| import OAI.Analysis.Brenier.AffineTies |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory |
| open scoped InnerProductSpace |
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| namespace Problem358 |
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| /-- A finite nonempty family of measurable scores has a measurable maximizing |
| index. The maximizing score is measurable as well; no measurability assumption |
| on individual index singletons is needed. -/ |
| theorem exists_measurable_finset_argmax {X ι : Type*} |
| [MeasurableSpace X] [MeasurableSpace ι] |
| (f : ι → X → ℝ) (hf : ∀ i, Measurable (f i)) : |
| ∀ (s : Finset ι), s.Nonempty → |
| ∃ g : X → ι, Measurable g ∧ (∀ x, g x ∈ s) ∧ |
| Measurable (fun x => f (g x) x) ∧ |
| ∀ x i, i ∈ s → f i x ≤ f (g x) x := by |
| classical |
| intro s |
| induction s using Finset.induction_on with |
| | empty => |
| intro hs |
| exact False.elim (Finset.not_nonempty_empty hs) |
| | @insert a s ha ih => |
| intro hs |
| by_cases hsn : s.Nonempty |
| · obtain ⟨g, hg, hgm, hfg, hmax⟩ := ih hsn |
| let p : X → Prop := fun x => f (g x) x ≤ f a x |
| let k : X → ι := fun x => if p x then a else g x |
| have hp : MeasurableSet {x | p x} := measurableSet_le hfg (hf a) |
| have hk : Measurable k := Measurable.ite hp measurable_const hg |
| have hfk : Measurable (fun x => f (k x) x) := by |
| have hval : (fun x => f (k x) x) = |
| (fun x => if p x then f a x else f (g x) x) := by |
| funext x |
| simp only [k] |
| split_ifs <;> rfl |
| rw [hval] |
| exact Measurable.ite hp (hf a) hfg |
| refine ⟨k, hk, ?_, hfk, ?_⟩ |
| · intro x |
| dsimp [k] |
| split_ifs |
| · exact Finset.mem_insert_self _ _ |
| · exact Finset.mem_insert_of_mem (hgm x) |
| · intro x i hi |
| dsimp [k] |
| split_ifs with hx |
| · rcases Finset.mem_insert.mp hi with rfl | hi |
| · exact le_rfl |
| · exact (hmax x i hi).trans hx |
| · rcases Finset.mem_insert.mp hi with rfl | hi |
| · exact (lt_of_not_ge hx).le |
| · exact hmax x i hi |
| · have hs0 : s = ∅ := Finset.not_nonempty_iff_eq_empty.mp hsn |
| subst s |
| refine ⟨fun _ => a, measurable_const, ?_, hf a, ?_⟩ |
| · intro x |
| simp |
| · intro x i hi |
| have hia : i = a := by simpa using hi |
| subst i |
| exact le_rfl |
|
|
| /-- A measurable choice of an index maximizing a finite measurable family. -/ |
| theorem exists_measurable_argmax {X ι : Type*} |
| [MeasurableSpace X] [MeasurableSpace ι] [Fintype ι] [Nonempty ι] |
| (f : ι → X → ℝ) (hf : ∀ i, Measurable (f i)) : |
| ∃ g : X → ι, Measurable g ∧ Measurable (fun x => f (g x) x) ∧ |
| ∀ x i, f i x ≤ f (g x) x := by |
| classical |
| obtain ⟨g, hg, _, hfg, hmax⟩ := |
| exists_measurable_finset_argmax f hf Finset.univ Finset.univ_nonempty |
| exact ⟨g, hg, hfg, fun x i => hmax x i (Finset.mem_univ i)⟩ |
|
|
| /-- Finite affine maxima have a globally measurable maximizing-slope map, |
| including on their tie sets. -/ |
| theorem exists_measurable_affine_maximizer {V ι : Type*} |
| [NormedAddCommGroup V] [InnerProductSpace ℝ V] |
| [MeasurableSpace V] [BorelSpace V] |
| [Fintype ι] [Nonempty ι] (y : ι → V) (c : ι → ℝ) : |
| ∃ T : V → V, Measurable T ∧ ∀ x, ∃ i, |
| T x = y i ∧ ∀ j, ⟪y j, x⟫_ℝ + c j ≤ ⟪y i, x⟫_ℝ + c i := by |
| classical |
| let : MeasurableSpace ι := ⊤ |
| have hf (i : ι) : Measurable (fun x : V => ⟪y i, x⟫_ℝ + c i) := by |
| fun_prop |
| obtain ⟨g, hg, _, hmax⟩ := exists_measurable_argmax |
| (fun i x => ⟪y i, x⟫_ℝ + c i) hf |
| have hy : Measurable y := Measurable.of_discrete |
| exact ⟨fun x => y (g x), hy.comp hg, fun x => ⟨g x, rfl, hmax x⟩⟩ |
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| end Problem358 |
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| end |
|
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| noncomputable section |
| open MeasureTheory |
| open scoped BigOperators ENNReal |
|
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| namespace Problem358 |
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| /-- The law of a finite choice of prescribed values is determined by the choice-cell masses. -/ |
| lemma map_finite_selector {α β ι : Type*} [MeasurableSpace α] [MeasurableSpace β] |
| [MeasurableSpace ι] [MeasurableSingletonClass ι] [Fintype ι] |
| [MeasurableSingletonClass β] (ρ : Measure α) (s : α → ι) |
| (hs : Measurable s) (y : ι → β) : |
| Measure.map (fun x => y (s x)) ρ = ∑ i, ρ (s ⁻¹' {i}) • Measure.dirac (y i) := by |
| have hy : Measurable y := measurable_of_countable y |
| change Measure.map (y ∘ s) ρ = _ |
| rw [← Measure.map_map hy hs, Measure.map_eq_sum ρ s hs, Measure.sum_fintype, |
| Measure.map_finset_sum hy.aemeasurable] |
| simp only [Measure.map_smul _ hy.aemeasurable, Measure.map_dirac] |
|
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| /-- A version of `map_finite_selector` using supplied cell masses. -/ |
| lemma map_finite_selector_of_masses {α β ι : Type*} [MeasurableSpace α] [MeasurableSpace β] |
| [MeasurableSpace ι] [MeasurableSingletonClass ι] [Fintype ι] |
| [MeasurableSingletonClass β] (ρ : Measure α) (s : α → ι) |
| (hs : Measurable s) (y : ι → β) (w : ι → ℝ≥0∞) |
| (hw : ∀ i, ρ (s ⁻¹' {i}) = w i) : |
| Measure.map (fun x => y (s x)) ρ = ∑ i, w i • Measure.dirac (y i) := by |
| rw [map_finite_selector ρ s hs y] |
| simp_rw [hw] |
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| end Problem358 |
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| end |
|
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| noncomputable section |
| open MeasureTheory |
| open scoped RealInnerProductSpace |
|
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| namespace Problem358.ParameterizedArgmax |
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| /-- Off the tie set, maximizing selectors are independent of every choice. -/ |
| theorem ae_eq_of_maximizers |
| {X ι : Type*} [MeasurableSpace X] (ρ : Measure X) |
| (f : ι → X → ℝ) (k l : X → ι) |
| (hsep : ∀ᵐ x ∂ρ, Function.Injective (fun i => f i x)) |
| (hk : ∀ᵐ x ∂ρ, ∀ i, f i x ≤ f (k x) x) |
| (hl : ∀ᵐ x ∂ρ, ∀ i, f i x ≤ f (l x) x) : k =ᵐ[ρ] l := by |
| filter_upwards [hsep, hk, hl] with x hsep hk hl |
| exact hsep (le_antisymm (hl (k x)) (hk (l x))) |
|
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| /-- Any pointwise family of prescribed-mass maximizing selectors can be replaced |
| by one jointly measurable selector. Null ties preserve every mass exactly; no |
| measurability of the original family of choices is assumed. -/ |
| theorem exists_joint_measurable_argmax_with_masses |
| {P X ι : Type*} [MeasurableSpace P] [MeasurableSpace X] |
| [MeasurableSpace ι] [Fintype ι] [Nonempty ι] |
| (ρ : P → Measure X) (f : ι → P × X → ℝ) (m : P → ι → ℝ) |
| (hf : ∀ i, Measurable (f i)) |
| (hsep : ∀ p, ∀ᵐ x ∂ρ p, Function.Injective (fun i => f i (p, x))) |
| (hex : ∀ p, ∃ k : X → ι, |
| (∀ᵐ x ∂ρ p, ∀ i, f i (p, x) ≤ f (k x) (p, x)) ∧ |
| ∀ i, (ρ p).real {x | k x = i} = m p i) : |
| ∃ k : P × X → ι, Measurable k ∧ |
| (∀ p x i, f i (p, x) ≤ f (k (p, x)) (p, x)) ∧ |
| ∀ p i, (ρ p).real {x | k (p, x) = i} = m p i := by |
| obtain ⟨k, hk, _, hmax⟩ := exists_measurable_argmax f hf |
| refine ⟨k, hk, fun p x i => hmax (p, x) i, ?_⟩ |
| intro p i |
| obtain ⟨l, hl, hmass⟩ := hex p |
| have heq : (fun x => k (p, x)) =ᵐ[ρ p] l := |
| ae_eq_of_maximizers (ρ p) (fun i x => f i (p, x)) (fun x => k (p, x)) l |
| (hsep p) (Filter.Eventually.of_forall (fun x => hmax (p, x))) hl |
| have hsets : {x | k (p, x) = i} =ᵐ[ρ p] {x | l x = i} := by |
| filter_upwards [heq] with x hx |
| exact congrArg (fun z => z = i) hx |
| have hmeasure := measure_congr hsets |
| calc |
| (ρ p).real {x | k (p, x) = i} = (ρ p).real {x | l x = i} := |
| congrArg ENNReal.toReal hmeasure |
| _ = m p i := hmass i |
|
|
| /-- Concrete affine version for the uniform source: joint measurability of the |
| sites/intercepts and injective sites discharge the score and null-tie premises. -/ |
| theorem exists_joint_measurable_affine_argmax_with_masses |
| {P ι : Type*} [MeasurableSpace P] [MeasurableSpace ι] |
| [Fintype ι] [Nonempty ι] {d : ℕ} (K : Set (E d)) |
| (y : P → ι → E d) (c m : P → ι → ℝ) |
| (hy : ∀ i, Measurable (fun p => y p i)) |
| (hc : ∀ i, Measurable (fun p => c p i)) |
| (hinj : ∀ p, Function.Injective (y p)) |
| (hex : ∀ p, ∃ k : E d → ι, |
| (∀ᵐ x ∂uniformMeasure K, ∀ i, |
| ⟪x, y p i⟫ + c p i ≤ ⟪x, y p (k x)⟫ + c p (k x)) ∧ |
| ∀ i, (uniformMeasure K).real {x | k x = i} = m p i) : |
| ∃ k : P × E d → ι, Measurable k ∧ |
| (∀ p x i, ⟪x, y p i⟫ + c p i ≤ |
| ⟪x, y p (k (p, x))⟫ + c p (k (p, x))) ∧ |
| ∀ p i, (uniformMeasure K).real {x | k (p, x) = i} = m p i := by |
| apply exists_joint_measurable_argmax_with_masses |
| (fun _ => uniformMeasure K) (fun i p => ⟪p.2, y p.1 i⟫ + c p.1 i) m |
| · intro i |
| exact (measurable_snd.inner ((hy i).comp measurable_fst)).add |
| ((hc i).comp measurable_fst) |
| · intro p |
| filter_upwards [uniform_ae_affine_tie_implies_slopes_eq K (y p) (c p)] with x hx |
| intro i j hij |
| exact hinj p (hx i j hij) |
| · exact hex |
|
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| end Problem358.ParameterizedArgmax |
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| end |
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| end OAI |
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