| import Mathlib |
| import OAI.Geometry.IntegralFillings.Currents.WeakMass |
|
|
| namespace OAI |
|
|
| section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
|
|
| namespace SharpIntegralFillings |
| attribute [local instance] Classical.propDecidable |
| universe u |
| section Foundations |
| variable {X : Type u} [MetricSpace X] [MeasurableSpace X] |
| open scoped BoundedContinuousFunction |
|
|
| variable [BorelSpace X] |
| lemma summable_actions_of_summable_masses [CompactSpace X] |
| {k : ℕ} {Ts : ℕ → Functional X k} (hTs : ∀ j, IsMetricCurrent (Ts j)) |
| (hsum : Summable fun j => mass (Ts j)) {b : X → ℝ} {π : Fin k → X → ℝ} |
| (h : Admissible b π) : Summable fun j => Ts j b π := by |
| choose K hK using h.2 |
| obtain ⟨M, hM⟩ := h.1.2 |
| apply (hsum.mul_left ((∏ i, (K i : ℝ)) * M)).of_norm_bounded |
| intro j |
| exact (hTs j).mass_bound_uniform h.1 K hK hM |
|
|
| end Foundations |
|
|
| lemma tendsto_of_lipschitz_dense {A : Type*} [PseudoMetricSpace A] |
| {ι : Type*} {l : Filter ι} {f : ι → A → ℝ} {g : A → ℝ} |
| {K : ℝ≥0} (hf : ∀ j, LipschitzWith K (f j)) (hg : LipschitzWith K g) |
| {κ : Type*} {D : κ → A} (hD : DenseRange D) |
| (hlim : ∀ i, Tendsto (fun j => f j (D i)) l (𝓝 (g (D i)))) |
| (x : A) : Tendsto (fun j => f j x) l (𝓝 (g x)) := by |
| apply Metric.tendsto_nhds.mpr |
| intro ε hε |
| let δ : ℝ := ε / (4 * ((K : ℝ) + 1)) |
| have hδ : 0 < δ := div_pos hε (by positivity) |
| obtain ⟨i, hi⟩ := hD.exists_dist_lt x hδ |
| have hdr : (K : ℝ) * dist x (D i) < ε / 4 := by |
| have hpos : 0 < (K : ℝ) + 1 := by positivity |
| have hm := (mul_lt_mul_of_pos_left hi hpos) |
| have heq : ((K : ℝ) + 1) * δ = ε / 4 := by |
| dsimp [δ] |
| field_simp |
| rw [heq] at hm |
| calc (K : ℝ) * dist x (D i) ≤ ((K : ℝ) + 1) * dist x (D i) := by |
| nlinarith [dist_nonneg (x := x) (y := D i)] |
| _ < _ := hm |
| filter_upwards [(Metric.tendsto_nhds.mp (hlim i)) (ε / 2) (half_pos hε)] with j hj |
| calc dist (f j x) (g x) ≤ |
| dist (f j x) (f j (D i)) + dist (f j (D i)) (g (D i)) + dist (g (D i)) (g x) := |
| (dist_triangle _ (g (D i)) _).trans (add_le_add (dist_triangle _ (f j (D i)) _) le_rfl) |
| _ < ε / 4 + ε / 2 + ε / 4 := by |
| apply add_lt_add |
| · exact add_lt_add ((hf j).dist_le_mul x (D i) |>.trans_lt hdr) hj |
| · rw [dist_comm] |
| exact (hg.dist_le_mul x (D i)).trans_lt hdr |
| _ = ε := by ring |
|
|
| namespace BorelCoefficients |
|
|
| variable {X : Type u} [MetricSpace X] [CompactSpace X] |
|
|
| def lipAlgebra : Subalgebra ℝ C(X, ℝ) where |
| carrier := {f | BoundedLip f} |
| algebraMap_mem' r := BoundedLip.const r |
| add_mem' hf hg := BoundedLip.add hf hg |
| mul_mem' hf hg := BoundedLip.mul hf hg |
|
|
| lemma lipAlgebra_separates : (lipAlgebra (X := X)).SeparatesPoints := by |
| intro x y hxy |
| let f : C(X, ℝ) := ⟨fun z => dist z x, continuous_id.dist continuous_const⟩ |
| have hf : BoundedLip f := by |
| refine ⟨⟨1, LipschitzWith.dist_left x⟩, ‖f‖, ?_⟩ |
| intro z |
| exact (ContinuousMap.norm_coe_le_norm f z) |
| refine ⟨f, ⟨f, hf, rfl⟩, ?_⟩ |
| change dist x x ≠ dist y x |
| simpa only [dist_self, ne_eq, eq_comm, dist_eq_zero] using hxy.symm |
|
|
| lemma lipAlgebra_dense : DenseRange (fun f : lipAlgebra (X := X) => (f : C(X, ℝ))) := by |
| rw [denseRange_iff_closure_range] |
| rw [Subtype.range_coe_subtype] |
| change closure (lipAlgebra (X := X) : Set C(X, ℝ)) = univ |
| have h := ContinuousMap.subalgebra_topologicalClosure_eq_top_of_separatesPoints |
| (lipAlgebra (X := X)) lipAlgebra_separates |
| exact congrArg (fun A : Subalgebra ℝ C(X, ℝ) => (A : Set C(X, ℝ))) h |
|
|
| variable [MeasurableSpace X] [BorelSpace X] (μ : Measure X) [IsFiniteMeasure μ] |
|
|
| noncomputable def lipToL1 : lipAlgebra (X := X) →ₗ[ℝ] Lp ℝ 1 μ := |
| (ContinuousMap.toLp 1 μ ℝ).toLinearMap.comp (lipAlgebra (X := X)).val.toLinearMap |
|
|
| lemma lipToL1_dense : DenseRange (lipToL1 μ) := by |
| apply (ContinuousMap.toLp_denseRange ℝ μ ℝ (by simp : (1 : ℝ≥0∞) ≠ ∞)).comp |
| lipAlgebra_dense (ContinuousMap.toLp 1 μ ℝ).continuous |
|
|
| lemma norm_lipToL1 (f : lipAlgebra (X := X)) : |
| ‖lipToL1 μ f‖ = ∫ x, |f.val x| ∂μ := by |
| rw [L1.norm_eq_integral_norm] |
| apply integral_congr_ae |
| filter_upwards [ContinuousMap.coeFn_toLp (p := 1) (𝕜 := ℝ) μ f.val] with x hx |
| change ‖(ContinuousMap.toLp 1 μ ℝ f.val) x‖ = |f.val x| |
| rw [hx, Real.norm_eq_abs] |
|
|
| lemma exists_lipschitz_approximation {b : X → ℝ} (hb : Integrable b μ) : |
| ∃ f : ℕ → lipAlgebra (X := X), |
| Tendsto (fun j => ∫ x, |(f j).val x-b x| ∂μ) atTop (𝓝 0) := by |
| obtain ⟨u,hu,hulim⟩ := mem_closure_iff_seq_limit.mp ((lipToL1_dense μ) hb.toL1) |
| choose f hf using hu |
| have hfun : (fun j => lipToL1 μ (f j)) = u := funext hf |
| have hf := hfun ▸ hulim |
| refine ⟨f,?_⟩ |
| have hnorm := (tendsto_iff_norm_sub_tendsto_zero).mp hf |
| convert hnorm using 1 |
| funext j |
| rw [L1.norm_eq_integral_norm] |
| apply integral_congr_ae |
| filter_upwards [Lp.coeFn_sub (lipToL1 μ (f j)) hb.toL1, |
| ContinuousMap.coeFn_toLp (p := 1) (𝕜 := ℝ) μ (f j).val, |
| hb.coeFn_toL1] with x hx hfx hbx |
| change |(f j).val x-b x| = ‖(lipToL1 μ (f j)-hb.toL1) x‖ |
| rw [hx] |
| change _ = ‖(ContinuousMap.toLp 1 μ ℝ (f j).val) x - (hb.toL1) x‖ |
| rw [hfx,hbx,Real.norm_eq_abs] |
|
|
| lemma test_inequality_of_lipschitz |
| {ν : Measure X} [IsFiniteMeasure ν] (hν : ν ≤ μ) |
| (A : (X → ℝ) → ℝ) {C : ℝ} |
| (hA : ∀ f g, Integrable f μ → Integrable g μ → |
| |A f-A g| ≤ C * ∫ x, |f x-g x| ∂μ) |
| (hbound : ∀ f, BoundedLip f → |A f| ≤ ∫ x, |f x| ∂ν) |
| {b : X → ℝ} (hb : Integrable b μ) : |
| |A b| ≤ ∫ x, |b x| ∂ν := by |
| obtain ⟨f,hf⟩ := exists_lipschitz_approximation μ hb |
| have hfi (j : ℕ) : Integrable ((f j).val) μ := |
| (f j).val.continuous.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) |
| have hflim : Tendsto (fun j => A (f j).val) atTop (𝓝 (A b)) := by |
| apply tendsto_iff_dist_tendsto_zero.mpr |
| apply squeeze_zero (fun j => dist_nonneg) (fun j => ?_) (by simpa using hf.const_mul C) |
| simpa only [Real.dist_eq] using hA _ _ (hfi j) hb |
| have hib : Integrable b ν := hb.mono_measure hν |
| have hfint : Tendsto (fun j => ∫ x, |(f j).val 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 => ?_) hf |
| rw [Real.dist_eq,←integral_sub ((hfi j).mono_measure hν).abs hib.abs] |
| calc |∫ x, |(f j).val x|-|b x| ∂ν| ≤ ∫ x, abs (|(f j).val x|-|b x|) ∂ν := by |
| simpa only [Real.norm_eq_abs] using norm_integral_le_integral_norm |
| (fun x => |(f j).val x|-|b x|) |
| _ ≤ ∫ x, |(f j).val x-b x| ∂ν := |
| integral_mono_ae (((hfi j).mono_measure hν).abs.sub hib.abs).abs |
| (((hfi j).mono_measure hν).sub hib).abs |
| (Eventually.of_forall fun x => abs_abs_sub_abs_le _ _) |
| _ ≤ ∫ x, |(f j).val x-b x| ∂μ := |
| integral_mono_measure hν (Eventually.of_forall fun _ => abs_nonneg _) |
| ((hfi j).sub hb).abs |
| exact le_of_tendsto_of_tendsto hflim.abs hfint |
| (Eventually.of_forall fun j => hbound _ (f j).property) |
|
|
| end BorelCoefficients |
| end SharpIntegralFillings |
|
|
| end |
|
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| end OAI |
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|
