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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
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