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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 | import OAI.Probability.InvariantIsing.Spectral.CompactFiniteTestControl
namespace OAI
/-! Convergence in measure from separating continuous tests on a compact state space. -/
noncomputable section
open Set Filter MeasureTheory
open scoped Topology
namespace InvariantIsing
theorem compact_tests_tendstoInMeasure {X ι Y Ω : Type*} [TopologicalSpace X] [CompactSpace X]
[PseudoMetricSpace Y] [MeasurableSpace Ω] (P : Measure Ω)
(F : ι → X → ℝ) (hF : ∀ i, Continuous (F i))
(hsep : ∀ x y, (∀ i, F i x = F i y) → x = y)
(G : X → Y) (hG : Continuous G) (x₀ : X) (Z : ℕ → Ω → X)
(hZ : ∀ i, TendstoInMeasure P (fun n ω => F i (Z n ω)) atTop (fun _ => F i x₀)) :
TendstoInMeasure P (fun n ω => G (Z n ω)) atTop (fun _ => G x₀) := by
classical
apply tendstoInMeasure_iff_dist.mpr
intro ε hε
obtain ⟨s,δ,hδ,hcontrol⟩ := compact_finite_test_control F hF hsep G hG x₀ hε
have hb (n : ℕ) : {ω | ε ≤ dist (G (Z n ω)) (G x₀)} ⊆
⋃ i ∈ s, {ω | δ i ≤ dist (F i (Z n ω)) (F i x₀)} := by
intro ω hω
have hh : ∃ i ∈ s, δ i ≤ dist (F i (Z n ω)) (F i x₀) := by
by_contra! hh
exact (not_lt_of_ge hω) (hcontrol (Z n ω) hh)
obtain ⟨i,hi,hii⟩ := hh
exact mem_iUnion.mpr ⟨i,mem_iUnion.mpr ⟨hi,hii⟩⟩
have ht := tendsto_finsetSum s (fun i _ => (tendstoInMeasure_iff_dist.mp (hZ i)) (δ i) (hδ i))
have ht' : Tendsto (fun n => ∑ i ∈ s, P {ω | δ i ≤ dist (F i (Z n ω)) (F i x₀)})
atTop (𝓝 0) := by simpa only [Finset.sum_const_zero] using ht
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_biUnion_finset_le _ _))
end InvariantIsing
end
end OAI
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