File size: 2,749 Bytes
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 47 48 49 50 51 52 53 54 55 | import OAI.Probability.IsingPerceptron.PartitionPowLe
namespace OAI
/-! Passing the Gaussian variance bound from finite leaf cutoffs to their limit. -/
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped Topology ENNReal
namespace InvariantIsing
/-- Fatou's lemma for centered squares. This permits finite-leaf Gaussian
variance estimates to pass to the countable cascade prior. -/
theorem variance_le_of_ae_tendsto_of_mean_tendsto {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} [IsProbabilityMeasure μ] {f : ℕ → Ω → ℝ} {g : Ω → ℝ} {C : ℝ}
(hf : ∀ n, MemLp (f n) 2 μ) (hg : MemLp g 2 μ)
(ht : ∀ᵐ ω ∂μ, Tendsto (fun n => f n ω) atTop (𝓝 (g ω)))
(hm : Tendsto (fun n => ∫ ω, f n ω ∂μ) atTop (𝓝 (∫ ω, g ω ∂μ)))
(hv : ∀ n, variance (f n) μ ≤ C) : variance g μ ≤ C := by
let F := fun n ω => (f n ω - ∫ z, f n z ∂μ) ^ 2
let G := fun ω => (g ω - ∫ z, g z ∂μ) ^ 2
have hFi (n : ℕ) : Integrable (F n) μ :=
((hf n).sub (memLp_const _)).integrable_sq
have hGi : Integrable G μ := (hg.sub (memLp_const _)).integrable_sq
have hlim : ∀ᵐ ω ∂μ, Tendsto (fun n => ENNReal.ofReal (F n ω)) atTop (𝓝 (ENNReal.ofReal (G ω))) := by
filter_upwards [ht] with ω hω
exact ENNReal.continuous_ofReal.continuousAt.tendsto.comp ((hω.sub hm).pow 2)
have hfatou : (∫⁻ ω, ENNReal.ofReal (G ω) ∂μ) ≤
liminf (fun n => ∫⁻ ω, ENNReal.ofReal (F n ω) ∂μ) atTop := by
calc
_ = ∫⁻ ω, liminf (fun n => ENNReal.ofReal (F n ω)) atTop ∂μ :=
lintegral_congr_ae (hlim.mono fun _ hω => hω.liminf_eq.symm)
_ ≤ _ := lintegral_liminf_le' (fun n => (hFi n).aemeasurable.ennreal_ofReal)
have hpoint (n : ℕ) : (∫⁻ ω, ENNReal.ofReal (F n ω) ∂μ) ≤ ENNReal.ofReal C := by
rw [← ofReal_integral_eq_lintegral_ofReal (hFi n) (ae_of_all _ fun _ => sq_nonneg _)]
change ENNReal.ofReal (∫ ω, (f n ω - ∫ z, f n z ∂μ) ^ 2 ∂μ) ≤ _
rw [← variance_eq_integral (hf n).aemeasurable]
exact ENNReal.ofReal_le_ofReal (hv n)
have hb : liminf (fun n => ∫⁻ ω, ENNReal.ofReal (F n ω) ∂μ) atTop ≤ ENNReal.ofReal C := by
refine liminf_le_of_le (by isBoundedDefault) (fun b hb => ?_)
obtain ⟨n, hn⟩ := hb.exists
exact hn.trans (hpoint n)
rw [← ofReal_integral_eq_lintegral_ofReal hGi (ae_of_all _ fun _ => sq_nonneg _)] at hfatou
change ENNReal.ofReal (∫ ω, (g ω - ∫ z, g z ∂μ) ^ 2 ∂μ) ≤ _ at hfatou
rw [← variance_eq_integral hg.aemeasurable] at hfatou
exact (ENNReal.ofReal_le_ofReal_iff ((variance_nonneg (f 0) μ).trans (hv 0))).mp (hfatou.trans hb)
end InvariantIsing
end
end OAI
|