| import Mathlib |
| import OAI.Geometry.IntegralFillings.Currents.ComponentMass |
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| namespace OAI |
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| 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 |
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| namespace SharpIntegralFillings |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
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|
| variable {X : Type*} [MeasurableSpace X] {μ : Measure X} |
| lemma tsum_lintegral_norm_ne_top {f : ℕ → X → ℝ} |
| (hf : ∀ i, Integrable (f i) μ) |
| (hS : Summable (fun i => ∫ x, ‖f i x‖ ∂μ)) : |
| ∑' i, ∫⁻ x, ‖f i x‖ₑ ∂μ ≠ ⊤ := by |
| have h i : ∫⁻ x, ‖f i x‖ₑ ∂μ = ‖∫ x, ‖f i x‖ ∂μ‖ₑ := by |
| dsimp [enorm] |
| rw [lintegral_coe_eq_integral _ (hf i).norm] |
| simp only [ENNReal.coe_nnreal_eq,coe_nnnorm,Real.norm_eq_abs] |
| rw [abs_of_nonneg (integral_nonneg (fun x => abs_nonneg (f i x)))] |
| rw [funext h] |
| exact ENNReal.tsum_coe_ne_top_iff_summable.2 (NNReal.summable_coe.1 hS.abs) |
| |
| lemma ae_summable_norm_of_integral {f : ℕ → X → ℝ} |
| (hf : ∀ i, Integrable (f i) μ) |
| (hS : Summable (fun i => ∫ x, ‖f i x‖ ∂μ)) : |
| ∀ᵐ x ∂μ, Summable (fun i => ‖f i x‖) := by |
| have ht := tsum_lintegral_norm_ne_top hf hS |
| rw [←lintegral_tsum (fun i => (hf i).aestronglyMeasurable.enorm)] at ht |
| filter_upwards [ae_lt_top' (AEMeasurable.tsum |
| (fun i => (hf i).aestronglyMeasurable.enorm)) ht] with x hx |
| exact tsum_enorm_ne_top_iff_summable_norm.mp hx.ne |
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| lemma integrable_series_norm {f : ℕ → X → ℝ} |
| (hf : ∀ i, Integrable (f i) μ) |
| (hS : Summable (fun i => ∫ x, ‖f i x‖ ∂μ)) : |
| Integrable (fun x => ∑' i, ‖f i x‖) μ := by |
| refine ⟨by fun_prop,?_⟩ |
| have hfin : ∫⁻ x, ∑' i, ‖f i x‖ₑ ∂μ < ⊤ := by |
| rw [lintegral_tsum (fun i => (hf i).aestronglyMeasurable.enorm)] |
| exact (tsum_lintegral_norm_ne_top hf hS).lt_top |
| change (∫⁻ x, ‖∑' i, ‖f i x‖‖ₑ ∂μ) < ⊤ |
| convert hfin using 1 |
| apply lintegral_congr_ae |
| filter_upwards [ae_summable_norm_of_integral hf hS] with x hx |
| simp_rw [←coe_nnnorm,←NNReal.coe_tsum,enorm_eq_nnnorm,NNReal.nnnorm_eq] |
| exact ENNReal.coe_tsum (NNReal.summable_coe.mp hx) |
| |
| lemma integrable_series {f : ℕ → X → ℝ} |
| (hf : ∀ i, Integrable (f i) μ) |
| (hS : Summable (fun i => ∫ x, ‖f i x‖ ∂μ)) : |
| Integrable (fun x => ∑' i, f i x) μ := by |
| apply (integrable_series_norm hf hS).mono' (by fun_prop) |
| filter_upwards [ae_summable_norm_of_integral hf hS] with x hx |
| exact norm_tsum_le_tsum_norm hx |
| |
| end SharpIntegralFillings |
| |
| namespace SharpIntegralFillings |
| open Set MeasureTheory Filter |
| open scoped Topology ENNReal NNReal |
| |
| variable {X : Type*} [MetricSpace X] [CompactSpace X] |
| [MeasurableSpace X] [BorelSpace X] {k : ℕ} |
| |
| lemma summable_actions_all {T : ℕ → Functional X k} |
| (hT : ∀ i, IsMetricCurrent (T i)) (hS : Summable (fun i => mass (T i))) |
| (b : X → ℝ) (π : Fin k → X → ℝ) : Summable (fun i => T i b π) := by |
| by_cases hab : Admissible b π |
| · exact summable_actions_of_summable_masses hT hS hab |
| · simp only [fun i => (hT i).offDomain b π hab,summable_zero] |
| |
| theorem isMetricCurrent_tsum {T : ℕ → Functional X k} |
| (hT : ∀ i, IsMetricCurrent (T i)) (hS : Summable (fun i => mass (T i))) : |
| IsMetricCurrent (fun b π => ∑' i, T i b π) := by |
| have hs := summable_actions_all hT 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).mass_bound_uniform 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] |
| · choose μ hfin hμ heq using fun i => (hT i).exists_controls_mass_eq |
| let : ∀ i, IsFiniteMeasure (μ i) := hfin |
| have hm : Summable (fun i => (μ i).real univ) := by simpa only [←heq] using hS |
| let := finite_sum_measure_of_summable_total μ hm |
| exact ⟨Measure.sum μ,inferInstance,controls_sum hμ (fun _ _ => rfl)⟩ |
|
|
| lemma mass_tsum_le {T : ℕ → Functional X k} |
| (hT : ∀ i, IsMetricCurrent (T i)) (hS : Summable (fun i => mass (T i))) : |
| mass (fun b π => ∑' i, T i b π) ≤ ∑' i, mass (T i) := by |
| choose μ hfin hμ heq using fun i => (hT i).exists_controls_mass_eq |
| let : ∀ i, IsFiniteMeasure (μ i) := hfin |
| have hm : Summable (fun i => (μ i).real univ) := by simpa only [←heq] using hS |
| let := finite_sum_measure_of_summable_total μ hm |
| apply (mass_le_measure (inferInstance : IsFiniteMeasure (Measure.sum μ)) (controls_sum hμ (fun _ _ => rfl))).trans_eq |
| rw [measureReal_def,Measure.sum_apply _ MeasurableSet.univ, ENNReal.tsum_toReal_eq] |
| · exact tsum_congr (fun i => (heq i).symm) |
| · intro i |
| exact measure_ne_top _ _ |
|
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| end SharpIntegralFillings |
| end |
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| end OAI |
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