.1 x| ∂μ := by |
| apply (hp K).induction_on (p := fun r : UnitTestClass (X := X) d K => |
| |TestClass.eval T r.val| ≤ ∫ x, |r.val.val.1 x| ∂μ) q' |
| · apply isClosed_le |
| · exact ((IsMetricCurrent.testClass_continuous hT K).comp continuous_subtype_val).abs |
| · exact (TestClass.continuous_integral_abs d K μ).comp continuous_subtype_val |
| · intro j |
| exact he K j |
| simpa only [q',TestClass.eval,hbq,hπq] using hh |
| |
| theorem ae_controls_of_testwise_bound {A : Type*} [MeasurableSpace A] (ν : Measure A) |
| {d : ℕ} {T : A → Functional X d} {μ : A → Measure X} |
| (hT : ∀ᵐ a ∂ν, IsMetricCurrent (T a)) |
| (hμ : ∀ᵐ a ∂ν, IsFiniteMeasure (μ a)) |
| (hb : ∀ b π, BoundedLip b → (∀ i, LipschitzWith 1 (π i)) → |
| ∀ᵐ a ∂ν, |T a b π| ≤ ∫ x, |b x| ∂μ a) : |
| ∀ᵐ a ∂ν, Controls (T a) (μ a) := by |
| obtain ⟨b,π,ht,hd⟩ := exists_countable_mass_tests (X := X) d |
| have hc : ∀ᵐ a ∂ν, ∀ K j, |T a (b K j) (π K j)| ≤ ∫ x, |b K j x| ∂μ a := |
| ae_all_iff.mpr fun K => ae_all_iff.mpr fun j => hb _ _ (ht K j).1 (ht K j).2 |
| filter_upwards [hT,hμ,hc] with a hta hma hca |
| exact hd _ hta _ hma hca |
| |
| end SharpIntegralFillings.Foundations |
| end |
| |
| end OAI |
| |
