| 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 |
|
|
