File size: 2,887 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 59 60 61 62 63 64 65 66 67 68 69 70 71 72 | import OAI.Probability.InvariantIsing.Spectral.CompactTestsInMeasure
namespace OAI
/-! A vanishing metric perturbation preserves convergence in probability. -/
noncomputable section
open MeasureTheory Filter Set
open scoped Topology
namespace InvariantIsing
theorem tendstoInMeasure_zero_of_abs_le {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
(f g : ℕ → Ω → ℝ) (hg : TendstoInMeasure P g atTop (fun _ => 0))
(hbound : ∀ n ω, |f n ω| ≤ |g n ω|) :
TendstoInMeasure P f atTop (fun _ => 0) := by
apply tendstoInMeasure_iff_dist.mpr
intro ε hε
apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds
(tendstoInMeasure_iff_dist.mp hg ε hε) (fun _ => bot_le)
intro n
apply measure_mono
intro ω hω
simp only [mem_ofPred_eq,Real.dist_eq,sub_zero] at hω ⊢
exact hω.trans (hbound n ω)
theorem tendstoInMeasure_div_pos {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
(f : ℕ → Ω → ℝ) (hf : TendstoInMeasure P f atTop (fun _ => 0))
{c : ℝ} (hc : 0 < c) :
TendstoInMeasure P (fun n ω => f n ω/c) atTop (fun _ => 0) := by
apply tendstoInMeasure_iff_dist.mpr
intro ε hε
convert (tendstoInMeasure_iff_dist.mp hf) (ε*c) (mul_pos hε hc) using 1
funext n
congr 1
ext ω
simp only [Real.dist_eq,sub_zero,abs_div,abs_of_pos hc]
exact le_div_iff₀ hc
theorem tendstoInMeasure_of_dist_le {Ω Y : Type*} [MeasurableSpace Ω] [PseudoMetricSpace Y]
(P : Measure Ω) (f g : ℕ → Ω → Y) (h : ℕ → Ω → ℝ) (z : Ω → Y)
(hg : TendstoInMeasure P g atTop z) (hh : TendstoInMeasure P h atTop (fun _ => 0))
(hbound : ∀ n ω, dist (f n ω) (g n ω) ≤ h n ω) :
TendstoInMeasure P f atTop z := by
apply tendstoInMeasure_iff_dist.mpr
intro ε hε
have hhalf : 0 < ε/2 := half_pos hε
have ht₁ := tendstoInMeasure_iff_dist.mp hh (ε/2) hhalf
have ht₂ := tendstoInMeasure_iff_dist.mp hg (ε/2) hhalf
have hb (n : ℕ) : {ω | ε ≤ dist (f n ω) (z ω)} ⊆
{ω | ε/2 ≤ dist (h n ω) 0} ∪ {ω | ε/2 ≤ dist (g n ω) (z ω)} := by
intro ω hω
by_cases hh' : ε/2 ≤ dist (h n ω) 0
· exact Or.inl hh'
· right
have hp : h n ω < ε/2 := lt_of_le_of_lt (by simpa only [Real.dist_eq,sub_zero] using le_abs_self (h n ω))
(lt_of_not_ge hh')
have hx := dist_triangle (f n ω) (g n ω) (z ω)
have hb' := hbound n ω
change ε ≤ dist (f n ω) (z ω) at hω
change ε/2 ≤ dist (g n ω) (z ω)
linarith
have ht : Tendsto (fun n => P {ω | ε/2 ≤ dist (h n ω) 0}+
P {ω | ε/2 ≤ dist (g n ω) (z ω)}) atTop (𝓝 0) := by
simpa only [zero_add] using ht₁.add ht₂
exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds ht
(fun _ => bot_le) (fun n => (measure_mono (hb n)).trans (measure_union_le _ _))
end InvariantIsing
end
end OAI
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