| import Mathlib |
| import OAI.Analysis.BiholderTransport.Duality.CompactDual |
| import OAI.Analysis.BiholderTransport.Regularity.ContinuousClosedRelationSelector |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped Topology NNReal BoundedContinuousFunction |
|
|
| namespace WeakMTWTransport |
|
|
| section CompactAE |
| variable {M : Type*} [MetricSpace M] [CompactSpace M] [Nonempty M] |
|
|
| omit [Nonempty M] in |
|
|
| lemma contact_gap_control_at (u v : M →ᵇ ℝ) (x y : M) |
| (hsingle : ∀ z, contactGap u v x z = 0 → z = y) |
| (hgap : ∀ z, 0 ≤ contactGap u v x z) (f : M →ᵇ ℝ) |
| {ε : ℝ} (hε : 0 < ε) : |
| ∃ δ : ℝ, 0 < δ ∧ ∀ z, contactGap u v x z ≤ δ → f y - f z ≤ ε := by |
| let G : M → ℝ := fun z => contactGap u v x z |
| let B : Set M := {z | ε ≤ f y - f z} |
| have hG : Continuous G := (continuous_const.add (continuous_cost_right x)).add v.continuous |
| have hB : IsCompact B := (isClosed_le continuous_const |
| (continuous_const.sub f.continuous)).isCompact |
| by_cases hne : B.Nonempty |
| · obtain ⟨z,hz,hmin⟩ := hB.exists_isMinOn hne hG.continuousOn |
| have hpos : 0 < G z := by |
| apply lt_of_le_of_ne (hgap z) |
| intro he |
| have heq := hsingle z he.symm |
| change ε ≤ f y - f z at hz |
| rw [heq,sub_self] at hz |
| exact (not_le.mpr hε) hz |
| refine ⟨G z / 2,half_pos hpos,?_⟩ |
| intro w hw |
| by_contra hn |
| have hm := hmin (le_of_lt (lt_of_not_ge hn)) |
| change G z ≤ contactGap u v x w at hm |
| linarith |
| · refine ⟨1,by norm_num,?_⟩ |
| intro z _ |
| by_contra hn |
| exact hne ⟨z,le_of_lt (lt_of_not_ge hn)⟩ |
|
|
| lemma transform_perturbation_upper_at (u v : M →ᵇ ℝ) (x y : M) |
| (hsingle : ∀ z, contactGap u v x z = 0 → z = y) |
| (hgap : ∀ z, 0 ≤ contactGap u v x z) (f : M →ᵇ ℝ) |
| {ε : ℝ} (hε : 0 < ε) : |
| ∃ τ : ℝ, 0 < τ ∧ ∀ t : ℝ, 0 < t → t ≤ τ → |
| cTransform (fun z => v z + t * f z) x ≤ u x - t * f y + t * ε := by |
| obtain ⟨δ,hδ,hcontrol⟩ := contact_gap_control_at u v x y hsingle hgap f hε |
| have hden : 0 < 2 * ‖f‖ + 1 := by positivity |
| refine ⟨δ / (2 * ‖f‖ + 1),div_pos hδ hden,?_⟩ |
| intro t ht htτ |
| have htd : t * (2 * ‖f‖ + 1) ≤ δ := (le_div_iff₀ hden).mp htτ |
| have htbound : t * (2 * ‖f‖) ≤ δ := by nlinarith |
| rw [cTransform_le_iff (v := fun z => v z + t * f z) |
| (v.continuous.add (continuous_const.mul f.continuous))] |
| intro z |
| have hg := hgap z |
| have hbound : t * (f y - f z) ≤ contactGap u v x z + t * ε := by |
| by_cases hs : contactGap u v x z ≤ δ |
| · have hh := mul_le_mul_of_nonneg_left (hcontrol z hs) ht.le |
| nlinarith |
| · have hf1 := BoundedContinuousFunction.norm_coe_le_norm f y |
| have hf2 := BoundedContinuousFunction.norm_coe_le_norm f z |
| rw [Real.norm_eq_abs,abs_le] at hf1 hf2 |
| have hw : f y - f z ≤ 2 * ‖f‖ := by linarith [hf1.2,hf2.1] |
| have hh := mul_le_mul_of_nonneg_left hw ht.le |
| have hte : 0 ≤ t * ε := mul_nonneg ht.le hε.le |
| linarith |
| dsimp [contactGap] at hbound |
| nlinarith |
|
|
| def transformedDifferenceQuotient (v f : M →ᵇ ℝ) (t : ℝ) : M →ᵇ ℝ := |
| t⁻¹ • (boundedCTransform (v + t • f) - boundedCTransform v) |
|
|
| lemma transformedDifferenceQuotient_apply (v f : M →ᵇ ℝ) (t : ℝ) (x : M) : |
| transformedDifferenceQuotient v f t x = |
| (cTransform (fun z => v z + t * f z) x - cTransform v x) / t := by |
| simp only [transformedDifferenceQuotient,BoundedContinuousFunction.smul_apply, |
| BoundedContinuousFunction.sub_apply,boundedCTransform_apply,smul_eq_mul] |
| rw [div_eq_mul_inv,mul_comm] |
| rfl |
|
|
| lemma transformedDifferenceQuotient_norm_le (v f : M →ᵇ ℝ) {t : ℝ} (ht : 0 < t) (x : M) : |
| ‖transformedDifferenceQuotient v f t x‖ ≤ ‖f‖ := by |
| have hh := boundedCTransform_nonexpansive (M := M) |>.dist_le_mul (v + t • f) v |
| simp only [NNReal.coe_one,one_mul,dist_eq_norm,add_sub_cancel_left,norm_smul, |
| Real.norm_eq_abs,abs_of_pos ht] at hh |
| have hp := BoundedContinuousFunction.norm_coe_le_norm |
| (boundedCTransform (v + t • f) - boundedCTransform v) x |
| have hbound := hp.trans hh |
| rw [transformedDifferenceQuotient,BoundedContinuousFunction.smul_apply,norm_smul, |
| Real.norm_eq_abs,abs_inv,abs_of_pos ht] |
| exact (inv_mul_le_iff₀ ht).mpr hbound |
|
|
| lemma transformedDifferenceQuotient_tendsto_at (v f : M →ᵇ ℝ) (x y : M) |
| (hcontact : contactGap (cTransform v) v x y = 0) |
| (hsingle : ∀ z, contactGap (cTransform v) v x z = 0 → z = y) |
| {s : ℕ → ℝ} (hs : ∀ j, 0 < s j) (hlim : Tendsto s atTop (𝓝 0)) : |
| Tendsto (fun j => transformedDifferenceQuotient v f (s j) x) atTop (𝓝 (-f y)) := by |
| rw [Metric.tendsto_nhds] |
| intro ε hε |
| obtain ⟨τ,hτ,hup⟩ := transform_perturbation_upper_at (boundedCTransform v) v x y |
| hsingle (cTransform_gap_nonneg v.continuous x) f (half_pos hε) |
| have hevent : ∀ᶠ j in atTop, s j ≤ τ := |
| (hlim.eventually (eventually_lt_nhds hτ)).mono fun _ h => h.le |
| filter_upwards [hevent] with j hj |
| rw [transformedDifferenceQuotient_apply,Real.dist_eq,sub_neg_eq_add,abs_lt] |
| have hu := hup (s j) (hs j) hj |
| have hl := cTransform_gap_nonneg |
| (v.continuous.add (continuous_const.mul f.continuous) : |
| Continuous (fun z => v z + s j * f z)) x y |
| change cTransform (fun z => v z + s j * f z) x ≤ |
| cTransform v x - s j * f y + s j * (ε/2) at hu |
| dsimp only [contactGap] at hcontact |
| change 0 ≤ cTransform (fun z => v z + s j * f z) x + cost x y + (v y + s j * f y) at hl |
| have hlo : -f y ≤ (cTransform (fun z => v z + s j * f z) x - cTransform v x) / s j := by |
| apply (le_div_iff₀ (hs j)).mpr |
| nlinarith |
| have hhi : (cTransform (fun z => v z + s j * f z) x - cTransform v x) / s j ≤ -f y + ε/2 := by |
| apply (div_le_iff₀ (hs j)).mpr |
| nlinarith |
| constructor <;> linarith |
|
|
| variable [MeasurableSpace M] [BorelSpace M] |
|
|
| lemma dual_minimum_ae_integral_eq (μ ν : Measure M) [IsProbabilityMeasure μ] |
| [IsProbabilityMeasure ν] (v : M →ᵇ ℝ) (T : M → M) |
| (hcontact : ∀ x, contactGap (cTransform v) v x (T x) = 0) |
| (hsingle : ∀ᵐ x ∂μ, ∀ y, contactGap (cTransform v) v x y = 0 → y = T x) |
| (hmin : ∀ a b : M →ᵇ ℝ, (∀ x y, 0 ≤ contactGap a b x y) → |
| dualObjective μ ν (boundedCTransform v,v) ≤ dualObjective μ ν (a,b)) |
| (f : M →ᵇ ℝ) : (∫ x, f (T x) ∂μ) = ∫ y, f y ∂ν := by |
| have hle (f : M →ᵇ ℝ) : (∫ x, f (T x) ∂μ) ≤ ∫ y, f y ∂ν := by |
| let s : ℕ → ℝ := fun j => 1 / ((j : ℝ) + 1) |
| have hs (j : ℕ) : 0 < s j := by dsimp [s]; positivity |
| have hslim : Tendsto s atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat |
| have hlim : Tendsto (fun j => ∫ x, transformedDifferenceQuotient v f (s j) x ∂μ) |
| atTop (𝓝 (∫ x, -f (T x) ∂μ)) := by |
| apply tendsto_integral_of_dominated_convergence (fun _ => ‖f‖) |
| · intro j |
| exact (transformedDifferenceQuotient v f (s j)).continuous.aestronglyMeasurable |
| · exact integrable_const _ |
| · intro j |
| exact Eventually.of_forall (transformedDifferenceQuotient_norm_le v f (hs j)) |
| · filter_upwards [hsingle] with x hx |
| exact transformedDifferenceQuotient_tendsto_at v f x (T x) (hcontact x) hx hs hslim |
| have hineq (j : ℕ) : -(∫ y, f y ∂ν) ≤ |
| ∫ x, transformedDifferenceQuotient v f (s j) x ∂μ := by |
| let a := boundedCTransform (v + s j • f) |
| let b : M →ᵇ ℝ := v + s j • f |
| have hi := hmin a b (cTransform_gap_nonneg b.continuous) |
| have hb : (∫ y, b y ∂ν) = (∫ y, v y ∂ν) + s j * ∫ y, f y ∂ν := by |
| change (∫ y, v y + s j * f y ∂ν) = _ |
| rw [integral_add (v.integrable ν) ((f.integrable ν).const_mul (s j)), |
| integral_const_mul] |
| have hq : (∫ x, transformedDifferenceQuotient v f (s j) x ∂μ) = |
| ((∫ x, a x ∂μ) - (∫ x, boundedCTransform v x ∂μ)) / s j := by |
| change (∫ x, (s j)⁻¹ * (a x - boundedCTransform v x) ∂μ) = _ |
| rw [integral_const_mul,integral_sub (a.integrable μ) ((boundedCTransform v).integrable μ)] |
| rw [div_eq_mul_inv,mul_comm] |
| rw [hq] |
| apply (le_div_iff₀ (hs j)).mpr |
| change (∫ x, boundedCTransform v x ∂μ) + (∫ y, v y ∂ν) ≤ |
| (∫ x, a x ∂μ) + (∫ y, b y ∂ν) at hi |
| rw [hb] at hi |
| nlinarith |
| have hbound := ge_of_tendsto' hlim hineq |
| rw [integral_neg] at hbound |
| linarith |
| apply le_antisymm (hle f) |
| have hneg := hle (-f) |
| simp only [BoundedContinuousFunction.neg_apply,integral_neg] at hneg |
| linarith |
|
|
| omit [CompactSpace M] [Nonempty M] [MeasurableSpace M] [BorelSpace M] in |
| lemma cost_contact_relation_isClosed (u v : M →ᵇ ℝ) : |
| IsClosed {z : M × M | contactGap u v z.1 z.2 = 0} := by |
| apply isClosed_eq _ continuous_const |
| exact ((u.continuous.comp continuous_fst).add |
| (((continuous_fst.dist continuous_snd).pow 2).div_const 2)).add |
| (v.continuous.comp continuous_snd) |
|
|
| lemma dual_minimum_ae_pushforward (μ ν : Measure M) [IsProbabilityMeasure μ] |
| [IsProbabilityMeasure ν] (v : M →ᵇ ℝ) (T : M → M) |
| (hcontact : ∀ x, contactGap (cTransform v) v x (T x) = 0) |
| (hsingle : ∀ᵐ x ∂μ, ∀ y, contactGap (cTransform v) v x y = 0 → y = T x) |
| (hmin : ∀ a b : M →ᵇ ℝ, (∀ x y, 0 ≤ contactGap a b x y) → |
| dualObjective μ ν (boundedCTransform v,v) ≤ dualObjective μ ν (a,b)) : |
| AEMeasurable T μ ∧ Measure.map T μ = ν := by |
| have hT : AEMeasurable T μ := aemeasurable_closed_relation_selector |
| (cost_contact_relation_isClosed (boundedCTransform v) v) hcontact hsingle |
| refine ⟨hT,?_⟩ |
| apply ext_of_forall_integral_eq_of_IsFiniteMeasure |
| intro f |
| rw [integral_map hT f.continuous.aestronglyMeasurable] |
| exact dual_minimum_ae_integral_eq μ ν v T hcontact hsingle hmin f |
|
|
| end CompactAE |
| end WeakMTWTransport |
|
|
| end |
|
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| end |
|
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| end |
|
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| end OAI |
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|
