| import OAI.Probability.FreeIsing.Estimates.FiniteCoordinateRepresentation |
| import OAI.Probability.FreeIsing.Spectral.KoopmanInvolutions |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory |
|
|
| namespace Problem367 |
|
|
| theorem dense_centered_finite_coordinate |
| {ι : Type*} [Countable ι] {X : ι → Type*} [∀ i, MeasurableSpace (X i)] |
| (μ : Measure (∀ i, X i)) [IsFiniteMeasure μ] : |
| Dense {f : centeredL2 μ | ∃ (s : Finset ι) (g : (∀ i : s, X i) → ℝ), |
| Measurable g ∧ (fun x => (f : Lp ℝ 2 μ) x) =ᵐ[μ] g ∘ s.restrict} := by |
| apply Metric.dense_iff.mpr |
| intro f ε hε |
| obtain ⟨s, g, hgm, hgLp, hint, happrox⟩ := |
| exists_finite_coordinate_Lp_approx μ (by norm_num : 1 ≤ (2 : ENNReal)) |
| (by norm_num : (2 : ENNReal) ≠ ⊤) (Lp.memLp (f : Lp ℝ 2 μ)) |
| (ENNReal.ofReal_pos.mpr hε) |
| let gLp : Lp ℝ 2 μ := hgLp.toLp (g ∘ s.restrict) |
| have hcenter : gLp ∈ centeredL2 μ := by |
| rw [mem_centeredL2_iff] |
| calc |
| (∫ x, gLp x ∂μ) = ∫ x, g (s.restrict x) ∂μ := |
| integral_congr_ae hgLp.coeFn_toLp |
| _ = ∫ x, (f : Lp ℝ 2 μ) x ∂μ := hint |
| _ = 0 := (mem_centeredL2_iff _).mp f.property |
| refine ⟨⟨gLp, hcenter⟩, ?_, s, g, hgm, hgLp.coeFn_toLp⟩ |
| change dist gLp (f : Lp ℝ 2 μ) < ε |
| rw [dist_comm, Lp.dist_def] |
| apply ENNReal.toReal_lt_of_lt_ofReal |
| rw [eLpNorm_congr_ae (Filter.EventuallyEq.rfl.sub hgLp.coeFn_toLp)] |
| exact happrox |
|
|
| end Problem367 |
|
|
| end |
|
|
| end OAI |
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|
