| import Mathlib |
| import OAI.Geometry.IntegralFillings.Slicing.CoordinateCoarea |
|
|
| 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 |
| open Set MeasureTheory Filter |
| open scoped ENNReal NNReal Topology |
|
|
| lemma finite_sum_measure_of_summable_total {X : Type*} [MeasurableSpace X] |
| (μ : ℕ → Measure X) [∀ i, IsFiniteMeasure (μ i)] |
| (hs : Summable (fun i => (μ i).real univ)) : IsFiniteMeasure (Measure.sum μ) := by |
| constructor |
| have heq : (Measure.sum μ) univ = ENNReal.ofReal (∑' i, (μ i).real univ) := by |
| rw [Measure.sum_apply _ MeasurableSet.univ, |
| ENNReal.ofReal_tsum_of_nonneg (fun i => measureReal_nonneg) hs] |
| congr 1 |
| funext i |
| exact (ENNReal.ofReal_toReal (measure_ne_top (μ i) univ)).symm |
| rw [heq] |
| exact ENNReal.ofReal_lt_top |
|
|
| section SumControls |
| variable {X : Type*} [MetricSpace X] [CompactSpace X] [MeasurableSpace X] [BorelSpace X] |
| {k : ℕ} {T : Functional X k} {S : ℕ → Functional X k} |
| {μ : ℕ → Measure X} [IsFiniteMeasure (Measure.sum μ)] |
|
|
| lemma controls_sum (hμ : ∀ i, Controls (S i) (μ i)) |
| (hT : ∀ b π, T b π = ∑' i, S i b π) : Controls T (Measure.sum μ) := by |
| intro b π hb hπ |
| have hbi : Integrable (fun x => |b x|) (Measure.sum μ) := |
| (BorelCoefficients.integrable_boundedLip _ hb).abs |
| have hs : Summable (fun i => ∫ x, |b x| ∂μ i) := by |
| simpa only [Real.norm_eq_abs,abs_abs] using hbi.summable_integral |
| have hsa : Summable (fun i => S i b π) := |
| hs.of_norm_bounded (fun i => by simpa only [Real.norm_eq_abs] using hμ i b π hb hπ) |
| rw [hT b π,integral_sum_measure hbi] |
| calc |
| |∑' i, S i b π| ≤ ∑' i, |S i b π| := by |
| simpa only [Real.norm_eq_abs] using |
| (norm_tsum_le_tsum_norm (f := fun i => S i b π) hsa.norm) |
| _ ≤ ∑' i, ∫ x, |b x| ∂μ i := hsa.abs.tsum_le_tsum (fun i => hμ i b π hb hπ) hs |
|
|
| end SumControls |
|
|
| lemma tendsto_integral_abs_of_l1 {X : Type*} [MeasurableSpace X] |
| {μ ν : Measure X} (hν : ν ≤ μ) {f : ℕ → X → ℝ} {b : X → ℝ} |
| (hf : ∀ j, Integrable (f j) μ) (hb : Integrable b μ) |
| (hlim : Tendsto (fun j => ∫ x, |f j x-b x| ∂μ) atTop (𝓝 0)) : |
| Tendsto (fun j => ∫ x, |f j x| ∂ν) atTop (𝓝 (∫ x, |b x| ∂ν)) := by |
| apply tendsto_iff_dist_tendsto_zero.mpr |
| apply squeeze_zero (fun j => dist_nonneg) (fun j => ?_) hlim |
| rw [Real.dist_eq,←integral_sub ((hf j).mono_measure hν).abs (hb.mono_measure hν).abs] |
| calc |∫ x, |f j x|-|b x| ∂ν| ≤ ∫ x, abs (|f j x|-|b x|) ∂ν := by |
| simpa only [Real.norm_eq_abs] using norm_integral_le_integral_norm |
| (fun x => |f j x|-|b x|) |
| _ ≤ ∫ x, |f j x-b x| ∂ν := |
| integral_mono_ae (((hf j).mono_measure hν).abs.sub (hb.mono_measure hν).abs).abs |
| (((hf j).mono_measure hν).sub (hb.mono_measure hν)).abs |
| (Eventually.of_forall fun x => abs_abs_sub_abs_le _ _) |
| _ ≤ ∫ x, |f j x-b x| ∂μ := |
| integral_mono_measure hν (Eventually.of_forall fun _ => abs_nonneg _) |
| ((hf j).sub hb).abs |
| end SharpIntegralFillings |
|
|
| namespace SharpIntegralFillings |
| open Set MeasureTheory Filter |
| open scoped ENNReal NNReal Topology |
|
|
| section ComponentControls |
| variable {X : Type*} [MetricSpace X] [CompactSpace X] [MeasurableSpace X] [BorelSpace X] |
| {k : ℕ} {T : Functional X k} {S : ℕ → Functional X k} |
| {μ : ℕ → Measure X} [IsFiniteMeasure (Measure.sum μ)] |
| {Z : ℕ → Set X} |
|
|
| lemma component_approximation_bound (hS : ∀ i, IsMetricCurrent (S i)) |
| (hμ : ∀ i, Controls (S i) (μ i)) |
| (hT : ∀ b π, T b π = ∑' i, S i b π) |
| (hZ : ∀ i, MeasurableSet (Z i)) (hd : Pairwise (fun i j => Disjoint (Z i) (Z j))) |
| (hc : ∀ i, ∀ᵐ x ∂μ i, x ∈ Z i) (i : ℕ) |
| {f b : X → ℝ} (hf : BoundedLip f) (hb : BoundedLip b) |
| {π : Fin k → X → ℝ} (hπ : ∀ j, LipschitzWith 1 (π j)) : |
| |T f π - S i b π| ≤ ∫ x, |f x - (Z i).indicator b x| ∂Measure.sum μ := by |
| classical |
| have hfi := BorelCoefficients.integrable_boundedLip (Measure.sum μ) hf |
| have hbi := (BorelCoefficients.integrable_boundedLip (Measure.sum μ) hb).indicator (hZ i) |
| let d : ℕ → ℝ := fun j => S j f π - if j = i then S i b π else 0 |
| have hbd (j : ℕ) : |d j| ≤ ∫ x, |f x-(Z i).indicator b x| ∂μ j := by |
| by_cases hji : j = i |
| · subst j |
| have heq : S i (fun x => f x - b x) π = S i f π - S i b π := by |
| simpa only [one_mul, neg_one_mul, sub_eq_add_neg] using |
| (hS i).linearFirst f b π 1 (-1) hf hb (fun l => ⟨1,hπ l⟩) |
| have hbound := hμ i (fun x => f x-b x) π (by simpa only [neg_one_mul, sub_eq_add_neg] using hf.add (hb.const_mul (-1))) hπ |
| rw [heq] at hbound |
| change |S i f π - if i = i then S i b π else 0| ≤ _ |
| rw [ite_eq_left rfl] |
| apply hbound.trans_eq |
| apply integral_congr_ae |
| filter_upwards [hc i] with x hx |
| rw [indicator_of_mem hx] |
| · have hbound := hμ j f π hf hπ |
| change |S j f π - if j = i then S i b π else 0| ≤ _ |
| rw [ite_eq_right hji,sub_zero] |
| apply hbound.trans_eq |
| apply integral_congr_ae |
| filter_upwards [hc j] with x hx |
| have hxi : x ∉ Z i := fun hxi => Set.disjoint_left.mp (hd hji) hx hxi |
| rw [indicator_of_notMem hxi,sub_zero] |
| have heint : Integrable (fun x => |f x-(Z i).indicator b x|) (Measure.sum μ) := |
| (hfi.sub hbi).abs |
| have he : Summable (fun j => ∫ x, |f x-(Z i).indicator b x| ∂μ j) := by |
| simpa only [Real.norm_eq_abs,abs_abs] using heint.summable_integral |
| have hdS : Summable d := he.of_norm_bounded (fun j => by simpa only [Real.norm_eq_abs] using hbd j) |
| have hfS : Summable (fun j => S j f π) := by |
| have hfs : Summable (fun j => ∫ x, |f x| ∂μ j) := by |
| simpa only [Real.norm_eq_abs,abs_abs] using hfi.abs.summable_integral |
| exact hfs.of_norm_bounded (fun j => by simpa only [Real.norm_eq_abs] using hμ j f π hf hπ) |
| have hsum : ∑' j, d j = T f π - S i b π := by |
| rw [hT f π] |
| dsimp only [d] |
| rw [hfS.tsum_sub (hasSum_ite_eq i (S i b π)).summable,tsum_ite_eq] |
| rw [←hsum,integral_sum_measure heint] |
| calc |∑' j, d j| ≤ ∑' j, |d j| := by |
| simpa only [Real.norm_eq_abs] using (norm_tsum_le_tsum_norm (f := d) hdS.norm) |
| _ ≤ ∑' j, ∫ x, |f x-(Z i).indicator b x| ∂μ j := hdS.abs.tsum_le_tsum hbd he |
|
|
| lemma controls_component_restrict (hS : ∀ i, IsMetricCurrent (S i)) |
| (hμ : ∀ i, Controls (S i) (μ i)) |
| (hT : ∀ b π, T b π = ∑' i, S i b π) |
| (hZ : ∀ i, MeasurableSet (Z i)) (hd : Pairwise (fun i j => Disjoint (Z i) (Z j))) |
| (hc : ∀ i, ∀ᵐ x ∂μ i, x ∈ Z i) |
| {ν : Measure X} [IsFiniteMeasure ν] (hν : Controls T ν) (i : ℕ) : |
| Controls (S i) (ν.restrict (Z i)) := by |
| intro b π hb hπ |
| let η := ν + Measure.sum μ |
| have hbη : Integrable ((Z i).indicator b) η := |
| (BorelCoefficients.integrable_boundedLip η hb).indicator (hZ i) |
| obtain ⟨f,hf⟩ := BorelCoefficients.exists_lipschitz_approximation η hbη |
| have hfi (j : ℕ) : Integrable (f j).val η := |
| BorelCoefficients.integrable_boundedLip η (f j).property |
| have hleμ : Measure.sum μ ≤ η := Measure.le_add_left le_rfl |
| have hleν : ν ≤ η := Measure.le_add_right le_rfl |
| have hlim : Tendsto (fun j => T (f j).val π) atTop (𝓝 (S i b π)) := by |
| apply tendsto_iff_dist_tendsto_zero.mpr |
| apply squeeze_zero (fun j => dist_nonneg) (fun j => ?_) hf |
| rw [Real.dist_eq] |
| apply (component_approximation_bound hS hμ hT hZ hd hc i (f j).property hb hπ).trans |
| exact integral_mono_measure hleμ (Eventually.of_forall fun _ => abs_nonneg _) |
| ((hfi j).sub hbη).abs |
| have hlimν := tendsto_integral_abs_of_l1 hleν hfi hbη hf |
| have hle := le_of_tendsto_of_tendsto hlim.abs hlimν |
| (Eventually.of_forall fun j => hν _ π (f j).property hπ) |
| apply hle.trans_eq |
| rw [←integral_indicator (hZ i)] |
| apply integral_congr_ae |
| exact Eventually.of_forall fun x => by |
| by_cases hx : x ∈ Z i <;> simp [hx] |
|
|
| end ComponentControls |
| end SharpIntegralFillings |
|
|
| namespace SharpIntegralFillings |
| open Set MeasureTheory Filter |
| open scoped ENNReal NNReal Topology |
|
|
| lemma sum_measure_le_of_component_minimal |
| {X : Type*} [MetricSpace X] [CompactSpace X] [MeasurableSpace X] [BorelSpace X] |
| {k : ℕ} {T : Functional X k} {S : ℕ → Functional X k} |
| {μ : ℕ → Measure X} [IsFiniteMeasure (Measure.sum μ)] |
| {Z : ℕ → Set X} |
| (hS : ∀ i, IsMetricCurrent (S i)) (hμ : ∀ i, Controls (S i) (μ i)) |
| (hmin : ∀ i (ν : Measure X), IsFiniteMeasure ν → Controls (S i) ν → μ i ≤ ν) |
| (hT : ∀ b π, T b π = ∑' i, S i b π) |
| (hZ : ∀ i, MeasurableSet (Z i)) (hd : Pairwise (fun i j => Disjoint (Z i) (Z j))) |
| (hc : ∀ i, ∀ᵐ x ∂μ i, x ∈ Z i) |
| {ν : Measure X} [IsFiniteMeasure ν] (hν : Controls T ν) : Measure.sum μ ≤ ν := by |
| have hle (i) : μ i ≤ ν.restrict (Z i) := |
| hmin i _ inferInstance (controls_component_restrict hS hμ hT hZ hd hc hν i) |
| have hsum : Measure.sum μ ≤ Measure.sum (fun i => ν.restrict (Z i)) := by |
| apply Measure.le_iff.mpr |
| intro B hB |
| simp only [Measure.sum_apply _ hB] |
| exact ENNReal.tsum_le_tsum (fun i => hle i B) |
| rw [←Measure.restrict_iUnion hd hZ] at hsum |
| exact hsum.trans Measure.restrict_le_self |
|
|
| end SharpIntegralFillings |
|
|
| end |
|
|
| end OAI |
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