openAI-math-dataset / lean /OAI /Probability /FreeIsing /Estimates /FiniteCoordinateRepresentation.lean
| 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 | |