File size: 3,118 Bytes
b317718 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 | import Mathlib
namespace OAI
noncomputable section
open MeasureTheory Filter Set
open scoped NNReal Topology
namespace Problem367.PredictableVersion
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
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
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)
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⟩
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⟩
end Problem367.PredictableVersion
end
end OAI
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