File size: 2,457 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 56 57 58 | import OAI.Probability.InvariantIsing.Pressure.RandomUniformIntegrability
import OAI.Probability.InvariantIsing.Pressure.ThermalLimitSqueeze
namespace OAI
/-! A soft-maximum squeeze for convergence in probability. -/
noncomputable section
open MeasureTheory Filter Set
open scoped Topology
namespace InvariantIsing
lemma tendstoInMeasure_continuous_comp {Ω : Type*} [MeasurableSpace Ω]
(P : Measure Ω) [IsFiniteMeasure P] (X : ℕ → Ω → ℝ) (L : ℝ)
(hX : ∀ n, Measurable (X n)) (h : TendstoInMeasure P X atTop (fun _ => L))
(f : ℝ → ℝ) (hf : Continuous f) :
TendstoInMeasure P (fun n ω => f (X n ω)) atTop (fun _ => f L) := by
apply (exists_seq_tendstoInMeasure_atTop_iff
(fun n => (hf.measurable.comp (hX n)).aestronglyMeasurable)).mpr
intro ns hns
obtain ⟨u,hu,he⟩ := (h.comp hns.tendsto_atTop).exists_seq_tendsto_ae
refine ⟨u,hu,he.mono fun ω hω => (hf.tendsto L).comp hω⟩
lemma thermal_squeeze_in_measure {Ω : Type*} [MeasurableSpace Ω]
(P : Measure Ω) (X : ℕ → ℕ → Ω → ℝ) (G : ℕ → Ω → ℝ)
(v : ℕ → ℝ) (L c : ℝ) (hc : 0 ≤ c)
(hv : Tendsto v atTop (𝓝 L))
(hp : ∀ k, TendstoInMeasure P (X k) atTop (fun _ => v k))
(hb : ∀ k n ω, X k n ω ≤ G n ω ∧ G n ω ≤ X k n ω+c/(k+1)) :
TendstoInMeasure P G atTop (fun _ => L) := by
rw [tendstoInMeasure_iff_norm]
intro ε hε
have hd : Tendsto (fun k : ℕ => c/(k+1)) atTop (𝓝 0) := by
simpa only [mul_one_div,mul_zero] using
(tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul c
have hvd : Tendsto (fun k => |v k-L|+c/(k+1)) atTop (𝓝 0) := by
simpa using ((hv.sub_const L).abs.add hd)
obtain ⟨k,hk⟩ := (hvd.eventually (gt_mem_nhds (half_pos hε))).exists
have hprob := (tendstoInMeasure_iff_norm.mp (hp k)) (ε/2) (half_pos hε)
apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hprob
(Filter.Eventually.of_forall fun _ => bot_le)
(Filter.Eventually.of_forall fun n => measure_mono ?_)
intro ω hω
simp only [mem_ofPred_eq,Real.norm_eq_abs] at hω ⊢
by_contra! hbad
have hB := hb k n ω
have hc' : 0 ≤ c/(k+1) := by positivity
have hG : |G n ω-L| ≤ |X k n ω-v k|+|v k-L|+c/(k+1) := by
rw [abs_le]
constructor <;> linarith [neg_abs_le (X k n ω-v k),le_abs_self (X k n ω-v k),
neg_abs_le (v k-L),le_abs_self (v k-L)]
linarith
end InvariantIsing
end
end OAI
|