| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory Filter Set |
| open scoped NNReal Topology |
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| namespace Problem367.PredictableVersion |
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| theorem exists_stronglyMeasurable_version_of_ae_tendsto |
| {A : Type*} [mA : MeasurableSpace A] (m : MeasurableSpace A) |
| (μ : Measure[mA] A) {f : ℕ → A → ℝ} {g : A → ℝ} |
| (hf : ∀ n, StronglyMeasurable[m] (f n)) |
| (hfg : ∀ᵐ a ∂μ, Tendsto (fun n => f n a) atTop (𝓝 (g a))) : |
| ∃ v : A → ℝ, StronglyMeasurable[m] v ∧ v =ᵐ[μ] g := by |
| refine ⟨fun a => limsup (fun n => f n a) atTop, ?_, ?_⟩ |
| · exact (Measurable.limsup (fun n => (hf n).measurable)).stronglyMeasurable |
| · filter_upwards [hfg] with a ha |
| exact ha.limsup_eq |
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| theorem exists_stronglyMeasurable_version_of_tendstoInMeasure |
| {A : Type*} [mA : MeasurableSpace A] (m : MeasurableSpace A) |
| (μ : Measure[mA] A) {f : ℕ → A → ℝ} {g : A → ℝ} |
| (hf : ∀ n, StronglyMeasurable[m] (f n)) |
| (hfg : TendstoInMeasure μ f atTop g) : |
| ∃ v : A → ℝ, StronglyMeasurable[m] v ∧ v =ᵐ[μ] g := by |
| obtain ⟨ns, _, hns⟩ := hfg.exists_seq_tendsto_ae |
| exact exists_stronglyMeasurable_version_of_ae_tendsto (mA := mA) m μ (fun n => hf (ns n)) hns |
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| theorem predictable_le_product {Ω : Type*} [mΩ : MeasurableSpace Ω] |
| (F : Filtration ℝ≥0 mΩ) : |
| F.predictable ≤ (inferInstance : MeasurableSpace (ℝ≥0 × Ω)) := by |
| apply measurableSpace_le_predictable_of_measurableSet |
| · intro A hA |
| exact (measurableSet_singleton (0 : ℝ≥0)).prod (F.le 0 A hA) |
| · intro t A hA |
| exact measurableSet_Ioi.prod (F.le t A hA) |
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| theorem exists_predictable_version_of_tendstoInMeasure |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] (F : Filtration ℝ≥0 mΩ) |
| (μ : Measure (ℝ≥0 × Ω)) {f : ℕ → ℝ≥0 → Ω → ℝ} {g : ℝ≥0 → Ω → ℝ} |
| (hf : ∀ n, IsStronglyPredictable F (f n)) |
| (hfg : TendstoInMeasure μ (fun n => Function.uncurry (f n)) atTop |
| (Function.uncurry g)) : |
| ∃ v : ℝ≥0 → Ω → ℝ, IsStronglyPredictable F v ∧ |
| Function.uncurry v =ᵐ[μ] Function.uncurry g := by |
| obtain ⟨v, hv, heq⟩ := exists_stronglyMeasurable_version_of_tendstoInMeasure |
| F.predictable μ hf hfg |
| exact ⟨Function.curry v, hv, heq⟩ |
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| theorem exists_predictable_version_of_tendsto_eLpNorm |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] (F : Filtration ℝ≥0 mΩ) |
| (μ : Measure (ℝ≥0 × Ω)) {f : ℕ → ℝ≥0 → Ω → ℝ} {g : ℝ≥0 → Ω → ℝ} |
| (hf : ∀ n, IsStronglyPredictable F (f n)) |
| (hg : MemLp (Function.uncurry g) 2 μ) |
| (hfg : Tendsto (fun n => eLpNorm |
| (Function.uncurry (f n) - Function.uncurry g) 2 μ) atTop (𝓝 0)) : |
| ∃ v : ℝ≥0 → Ω → ℝ, IsStronglyPredictable F v ∧ |
| MemLp (Function.uncurry v) 2 μ ∧ |
| Function.uncurry v =ᵐ[μ] Function.uncurry g := by |
| obtain ⟨v, hv, heq⟩ := exists_predictable_version_of_tendstoInMeasure F μ hf |
| (tendstoInMeasure_of_tendsto_eLpNorm (by norm_num) hfg) |
| exact ⟨v, hv, hg.ae_eq heq.symm, heq⟩ |
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| end Problem367.PredictableVersion |
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| end |
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| end OAI |
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