openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Sampling /CharacterSamplingLimit.lean
| 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 | |