| import Mathlib |
| import OAI.Geometry.IntegralFillings.Currents.DenseTests |
| import OAI.Geometry.IntegralFillings.Charts.DensityPush |
|
|
| namespace OAI |
|
|
| section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
|
|
| namespace SharpIntegralFillings |
| open MeasureTheory |
|
|
| section BorelChartTests |
| variable {Z X : Type*} [MeasurableSpace Z] [MetricSpace X] [CompactSpace X] |
| [MeasurableSpace X] [BorelSpace X] |
| theorem signed_chart_control_borel (μ : Measure Z) (ν : Measure X) [IsFiniteMeasure ν] |
| {φ : Z → X} (hφ : Measurable φ) {w : Z → ℝ} (hw : Integrable w μ) |
| (hcontrol : ∀ b : X → ℝ, BoundedLip b → |
| |∫ z, w z * b (φ z) ∂μ| ≤ ∫ x, |b x| ∂ν) |
| {b : X → ℝ} (hb : Measurable b) (hB : ∃ M : ℝ, ∀ x, |b x| ≤ M) : |
| |∫ z, w z * b (φ z) ∂μ| ≤ ∫ x, |b x| ∂ν := by |
| let τ := densityPush μ φ (fun z => |w z|) |
| let : IsFiniteMeasure τ := densityPush_finite μ φ hw.abs |
| let η := ν+τ |
| have hνη : ν ≤ η := Measure.le_add_right le_rfl |
| have hτη : τ ≤ η := Measure.le_add_left le_rfl |
| have hbi : Integrable b η := by |
| obtain ⟨M,hM⟩ := hB |
| exact Integrable.of_bound hb.aestronglyMeasurable M |
| (Eventually.of_forall fun x => by simpa only [Real.norm_eq_abs] using hM x) |
| obtain ⟨f,hf⟩ := BorelCoefficients.exists_lipschitz_approximation η hbi |
| have hfi (j : ℕ) : Integrable ((f j).val) η := |
| (f j).val.continuous.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) |
| have hwb : Integrable (fun z => w z*b (φ z)) μ := |
| integrable_weighted_comp μ hφ hw hb hB |
| have hwf (j : ℕ) : Integrable (fun z => w z*(f j).val (φ z)) μ := |
| integrable_weighted_comp μ hφ hw (f j).val.continuous.measurable (f j).property.2 |
| have hflim : Tendsto (fun j => ∫ z, w z*(f j).val (φ z) ∂μ) atTop |
| (𝓝 (∫ z, w z*b (φ z) ∂μ)) := by |
| apply tendsto_iff_dist_tendsto_zero.mpr |
| apply squeeze_zero (fun j => dist_nonneg) (fun j => ?_) hf |
| rw [Real.dist_eq,←integral_sub (hwf j) hwb] |
| calc |∫ z, w z*(f j).val (φ z)-w z*b (φ z) ∂μ| ≤ |
| ∫ z, |w z| * |(f j).val (φ z)-b (φ z)| ∂μ := by |
| simpa only [←mul_sub,Real.norm_eq_abs,abs_mul] using |
| norm_integral_le_integral_norm |
| (fun z => w z*(f j).val (φ z)-w z*b (φ z)) |
| _ = ∫ x, |(f j).val x-b x| ∂τ := |
| (integral_densityPush_measurable μ hφ hw.abs |
| (Eventually.of_forall fun _ => abs_nonneg _) |
| ((f j).val.continuous.measurable.sub hb).abs).symm |
| _ ≤ ∫ x, |(f j).val x-b x| ∂η := |
| integral_mono_measure hτη (Eventually.of_forall fun _ => abs_nonneg _) |
| ((hfi j).sub hbi).abs |
| have hib : Integrable b ν := hbi.mono_measure hνη |
| 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 hνη).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 hνη).abs.sub hib.abs).abs |
| (((hfi j).mono_measure hνη).sub hib).abs |
| (Eventually.of_forall fun x => abs_abs_sub_abs_le _ _) |
| _ ≤ ∫ x, |(f j).val x-b x| ∂η := |
| integral_mono_measure hνη (Eventually.of_forall fun _ => abs_nonneg _) |
| ((hfi j).sub hbi).abs |
| exact le_of_tendsto_of_tendsto hflim.abs hfint |
| (Eventually.of_forall fun j => hcontrol _ (f j).property) |
| end BorelChartTests |
| end SharpIntegralFillings |
|
|
| namespace SharpIntegralFillings |
| open MeasureTheory Set |
|
|
| variable {Z X : Type*} [MeasurableSpace Z] [MetricSpace X] [CompactSpace X] |
| [MeasurableSpace X] [BorelSpace X] |
|
|
| lemma signed_chart_control_dominates_of_measurable |
| (μ : Measure Z) (ν : Measure X) [IsFiniteMeasure ν] |
| {φ : Z → X} (hφ : MeasurableEmbedding φ) {w : Z → ℝ} |
| (hwm : Measurable w) (hw : Integrable w μ) |
| (hcontrol : ∀ b : X → ℝ, BoundedLip b → |
| |∫ z, w z * b (φ z) ∂μ| ≤ ∫ x, |b x| ∂ν) : |
| densityPush μ φ (fun z => |w z|) ≤ ν := by |
| classical |
| let τ := densityPush μ φ (fun z => |w z|) |
| let : IsFiniteMeasure τ := densityPush_finite μ φ hw.abs |
| let σ : Z → ℝ := fun z => if w z < 0 then -1 else 1 |
| have hσm : Measurable σ := |
| Measurable.piecewise (measurableSet_lt hwm measurable_const) measurable_const measurable_const |
| have hσabs (z : Z) : |σ z| = 1 := by simp [σ]; split <;> norm_num |
| have hσmul (z : Z) : w z * σ z = |w z| := by |
| by_cases hz : w z < 0 |
| · simp [σ,hz,abs_of_neg hz] |
| · simp [σ,hz,abs_of_nonneg (le_of_not_gt hz)] |
| let σe : X → ℝ := Function.extend φ σ (fun _ => 0) |
| have hσem : Measurable σe := hφ.measurable_extend hσm measurable_const |
| have hσee (z : Z) : σe (φ z) = σ z := hφ.injective.extend_apply σ (fun _ => 0) z |
| have hσeb (x : X) : |σe x| ≤ 1 := by |
| by_cases hx : ∃ z, φ z = x |
| · obtain ⟨z,rfl⟩ := hx |
| rw [hσee,hσabs] |
| · rw [show σe x = 0 from Function.extend_apply' σ (fun _ => 0) x hx] |
| norm_num |
| apply Measure.le_iff.mpr |
| intro s hs |
| let b := s.indicator σe |
| have hbm : Measurable b := hσem.indicator hs |
| have hbb (x : X) : |b x| ≤ s.indicator (fun _ => (1 : ℝ)) x := by |
| by_cases hx : x ∈ s |
| · simpa only [b,indicator_of_mem hx] using hσeb x |
| · simp [b,hx] |
| have hB : ∃ M : ℝ, ∀ x, |b x| ≤ M := ⟨1,fun x => (hbb x).trans (by |
| by_cases hx : x ∈ s <;> simp [hx])⟩ |
| have hbi : Integrable b ν := Integrable.of_bound hbm.aestronglyMeasurable 1 |
| (Eventually.of_forall fun x => by |
| simpa only [Real.norm_eq_abs] using (hbb x).trans (by by_cases hx : x ∈ s <;> simp [hx])) |
| have heq : (∫ z, w z*b (φ z) ∂μ) = τ.real s := by |
| rw [←integral_indicator_one hs] |
| rw [show τ = densityPush μ φ (fun z => |w z|) from rfl, |
| integral_densityPush_measurable μ hφ.measurable hw.abs |
| (Eventually.of_forall fun _ => abs_nonneg _) |
| (b := s.indicator (1 : X → ℝ)) ((measurable_one : Measurable (1 : X → ℝ)).indicator hs)] |
| apply integral_congr_ae |
| apply Eventually.of_forall |
| intro z |
| by_cases hz : φ z ∈ s |
| · simpa only [b,indicator_of_mem hz,Pi.one_apply,mul_one,hσee] using hσmul z |
| · simp [b,hz] |
| have hc := signed_chart_control_borel μ ν hφ.measurable hw hcontrol hbm hB |
| rw [heq,abs_of_nonneg (show 0 ≤ τ.real s from ENNReal.toReal_nonneg)] at hc |
| have hupper : (∫ x, |b x| ∂ν) ≤ ν.real s := by |
| rw [←integral_indicator_one hs] |
| exact integral_mono_ae hbi.abs ((integrable_const (1 : ℝ)).indicator hs) |
| (Eventually.of_forall hbb) |
| exact (ENNReal.toReal_le_toReal (measure_ne_top τ s) (measure_ne_top ν s)).mp |
| (hc.trans hupper) |
| |
| lemma signed_chart_control_dominates_on_of_measurable |
| (μ : Measure Z) (ν : Measure X) [IsFiniteMeasure ν] |
| {φ : Z → X} (hφ : Measurable φ) {s₀ : Set Z} |
| (hμ : ∀ᵐ z ∂μ, z ∈ s₀) |
| (he : MeasurableEmbedding (fun z : s₀ => φ (z : Z))) {w : Z → ℝ} |
| (hwm : Measurable w) (hw : Integrable w μ) |
| (hcontrol : ∀ b : X → ℝ, BoundedLip b → |
| |∫ z, w z * b (φ z) ∂μ| ≤ ∫ x, |b x| ∂ν) : |
| densityPush μ φ (fun z => |w z|) ≤ ν := by |
| classical |
| let τ := densityPush μ φ (fun z => |w z|) |
| let : IsFiniteMeasure τ := densityPush_finite μ φ hw.abs |
| let σ : Z → ℝ := fun z => if w z < 0 then -1 else 1 |
| have hσm : Measurable σ := |
| Measurable.piecewise (measurableSet_lt hwm measurable_const) measurable_const measurable_const |
| have hσabs (z : Z) : |σ z| = 1 := by simp [σ]; split <;> norm_num |
| have hσmul (z : Z) : w z * σ z = |w z| := by |
| by_cases hz : w z < 0 |
| · simp [σ,hz,abs_of_neg hz] |
| · simp [σ,hz,abs_of_nonneg (le_of_not_gt hz)] |
| let σe : X → ℝ := Function.extend (fun z : s₀ => φ (z : Z)) (fun z : s₀ => σ (z : Z)) (fun _ => 0) |
| have hσem : Measurable σe := he.measurable_extend (hσm.comp measurable_subtype_coe) measurable_const |
| have hσee (z : Z) (hz : z ∈ s₀) : σe (φ z) = σ z := |
| he.injective.extend_apply (fun z : s₀ => σ (z : Z)) (fun _ => 0) ⟨z,hz⟩ |
| have hσeb (x : X) : |σe x| ≤ 1 := by |
| by_cases hx : ∃ z : s₀, φ (z : Z) = x |
| · obtain ⟨z,rfl⟩ := hx |
| rw [hσee z z.property,hσabs] |
| · rw [show σe x = 0 from Function.extend_apply' (fun z : s₀ => σ (z : Z)) (fun _ => 0) x hx] |
| norm_num |
| apply Measure.le_iff.mpr |
| intro s hs |
| let b := s.indicator σe |
| have hbm : Measurable b := hσem.indicator hs |
| have hbb (x : X) : |b x| ≤ s.indicator (fun _ => (1 : ℝ)) x := by |
| by_cases hx : x ∈ s |
| · simpa only [b,indicator_of_mem hx] using hσeb x |
| · simp [b,hx] |
| have hB : ∃ M : ℝ, ∀ x, |b x| ≤ M := ⟨1,fun x => (hbb x).trans (by |
| by_cases hx : x ∈ s <;> simp [hx])⟩ |
| have hbi : Integrable b ν := Integrable.of_bound hbm.aestronglyMeasurable 1 |
| (Eventually.of_forall fun x => by |
| simpa only [Real.norm_eq_abs] using (hbb x).trans (by by_cases hx : x ∈ s <;> simp [hx])) |
| have heq : (∫ z, w z*b (φ z) ∂μ) = τ.real s := by |
| rw [←integral_indicator_one hs] |
| rw [show τ = densityPush μ φ (fun z => |w z|) from rfl, |
| integral_densityPush_measurable μ hφ hw.abs |
| (Eventually.of_forall fun _ => abs_nonneg _) |
| (b := s.indicator (1 : X → ℝ)) ((measurable_one : Measurable (1 : X → ℝ)).indicator hs)] |
| apply integral_congr_ae |
| filter_upwards [hμ] with z hzs |
| by_cases hz : φ z ∈ s |
| · simpa only [b,indicator_of_mem hz,Pi.one_apply,mul_one,hσee z hzs] using hσmul z |
| · simp [b,hz] |
| have hc := signed_chart_control_borel μ ν hφ hw hcontrol hbm hB |
| rw [heq,abs_of_nonneg (show 0 ≤ τ.real s from ENNReal.toReal_nonneg)] at hc |
| have hupper : (∫ x, |b x| ∂ν) ≤ ν.real s := by |
| rw [←integral_indicator_one hs] |
| exact integral_mono_ae hbi.abs ((integrable_const (1 : ℝ)).indicator hs) |
| (Eventually.of_forall hbb) |
| exact (ENNReal.toReal_le_toReal (measure_ne_top τ s) (measure_ne_top ν s)).mp |
| (hc.trans hupper) |
|
|
| end SharpIntegralFillings |
| end |
|
|
| end OAI |
|
|
