File size: 2,803 Bytes
f61e547 | 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 | import OAI.Probability.FreeIsing.Stochastic.FiniteCoordinateFiltration
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped ENNReal Topology
namespace Problem367
theorem cylinderEvents_eq_comap_restrict
{ι : Type*} {X : ι → Type*} [∀ i, MeasurableSpace (X i)] (D : Set ι) :
cylinderEvents (X := X) D = MeasurableSpace.pi.comap D.domRestrict := by
simp only [MeasurableSpace.pi, MeasurableSpace.comap_iSup, MeasurableSpace.comap_comp]
simp only [cylinderEvents, iSup_subtype]
rfl
theorem exists_coordinate_condExp_rep
{ι : Type*} {X : ι → Type*} [∀ i, MeasurableSpace (X i)]
(μ : Measure (∀ i, X i)) (D : Set ι) (f : (∀ i, X i) → ℝ) :
∃ g : (∀ i : D, X i) → ℝ, Measurable g ∧
μ[f | cylinderEvents D] = g ∘ D.domRestrict := by
have hm := stronglyMeasurable_condExp (μ := μ) (f := f) (m := cylinderEvents D)
rw [cylinderEvents_eq_comap_restrict] at hm
obtain ⟨g, hg, h⟩ := hm.exists_eq_measurable_comp
exact ⟨g, hg.measurable, by simpa only [cylinderEvents_eq_comap_restrict] using h⟩
theorem exists_finite_coordinate_Lp_approx
{ι : Type*} [Countable ι] {X : ι → Type*} [∀ i, MeasurableSpace (X i)]
(μ : Measure (∀ i, X i)) [IsFiniteMeasure μ]
{f : (∀ i, X i) → ℝ} {p : ℝ≥0∞}
(hp : 1 ≤ p) (hpt : p ≠ ∞) (hf : MemLp f p μ)
{ε : ℝ≥0∞} (hε : 0 < ε) :
∃ (s : Finset ι) (g : (∀ i : s, X i) → ℝ),
Measurable g ∧ MemLp (g ∘ s.restrict) p μ ∧
(∫ x, g (s.restrict x) ∂μ) = ∫ x, f x ∂μ ∧
eLpNorm (f - g ∘ s.restrict) p μ < ε := by
let : Encodable ι := Encodable.ofCountable ι
let s : ℕ → Finset ι := fun n =>
((Set.finite_Iic n).preimage Encodable.encode_injective.injOn).toFinset
have hs : Monotone (fun n => (s n : Set ι)) := by
intro n m hnm i hi
simp only [s, Finset.mem_coe, Set.Finite.mem_toFinset, Set.mem_preimage,
Set.mem_Iic] at hi ⊢
exact hi.trans hnm
have hex : ∀ i : ι, ∃ n, i ∈ (s n : Set ι) := by
intro i
refine ⟨Encodable.encode i, ?_⟩
simp [s]
have hlim := tendsto_Lp_coordinate_condExp μ (fun n => (s n : Set ι)) hs hex hp hpt hf
obtain ⟨n, hn⟩ := (hlim.eventually (eventually_lt_nhds hε)).exists
obtain ⟨g, hg, hrep⟩ := exists_coordinate_condExp_rep μ (s n : Set ι) f
refine ⟨s n, g, hg, ?_, ?_, ?_⟩
· change MemLp (g ∘ (s n : Set ι).domRestrict) p μ
rw [← hrep]
exact MemLp.condExp hp hf
· change (∫ x, (g ∘ (s n : Set ι).domRestrict) x ∂μ) = _
rw [← hrep]
exact integral_condExp cylinderEvents_le_pi
change eLpNorm (f - g ∘ (s n : Set ι).domRestrict) p μ < ε
rw [← hrep, eLpNorm_sub_comm]
exact hn
end Problem367
end
end OAI
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