File size: 3,364 Bytes
f1f7953 | 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 73 74 | import OAI.Analysis.IrrationalRotation.Sampling.CharacterLawConvergence
import OAI.Analysis.IrrationalRotation.Analysis.WeakConvergenceAEContinuous
namespace OAI
noncomputable section
namespace Invariant233.CharacterLaws
open MeasureTheory Filter
open scoped Topology
theorem tendsto_integral_character_phase_of_ae_continuous
{Z : Type*} [PseudoEMetricSpace Z] [MeasurableSpace Z] [BorelSpace Z]
(q : ℕ → ℕ → ℕ) (m : ℕ)
(hq : ∀ j i, 0 < q j i) (hmono : ∀ j, Monotone (q j))
(hgap : ∀ i, i + 1 < m → Tendsto
(fun j => (q j (i + 1) : ℝ) / (q j i : ℝ)) atTop atTop)
{a : ℕ → Z} {b : Z} (ha : Tendsto a atTop (𝓝 b))
{f : Z × UnitAddTorus (Fin m) → ℝ} (hf : Measurable f)
(hc : ∀ᵐ y : UnitAddTorus (Fin m), ContinuousAt f (b, y))
{B : ℝ} (hB : 0 ≤ B) (hb : ∀ z, |f z| ≤ B) :
Tendsto (fun j => ∫ x : AddCircle (1 : ℝ),
f (a j, fun i : Fin m => (q j i.val : ℤ) • x)) atTop
(𝓝 (∫ y : UnitAddTorus (Fin m), f (b, y))) := by
have hclaw : ∀ᵐ z ∂(haarPhaseLaw m b : Measure _), ContinuousAt f z := by
change ∀ᵐ z ∂Measure.map (fun y : UnitAddTorus (Fin m) => (b, y)) volume,
ContinuousAt f z
apply (ae_map_iff
(by fun_prop : Continuous (fun y : UnitAddTorus (Fin m) => (b, y))).measurable.aemeasurable
(measurableSet_of_continuousAt f)).mpr
exact hc
have h := WeakConvergenceAEContinuous.tendsto_integral_of_ae_continuous
(tendsto_characterPhaseLaw q m hq hmono hgap ha) hf hclaw hB hb
have hmap (j : ℕ) :
(∫ z, f z ∂(characterPhaseLaw (fun i : Fin m => q j i.val) (a j) : Measure _)) =
∫ x : AddCircle (1 : ℝ), f (a j, fun i : Fin m => (q j i.val : ℤ) • x) := by
change (∫ z, f z ∂Measure.map
(fun x => (a j, characterMap (fun i : Fin m => q j i.val) x)) volume) = _
rw [integral_map
(by fun_prop : Continuous
(fun x => (a j, characterMap (fun i : Fin m => q j i.val) x))).measurable.aemeasurable
hf.aestronglyMeasurable]
rfl
have hlim : (∫ z, f z ∂(haarPhaseLaw m b : Measure _)) =
∫ y : UnitAddTorus (Fin m), f (b, y) := by
change (∫ z, f z ∂Measure.map (fun y : UnitAddTorus (Fin m) => (b, y)) volume) = _
rw [integral_map
(by fun_prop : Continuous (fun y : UnitAddTorus (Fin m) => (b, y))).measurable.aemeasurable
hf.aestronglyMeasurable]
simpa only [hmap, hlim] using h
theorem eventually_integral_character_phase_lt
{Z : Type*} [PseudoEMetricSpace Z] [MeasurableSpace Z] [BorelSpace Z]
(q : ℕ → ℕ → ℕ) (m : ℕ)
(hq : ∀ j i, 0 < q j i) (hmono : ∀ j, Monotone (q j))
(hgap : ∀ i, i + 1 < m → Tendsto
(fun j => (q j (i + 1) : ℝ) / (q j i : ℝ)) atTop atTop)
{a : ℕ → Z} {b : Z} (ha : Tendsto a atTop (𝓝 b))
{f : Z × UnitAddTorus (Fin m) → ℝ} (hf : Measurable f)
(hc : ∀ᵐ y : UnitAddTorus (Fin m), ContinuousAt f (b, y))
{B δ : ℝ} (hB : 0 ≤ B) (hb : ∀ z, |f z| ≤ B)
(hδ : (∫ y : UnitAddTorus (Fin m), f (b, y)) < δ) :
∀ᶠ j in atTop, (∫ x : AddCircle (1 : ℝ),
f (a j, fun i : Fin m => (q j i.val : ℤ) • x)) < δ :=
(tendsto_integral_character_phase_of_ae_continuous
q m hq hmono hgap ha hf hc hB hb).eventually_lt_const hδ
end Invariant233.CharacterLaws
end
end OAI
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