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a89d282 | 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 | import OAI.Probability.InvariantIsing.Spectral.SpectralBallQuantizer
import OAI.Probability.InvariantIsing.Spectral.SpectralQuantizationWeak
namespace OAI
/-! Concrete finite quantizations of a compactly supported spectral law. -/
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set Metric
open scoped Topology
namespace InvariantIsing
theorem exists_compact_spectral_quantizers
(μ : ProbabilityMeasure ℝ) (K : Set ℝ) (hK : IsCompact K)
(hμ : ∀ᵐ x ∂(μ : Measure ℝ), x∈K) (a : ℝ) (ha : a∈K) :
∃ n : ℕ → ℕ, ∃ c r : (k : ℕ) → Fin (n k) → ℝ,
(∀ k i, c k i∈K) ∧
(∀ k i, 0 < r k i ∧ r k i < (1 : ℝ)/(k+1)) ∧
(∀ k i, (μ : Measure ℝ) (frontier (ball (c k i) (r k i)))=0) ∧
(∀ k, K ⊆ ⋃ i, ball (c k i) (r k i)) ∧
(∀ k, ∀ᵐ x ∂(μ : Measure ℝ),
dist (spectralBallQuantizer (c k) (r k) a x) x ≤ (1 : ℝ)/(k+1)) ∧
(∀ k x, spectralBallQuantizer (c k) (r k) a x∈K) ∧
Tendsto (fun k => μ.map (spectralBallQuantizer (c k) (r k) a)) atTop (𝓝 μ) := by
have hex (k : ℕ) := spectral_null_boundary_cover (μ : Measure ℝ) K hK ((1 : ℝ)/(k+1)) (by positivity)
choose n c r hc hr hn hb using hex
have hclose (k : ℕ) : ∀ᵐ x ∂(μ : Measure ℝ),
dist (spectralBallQuantizer (c k) (r k) a x) x ≤ (1 : ℝ)/(k+1) := by
filter_upwards [hμ] with x hx
exact spectralBallQuantizer_close (c k) (r k) a _ x (fun i => (hr k i).2.le) (hb k hx)
refine ⟨n,c,r,hc,?_,hn,hb,hclose,?_,?_⟩
· intro k i
exact ⟨lt_trans (by positivity : (0:ℝ) < ((1:ℝ)/(k+1))/2) (hr k i).1,(hr k i).2⟩
· intro k x
exact spectralBallQuantizer_mem (c k) (r k) a K (hc k) ha x
· exact spectral_quantization_weak μ (fun k => spectralBallQuantizer (c k) (r k) a)
(fun k => measurable_spectralBallQuantizer _ _ _)
(fun k => (1:ℝ)/(k+1)) (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)) hclose
end InvariantIsing
end
end OAI
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