openAI-math-dataset / lean /OAI /GroupTheory /PolycyclicRecognition /Analysis /CellOccupationError.lean
| import OAI.GroupTheory.PolycyclicRecognition.Algebra.OccupationWeakLimit | |
| import Mathlib.MeasureTheory.Integral.Bochner.Set | |
| namespace OAI | |
| /-! # Finite-cell errors for occupation observables | |
| The observable may vary with the scale. Only its common bound matters, | |
| so the estimate survives projection from nonunique factor coordinates. | |
| -/ | |
| namespace PolycyclicRecognition | |
| open MeasureTheory Set Filter | |
| open scoped Topology symmDiff | |
| /-- Changing a finite sampling cell changes a bounded observable integral | |
| by at most its bound times the measure of the symmetric difference. -/ | |
| theorem norm_setIntegral_sub_le_symmDiff | |
| {X E : Type*} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| (μ : Measure X) {s t : Set X} (hs : MeasurableSet s) (ht : MeasurableSet t) | |
| (hfs : μ s < ⊤) (hft : μ t < ⊤) | |
| (f : X → E) (hf : AEStronglyMeasurable f μ) | |
| {C : ℝ} (hC : 0 ≤ C) (hb : ∀ x, ‖f x‖ ≤ C) : | |
| ‖(∫ x in s, f x ∂μ) - ∫ x in t, f x ∂μ‖ ≤ C * (μ (s ∆ t)).toReal := by | |
| have his : IntegrableOn f s μ := | |
| IntegrableOn.of_bound hfs hf.restrict C (Eventually.of_forall hb) | |
| have hit : IntegrableOn f t μ := | |
| IntegrableOn.of_bound hft hf.restrict C (Eventually.of_forall hb) | |
| have hd : μ (s ∆ t) < ⊤ := lt_of_le_of_lt (measure_mono symmDiff_subset_union) | |
| ((measure_union_le s t).trans_lt (ENNReal.add_lt_top.mpr ⟨hfs, hft⟩)) | |
| have hic : IntegrableOn (fun _ : X => C) (s ∆ t) μ := | |
| IntegrableOn.of_bound hd aestronglyMeasurable_const C | |
| (Eventually.of_forall (fun _ => by simp [Real.norm_eq_abs, abs_of_nonneg hC])) | |
| rw [← integral_indicator hs, ← integral_indicator ht, | |
| ← integral_sub (his.integrable_indicator hs) (hit.integrable_indicator ht)] | |
| have hbound : ∀ x, ‖s.indicator f x - t.indicator f x‖ ≤ | |
| (s ∆ t).indicator (fun _ => C) x := by | |
| intro x | |
| by_cases hxs : x ∈ s <;> by_cases hxt : x ∈ t <;> | |
| simp [Set.indicator_of_mem, Set.indicator_of_notMem, hxs, hxt, | |
| Set.mem_symmDiff, hb x] | |
| calc | |
| _ ≤ ∫ x, (s ∆ t).indicator (fun _ => C) x ∂μ := | |
| norm_integral_le_of_norm_le (hic.integrable_indicator (hs.symmDiff ht)) | |
| (Eventually.of_forall hbound) | |
| _ = C * (μ (s ∆ t)).toReal := by | |
| rw [integral_indicator (hs.symmDiff ht), setIntegral_const] | |
| simp only [smul_eq_mul, measureReal_def, mul_comm] | |
| /-- A measure-preserving change of sampling coordinates has an error | |
| controlled solely by the symmetric difference of the finite cell. -/ | |
| theorem norm_cell_translation_integral_error_le | |
| {X : Type*} [MeasurableSpace X] (μ : Measure X) | |
| (T : X → X) (hT : MeasurePreserving T μ μ) | |
| {s : Set X} (hs : MeasurableSet s) (hfin : μ s < ⊤) | |
| (f : X → ℝ) (hf : AEStronglyMeasurable f μ) | |
| {C : ℝ} (hC : 0 ≤ C) (hb : ∀ x, ‖f x‖ ≤ C) : | |
| ‖(∫ x in s, f (T x) ∂μ) - ∫ x in s, f x ∂μ‖ ≤ | |
| C * (μ ((T ⁻¹' s) ∆ s)).toReal := by | |
| have hpre : μ (T ⁻¹' s) = μ s := hT.measure_preimage hs.nullMeasurableSet | |
| have heq : (∫ x in T ⁻¹' s, f (T x) ∂μ) = ∫ x in s, f x ∂μ := by | |
| have hfm : AEStronglyMeasurable f (Measure.map T μ) := by rwa [hT.map_eq] | |
| have hh := setIntegral_map hs hfm hT.measurable.aemeasurable | |
| rw [hT.map_eq] at hh | |
| exact hh.symm | |
| rw [← heq, norm_sub_rev] | |
| exact norm_setIntegral_sub_le_symmDiff μ (hT.measurable hs) hs | |
| (hpre ▸ hfin) hfin (fun x => f (T x)) | |
| (hf.comp_measurePreserving hT) hC (fun x => hb (T x)) | |
| /-- Vanishing boundary loss makes every uniformly bounded, scale-dependent | |
| sampling observable asymptotically translation invariant. -/ | |
| theorem cell_translation_integral_error_tendsto_zero | |
| {X ι : Type*} [MeasurableSpace X] (μ : Measure X) | |
| (T : ι → X → X) (hT : ∀ i, MeasurePreserving (T i) μ μ) | |
| {s : Set X} (hs : MeasurableSet s) (hfin : μ s < ⊤) | |
| (f : ι → X → ℝ) (hf : ∀ i, AEStronglyMeasurable (f i) μ) | |
| {C : ℝ} (hC : 0 ≤ C) (hb : ∀ i x, ‖f i x‖ ≤ C) | |
| {l : Filter ι} | |
| (hcell : Tendsto (fun i => μ (((T i) ⁻¹' s) ∆ s)) l (𝓝 0)) : | |
| Tendsto (fun i => (∫ x in s, f i (T i x) ∂μ) - ∫ x in s, f i x ∂μ) | |
| l (𝓝 0) := by | |
| have ht := (ENNReal.tendsto_toReal ENNReal.zero_ne_top).comp hcell | |
| have hlim : Tendsto (fun i => C * (μ (((T i) ⁻¹' s) ∆ s)).toReal) l (𝓝 0) := by | |
| simpa using tendsto_const_nhds.mul ht | |
| apply tendsto_zero_iff_norm_tendsto_zero.mpr | |
| exact squeeze_zero' (Eventually.of_forall (fun _ => norm_nonneg _)) | |
| (Eventually.of_forall (fun i => | |
| norm_cell_translation_integral_error_le μ (T i) (hT i) hs hfin | |
| (f i) (hf i) hC (hb i))) hlim | |
| end PolycyclicRecognition | |
| end OAI | |