| 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 |
| namespace BorelCoefficients |
| variable {X : Type*} [MetricSpace X] [CompactSpace X] |
| variable [MeasurableSpace X] [BorelSpace X] (μ : Measure X) [IsFiniteMeasure μ] |
| variable {k : ℕ} {T : Functional X k} |
| lemma borelAction_sub (hT : IsMetricCurrent T) {b c : X → ℝ} |
| (hb : Integrable b μ) (hc : Integrable c μ) (π : Fin k → X → ℝ) |
| (hπ : ∀ i, ∃ K : ℝ≥0, LipschitzWith K (π i)) : |
| borelAction μ hT (fun x => b x-c x) π = |
| borelAction μ hT b π-borelAction μ hT c π := by |
| have he : (fun x => b x-c x) = b+(-1 : ℝ) • c := by ext x |
| rw [he,borelAction_add μ hT hb (hc.smul (-1)) π hπ, |
| borelAction_smul μ hT hc π (-1) hπ] |
| ring |
|
|
| lemma borelAction_difference_bound (hT : IsMetricCurrent T) (hμ : Controls T μ) |
| {b c : X → ℝ} (hb : Integrable b μ) (hc : Integrable c μ) |
| (π : Fin k → X → ℝ) (K : Fin k → ℝ≥0) |
| (hK : ∀ i, LipschitzWith (K i) (π i)) : |
| |borelAction μ hT b π-borelAction μ hT c π| ≤ |
| (∏ i, (K i : ℝ)) * ∫ x, |b x-c x| ∂μ := by |
| rw [←borelAction_sub μ hT hb hc π (fun i => ⟨K i,hK i⟩)] |
| exact borelAction_bound μ hT hμ (hb.sub hc) π K hK |
|
|
| lemma borelAction_tendsto (hT : IsMetricCurrent T) (hμ : Controls T μ) |
| {b : X → ℝ} {f : ℕ → X → ℝ} (hb : Integrable b μ) |
| (hf : ∀ j, Integrable (f j) μ) |
| (hlim : Tendsto (fun j => ∫ x, |f j x-b x| ∂μ) atTop (𝓝 0)) |
| (π : Fin k → X → ℝ) (hπ : ∀ i, ∃ K : ℝ≥0, LipschitzWith K (π i)) : |
| Tendsto (fun j => borelAction μ hT (f j) π) atTop (𝓝 (borelAction μ hT b π)) := by |
| choose K hK using hπ |
| apply tendsto_iff_dist_tendsto_zero.mpr |
| apply squeeze_zero (fun _ => dist_nonneg) |
| (fun j => (borelAction_difference_bound μ hT hμ (hf j) hb π K hK)) |
| simpa only [mul_zero] using hlim.const_mul (∏ i, (K i : ℝ)) |
|
|
| lemma borelAction_independent (hT : IsMetricCurrent T) (hμ : Controls T μ) |
| {ν : Measure X} [IsFiniteMeasure ν] (hν : Controls T ν) |
| {b : X → ℝ} (hbμ : Integrable b μ) (hbν : Integrable b ν) |
| (π : Fin k → X → ℝ) (hπ : ∀ i, ∃ K : ℝ≥0, LipschitzWith K (π i)) : |
| borelAction μ hT b π = borelAction ν hT b π := by |
| obtain ⟨f,hf⟩ := exists_lipschitz_approximation (μ+ν) (hbμ.add_measure hbν) |
| have hi (j) := integrable_boundedLip (μ+ν) (f j).property |
| have hlμ : Tendsto (fun j => ∫ x, |(f j).val x-b x| ∂μ) atTop (𝓝 0) := by |
| apply squeeze_zero (fun _ => integral_nonneg fun _ => abs_nonneg _) |
| (fun j => integral_mono_measure (Measure.le_add_right le_rfl) |
| (Eventually.of_forall fun _ => abs_nonneg _) ((hi j).sub (hbμ.add_measure hbν)).abs) hf |
| have hlν : Tendsto (fun j => ∫ x, |(f j).val x-b x| ∂ν) atTop (𝓝 0) := by |
| apply squeeze_zero (fun _ => integral_nonneg fun _ => abs_nonneg _) |
| (fun j => integral_mono_measure (Measure.le_add_left le_rfl) |
| (Eventually.of_forall fun _ => abs_nonneg _) ((hi j).sub (hbμ.add_measure hbν)).abs) hf |
| have hμlim := borelAction_tendsto μ hT hμ hbμ |
| (fun j => integrable_boundedLip μ (f j).property) hlμ π hπ |
| have hνlim := borelAction_tendsto ν hT hν hbν |
| (fun j => integrable_boundedLip ν (f j).property) hlν π hπ |
| simp only [borelAction_eq μ hT hμ ⟨(f _).property,hπ⟩] at hμlim |
| simp only [borelAction_eq ν hT hν ⟨(f _).property,hπ⟩] at hνlim |
| exact tendsto_nhds_unique hμlim hνlim |
|
|
| end BorelCoefficients |
| end SharpIntegralFillings |
| |
| namespace SharpIntegralFillings |
| attribute [local instance] Classical.propDecidable |
| end SharpIntegralFillings |
| end |
|
|
| end OAI |
|
|
