| import Mathlib.Analysis.SpecialFunctions.Exp |
| import Mathlib.Data.Set.Countable |
| import Mathlib.MeasureTheory.Measure.Dirac.Basic |
| import Mathlib.Dynamics.Ergodic.Ergodic |
| import Mathlib.Dynamics.Ergodic.Conservative |
|
|
| namespace OAI |
|
|
| open MeasureTheory Set Filter |
| open scoped BigOperators |
|
|
| namespace InvariantProjection.PolarStructure |
|
|
| theorem countable_finite_drop_values {ι : Type*} [Countable ι] (c : ι → ℝ) : |
| (Set.range (fun s : Finset ι => Real.exp (-(∑ i ∈ s, c i)))).Countable := |
| Set.countable_range _ |
|
|
| theorem ae_countable_range_of_finite_drops {X ι : Type*} [MeasurableSpace X] |
| [Countable ι] {μ : Measure X} (c : ι → ℝ) (f : X → ℝ) |
| (hf : ∀ᵐ x ∂μ, ∃ s : Finset ι, f x = Real.exp (-(∑ i ∈ s, c i))) : |
| ∃ S : Set ℝ, S.Countable ∧ ∀ᵐ x ∂μ, f x ∈ S := by |
| refine ⟨Set.range (fun s : Finset ι => Real.exp (-(∑ i ∈ s, c i))), |
| countable_finite_drop_values c, ?_⟩ |
| filter_upwards [hf] with x hx |
| obtain ⟨s, hs⟩ := hx |
| exact ⟨s, hs.symm⟩ |
|
|
| theorem ae_countable_joint_range {X I Y : Type*} [MeasurableSpace X] [Finite I] |
| {μ : Measure X} (f : I → X → Y) |
| (hf : ∀ i, ∃ S : Set Y, S.Countable ∧ ∀ᵐ x ∂μ, f i x ∈ S) : |
| ∃ R : Set (I → Y), R.Countable ∧ ∀ᵐ x ∂μ, (fun i => f i x) ∈ R := by |
| classical |
| choose S hS using hf |
| refine ⟨{v | ∀ i, v i ∈ S i}, Set.countable_pi (fun i => (hS i).1), ?_⟩ |
| exact ae_all_iff.2 (fun i => (hS i).2) |
|
|
| theorem ae_countable_joint_range_of_measurePreserving |
| {X I Y : Type*} [MeasurableSpace X] [Finite I] {μ : Measure X} |
| (σ : I → X → X) (hσ : ∀ i, MeasurePreserving (σ i) μ μ) (f : X → Y) |
| (hf : ∃ S : Set Y, S.Countable ∧ ∀ᵐ x ∂μ, f x ∈ S) : |
| ∃ R : Set (I → Y), R.Countable ∧ ∀ᵐ x ∂μ, (fun i => f (σ i x)) ∈ R := by |
| apply ae_countable_joint_range |
| intro i |
| obtain ⟨S, hS, hf⟩ := hf |
| exact ⟨S, hS, (hσ i).quasiMeasurePreserving.ae hf⟩ |
|
|
| theorem ae_nonzero_fiber_of_ae_countable_range |
| {X Y : Type*} [MeasurableSpace X] {μ : Measure X} {f : X → Y} |
| {S : Set Y} (hS : S.Countable) (hf : ∀ᵐ x ∂μ, f x ∈ S) : |
| ∀ᵐ x ∂μ, μ (f ⁻¹' {f x}) ≠ 0 := by |
| have := hS.to_subtype |
| have hnull : ∀ y : S, ∀ᵐ x ∂μ, μ (f ⁻¹' {(y : Y)}) = 0 → f x ≠ y := by |
| intro y |
| by_cases hy : μ (f ⁻¹' {(y : Y)}) = 0 |
| · filter_upwards [measure_eq_zero_iff_ae_notMem.1 hy] with x hx |
| intro _ |
| simpa using hx |
| · exact Filter.Eventually.of_forall (fun _ h => (hy h).elim) |
| filter_upwards [hf, ae_all_iff.2 hnull] with x hx hforall |
| exact fun hzero => hforall ⟨f x, hx⟩ hzero rfl |
| |
| theorem measure_eq_sum_dirac_of_ae_mem_countable |
| {Y : Type*} [MeasurableSpace Y] [MeasurableSingletonClass Y] |
| {ν : Measure Y} {S : Set Y} (hS : S.Countable) (hν : ∀ᵐ y ∂ν, y ∈ S) : |
| ν = Measure.sum (fun y : S => ν {(y : Y)} • Measure.dirac (y : Y)) := by |
| classical |
| have := hS.to_subtype |
| ext t ht |
| rw [Measure.sum_apply _ ht] |
| calc |
| ν t = ν (S ∩ t) := (Measure.measure_inter_eq_of_ae hν).symm |
| _ = ν (⋃ y : S, t ∩ {(y : Y)}) := by |
| congr 1 |
| ext y |
| simp only [Set.mem_inter_iff, Set.mem_iUnion, Set.mem_singleton_iff] |
| constructor |
| · rintro ⟨hyS, hyt⟩ |
| exact ⟨⟨y, hyS⟩, hyt, rfl⟩ |
| · rintro ⟨z, hyt, rfl⟩ |
| exact ⟨z.property, hyt⟩ |
| _ = ∑' y : S, ν (t ∩ {(y : Y)}) := by |
| apply measure_iUnion |
| · intro y z hyz |
| apply Set.disjoint_left.2 |
| intro x hx hy |
| simp only [Set.mem_inter_iff, Set.mem_singleton_iff] at hx hy |
| exact hyz (Subtype.ext (hx.2.symm.trans hy.2)) |
| · intro y |
| exact ht.inter (measurableSet_singleton _) |
| _ = ∑' y : S, (ν {(y : Y)} • Measure.dirac (y : Y)) t := by |
| apply tsum_congr |
| intro y |
| by_cases hy : (y : Y) ∈ t |
| · simp [Measure.smul_apply, ht, hy, |
| Set.inter_eq_right.mpr (Set.singleton_subset_iff.2 hy)] |
| · have hdisj : t ∩ {(y : Y)} = ∅ := by |
| ext x |
| simp only [Set.mem_inter_iff, Set.mem_singleton_iff, Set.mem_empty_iff_false, |
| iff_false, not_and] |
| intro hx hxy |
| exact hy (hxy ▸ hx) |
| simp [Measure.smul_apply, ht, hy, hdisj] |
| |
| theorem map_eq_sum_dirac_of_ae_countable_range |
| {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] [MeasurableSingletonClass Y] |
| {μ : Measure X} {f : X → Y} (hf : AEMeasurable f μ) |
| {S : Set Y} (hS : S.Countable) (h : ∀ᵐ x ∂μ, f x ∈ S) : |
| μ.map f = Measure.sum (fun y : S => μ (f ⁻¹' {(y : Y)}) • Measure.dirac (y : Y)) := by |
| have hmap : ∀ᵐ y ∂μ.map f, y ∈ S := (ae_map_iff hf hS.measurableSet).2 h |
| rw [measure_eq_sum_dirac_of_ae_mem_countable hS hmap] |
| congr 1 |
| funext y |
| rw [Measure.map_apply_of_aemeasurable hf (measurableSet_singleton _)] |
|
|
| theorem mem_iff_of_eq_of_measurable_comap |
| {X Y : Type*} [MeasurableSpace Y] (f : X → Y) {t : Set X} |
| (ht : MeasurableSet[MeasurableSpace.comap f inferInstance] t) |
| {x z : X} (hxz : f x = f z) : x ∈ t ↔ z ∈ t := by |
| obtain ⟨B, -, rfl⟩ := MeasurableSpace.measurableSet_comap.1 ht |
| change f x ∈ B ↔ f z ∈ B |
| rw [hxz] |
|
|
| theorem fiber_is_atom_in_comap |
| {X Y : Type*} [MeasurableSpace Y] (f : X → Y) (y : Y) |
| {t : Set X} (ht : MeasurableSet[MeasurableSpace.comap f inferInstance] t) |
| (htsub : t ⊆ f ⁻¹' {y}) : t = ∅ ∨ t = f ⁻¹' {y} := by |
| by_cases ht0 : t = ∅ |
| · exact Or.inl ht0 |
| · right |
| obtain ⟨x, hx⟩ := Set.nonempty_iff_ne_empty.2 ht0 |
| apply Set.Subset.antisymm htsub |
| intro z hz |
| have hxy : f x = y := htsub hx |
| have hzy : f z = y := hz |
| exact (mem_iff_of_eq_of_measurable_comap f ht (hxy.trans hzy.symm)).1 hx |
|
|
| theorem countable_atomic_fiber_partition |
| {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] |
| [MeasurableSingletonClass Y] {μ : Measure X} {f : X → Y} |
| (hf : Measurable f) {S : Set Y} (hS : S.Countable) |
| (hrange : ∀ᵐ x ∂μ, f x ∈ S) : |
| ∃ C : Set Y, C.Countable ∧ (∀ᵐ x ∂μ, f x ∈ C) ∧ |
| (∀ y ∈ C, MeasurableSet (f ⁻¹' {y}) ∧ μ (f ⁻¹' {y}) ≠ 0 ∧ |
| ∀ t : Set X, MeasurableSet[MeasurableSpace.comap f inferInstance] t → |
| t ⊆ f ⁻¹' {y} → t = ∅ ∨ t = f ⁻¹' {y}) := by |
| refine ⟨{y ∈ S | μ (f ⁻¹' {y}) ≠ 0}, hS.mono (fun _ h => h.1), ?_, ?_⟩ |
| · filter_upwards [hrange, ae_nonzero_fiber_of_ae_countable_range hS hrange] with x hx hpos |
| exact ⟨hx, hpos⟩ |
| · intro y hy |
| exact ⟨hf (measurableSet_singleton y), hy.2, fun _ ht hsub => |
| fiber_is_atom_in_comap f y ht hsub⟩ |
| |
| theorem finite_family_atomic_fiber_partition |
| {X I Y : Type*} [MeasurableSpace X] [Finite I] [MeasurableSpace Y] |
| [MeasurableSingletonClass Y] {μ : Measure X} (f : I → X → Y) |
| (hf : ∀ i, Measurable (f i)) |
| (hrange : ∀ i, ∃ S : Set Y, S.Countable ∧ ∀ᵐ x ∂μ, f i x ∈ S) : |
| ∃ C : Set (I → Y), C.Countable ∧ (∀ᵐ x ∂μ, (fun i => f i x) ∈ C) ∧ |
| (∀ v ∈ C, MeasurableSet {x | (fun i => f i x) = v} ∧ |
| μ {x | (fun i => f i x) = v} ≠ 0 ∧ |
| ∀ t : Set X, MeasurableSet[⨆ i, MeasurableSpace.comap (f i) inferInstance] t → |
| t ⊆ {x | (fun i => f i x) = v} → |
| t = ∅ ∨ t = {x | (fun i => f i x) = v}) := by |
| obtain ⟨S, hS, hvalue⟩ := ae_countable_joint_range f hrange |
| obtain ⟨C, hC, hcover, hatom⟩ := countable_atomic_fiber_partition |
| (Measurable.of_eval hf) hS hvalue |
| refine ⟨C, hC, hcover, ?_⟩ |
| intro v hv |
| obtain ⟨hm, hp, hsmall⟩ := hatom v hv |
| refine ⟨hm, hp, ?_⟩ |
| intro t ht hsub |
| apply hsmall t _ hsub |
| rwa [MeasurableSpace.comap_process_pi] |
| |
| theorem not_ae_invariant_of_totalErgodic |
| {X : Type*} [MeasurableSpace X] {μ : Measure X} [IsProbabilityMeasure μ] |
| {σ : X → X} (hσ : ∀ n : ℕ, 0 < n → Ergodic σ^[n] μ) |
| {s : Set X} (hs : NullMeasurableSet s μ) (hs0 : μ s ≠ 0) (hs1 : μ s ≠ 1) |
| {n : ℕ} (hn : 0 < n) : ¬ (σ^[n] ⁻¹' s =ᵐ[μ] s) := by |
| intro heq |
| rcases (hσ n hn).quasiErgodic.ae_empty_or_univ₀ hs heq with h | h |
| · exact hs0 (by simpa using measure_congr h) |
| · exact hs1 (by simpa using measure_congr h) |
|
|
| theorem exists_positive_proper_orbit_overlap |
| {X : Type*} [MeasurableSpace X] {μ : Measure X} [IsProbabilityMeasure μ] |
| {σ : X → X} (hσ : ∀ n : ℕ, 0 < n → Ergodic σ^[n] μ) |
| {s : Set X} (hs : NullMeasurableSet s μ) (hs0 : μ s ≠ 0) (hs1 : μ s ≠ 1) : |
| ∃ n : ℕ, 0 < n ∧ μ (s ∩ σ^[n] ⁻¹' s) ≠ 0 ∧ μ (s \ σ^[n] ⁻¹' s) ≠ 0 := by |
| have hpres : MeasurePreserving σ μ μ := by |
| simpa using (hσ 1 (by omega)).toMeasurePreserving |
| obtain ⟨n, hn, hinter⟩ := |
| hpres.conservative.exists_gt_measure_inter_ne_zero hs hs0 0 |
| refine ⟨n, hn, hinter, ?_⟩ |
| intro hdiff |
| have hle : s ≤ᵐ[μ] σ^[n] ⁻¹' s := ae_le_set.2 hdiff |
| rcases (hσ n hn).ae_empty_or_univ_of_ae_le_preimage hs hle with h | h |
| · exact hs0 (by simpa using measure_congr h) |
| · exact hs1 (by simpa using measure_congr h) |
| |
| theorem measure_eq_one_of_atom_orbit |
| {X : Type*} [MeasurableSpace X] {μ : Measure X} [IsProbabilityMeasure μ] |
| {σ : X → X} (hσ : ∀ n : ℕ, 0 < n → Ergodic σ^[n] μ) |
| {s : Set X} (hs : NullMeasurableSet s μ) (hs0 : μ s ≠ 0) |
| (horbit : ∀ n : ℕ, 0 < n → |
| (σ^[n] ⁻¹' s =ᵐ[μ] s) ∨ AEDisjoint μ s (σ^[n] ⁻¹' s)) : μ s = 1 := by |
| by_contra hs1 |
| obtain ⟨n, hn, hinter, -⟩ := exists_positive_proper_orbit_overlap hσ hs hs0 hs1 |
| rcases horbit n hn with heq | hdisj |
| · exact not_ae_invariant_of_totalErgodic hσ hs hs0 hs1 hn heq |
| · exact hinter hdisj |
| |
| theorem invariant_family_has_no_measure_atoms |
| {X : Type*} [MeasurableSpace X] {μ : Measure X} [IsProbabilityMeasure μ] |
| {σ : X → X} (hσ : ∀ n : ℕ, 0 < n → Ergodic σ^[n] μ) |
| (A : Set (Set X)) (hmeas : ∀ s ∈ A, NullMeasurableSet s μ) |
| (hinter : ∀ s ∈ A, ∀ t ∈ A, s ∩ t ∈ A) |
| (hpre : ∀ s ∈ A, σ ⁻¹' s ∈ A) |
| (hnontrivial : ∃ b ∈ A, μ b ≠ 0 ∧ μ b ≠ 1) : |
| ∀ s ∈ A, μ s ≠ 0 → ∃ t ∈ A, t ⊆ s ∧ μ t ≠ 0 ∧ μ (s \ t) ≠ 0 := by |
| intro s hs hs0 |
| by_cases hs1 : μ s = 1 |
| · obtain ⟨b, hb, hb0, hb1⟩ := hnontrivial |
| have hsfull : ∀ᵐ x ∂μ, x ∈ s := by |
| exact ae_iff.2 ((prob_compl_eq_zero_iff₀ (hmeas s hs)).2 hs1) |
| refine ⟨s ∩ b, hinter s hs b hb, Set.inter_subset_left, ?_, ?_⟩ |
| · rwa [Measure.measure_inter_eq_of_ae hsfull] |
| · have hset : s \ (s ∩ b) = s ∩ bᶜ := by |
| ext x |
| simp only [Set.mem_sdiff, Set.mem_inter_iff, Set.mem_compl_iff] |
| tauto |
| rw [hset, Measure.measure_inter_eq_of_ae hsfull] |
| exact fun h => hb1 ((prob_compl_eq_zero_iff₀ (hmeas b hb)).1 h) |
| · obtain ⟨n, hn, hn0, hndiff⟩ := |
| exists_positive_proper_orbit_overlap hσ (hmeas s hs) hs0 hs1 |
| have hiter : ∀ k : ℕ, σ^[k] ⁻¹' s ∈ A := by |
| intro k |
| induction k with |
| | zero => simpa using hs |
| | succ k ih => |
| simpa only [Function.iterate_succ, Set.preimage_comp] using hpre _ ih |
| refine ⟨s ∩ σ^[n] ⁻¹' s, hinter s hs _ (hiter n), Set.inter_subset_left, hn0, ?_⟩ |
| have hset : s \ (s ∩ σ^[n] ⁻¹' s) = s \ σ^[n] ⁻¹' s := by |
| ext x |
| simp only [Set.mem_sdiff, Set.mem_inter_iff] |
| tauto |
| rwa [hset] |
|
|
| end InvariantProjection.PolarStructure |
|
|
| end OAI |
|
|
