File size: 1,481 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 | 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
|