| import OAI.Probability.IsingPerceptron.PartitionPowLe |
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| namespace OAI |
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| /-! Passing the Gaussian variance bound from finite leaf cutoffs to their limit. -/ |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory Filter |
| open scoped Topology ENNReal |
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| namespace InvariantIsing |
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| /-- 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) |
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| end InvariantIsing |
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| end |
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| end OAI |
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