File size: 5,265 Bytes
f205175 | 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 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 | import Mathlib
import OAI.Probability.CriticalPercolation.LinearGroups.LocalProjectiveAction
namespace OAI
section
noncomputable section
open Set Function Filter MeasureTheory NormedSpace Module Metric TopologicalSpace
open scoped Classical Topology ENNReal
namespace CriticalPercolation.LocalProjective
variable {K V : Type*} [NontriviallyNormedField K] [NormedAddCommGroup V] [NormedSpace K V]
variable [ProperSpace K] [FiniteDimensional K V]
lemma subspace_full_of_singular_limit
(μ : Measure (Space K V)) [IsProbabilityMeasure μ]
(a : ℕ → V ≃L[K] V) (c : ℕ → K) (hc : ∀ n, c n≠0)
(A : V →L[K] V) (hA : A≠0)
(hlim : Tendsto (fun n => c n • (a n).toContinuousLinearMap) atTop (𝓝 A))
(hμ : ∀ n, Measure.map (action (a n)) μ=μ)
(hker : μ (subspace A.toLinearMap.ker)=0) :
∀ᵐ q ∂μ, q∈subspace A.toLinearMap.range := by
let : MetricSpace (Space K V) := metrizableSpaceMetric _
let L : Set (Space K V) := subspace A.toLinearMap.range
have hL : IsClosed L := closed_subspace _
have hLne : L.Nonempty := by
have hex : ∃ x : V, A x≠0 := by
by_contra hn
push Not at hn
exact hA (ContinuousLinearMap.ext hn)
obtain ⟨x,hx⟩ := hex
exact ⟨point K V ⟨A x,hx⟩,(point_mem_subspace _ _).mpr ⟨x,rfl⟩⟩
let f : Space K V → ℝ := fun q => min 1 (infDist q L)
have hfcont : Continuous f := continuous_const.min (continuous_infDist_pt L)
have hfnonneg (q : Space K V) : 0≤f q := le_min zero_le_one infDist_nonneg
have hfbound (q : Space K V) : ‖f q‖≤1 := by
rw [Real.norm_eq_abs,abs_of_nonneg (hfnonneg q)]
exact min_le_left _ _
have hfi : Integrable f μ := (integrable_const (1:ℝ)).mono' hfcont.aestronglyMeasurable
(Filter.Eventually.of_forall hfbound)
have hx : ∀ᵐ q ∂μ, q∉subspace A.toLinearMap.ker := by
rw [ae_iff]
simpa only [not_not,Set.ofPred_mem_eq] using hker
have hpoint : ∀ᵐ q ∂μ, Tendsto (fun n => f (action (a n) q)) atTop (𝓝 0) := by
filter_upwards [hx] with q hq
obtain ⟨v,rfl⟩ := point_surjective K V q
have hv : A (v:V)≠0 := by
simpa only [point_mem_subspace,LinearMap.mem_ker,ContinuousLinearMap.coe_coe] using hq
let w : ℕ → NZ V := fun n => ⟨(c n • (a n).toContinuousLinearMap) (v:V),
smul_ne_zero (hc n) (nzImage (a n) v).property⟩
have hev : Tendsto (fun n => (c n • (a n).toContinuousLinearMap) (v:V)) atTop (𝓝 (A v)) :=
((continuous_id.clm_apply continuous_const).tendsto A).comp hlim
have hw : Tendsto w atTop (𝓝 ⟨A v,hv⟩) := tendsto_subtype_rng.mpr hev
have hp := (continuous_point K V).continuousAt.tendsto.comp hw
have he (n : ℕ) : point K V (w n)=action (a n) (point K V v) := by
rw [action_point,point_eq_iff]
exact ⟨c n,rfl⟩
change Tendsto (fun n => point K V (w n)) atTop (𝓝 (point K V ⟨A v,hv⟩)) at hp
simp_rw [he] at hp
have hf := hfcont.continuousAt.tendsto.comp hp
have hin : point K V ⟨A v,hv⟩∈L := (point_mem_subspace _ _).mpr ⟨v,rfl⟩
have hz : f (point K V ⟨A v,hv⟩)=0 := by
simp only [f,infDist_zero_of_mem hin,min_eq_right zero_le_one]
rw [hz] at hf
exact hf
have hconv := tendsto_integral_of_dominated_convergence (μ:=μ)
(F:=fun n q => f (action (a n) q)) (f:=fun _ => (0:ℝ)) (fun _ => (1:ℝ))
(fun n => (hfcont.comp (continuous_action (a n))).aestronglyMeasurable)
(integrable_const 1) (fun n => Filter.Eventually.of_forall (fun q => hfbound _)) hpoint
have he (n : ℕ) : (∫ q, f (action (a n) q) ∂μ)=∫ q, f q ∂μ := by
rw [←integral_map (continuous_action (a n)).measurable.aemeasurable hfcont.aestronglyMeasurable,
hμ n]
simp_rw [he] at hconv
have hint : (∫ q, f q ∂μ)=0 := by
exact tendsto_nhds_unique tendsto_const_nhds (by simpa only [integral_zero] using hconv)
have hz := (integral_eq_zero_iff_of_nonneg hfnonneg hfi).mp hint
filter_upwards [hz] with q hq
apply (hL.mem_iff_infDist_zero hLne).mpr
change min 1 (infDist q L)=0 at hq
rcases min_choice (1:ℝ) (infDist q L) with he|he
· rw [he] at hq
norm_num at hq
· exact he.symm.trans hq
lemma injective_of_nonzero_projective_limit
(μ : Measure (Space K V)) [IsProbabilityMeasure μ]
(hn : ∀ W : Submodule K V, W≠⊤ → μ (subspace W)=0)
(a : ℕ → V ≃L[K] V) (c : ℕ → K) (hc : ∀ n, c n≠0)
(A : V →L[K] V) (hA : A≠0)
(hlim : Tendsto (fun n => c n • (a n).toContinuousLinearMap) atTop (𝓝 A))
(hμ : ∀ n, Measure.map (action (a n)) μ=μ) : Injective A := by
have hk : A.toLinearMap.ker≠⊤ := by
intro he
apply hA
ext x
have hx : x∈A.toLinearMap.ker := he ▸ Submodule.mem_top
exact hx
have hae := subspace_full_of_singular_limit μ a c hc A hA hlim hμ (hn _ hk)
have hr : A.toLinearMap.range=⊤ := by
by_contra hr
have hnot : ∀ᵐ q ∂μ, q∉subspace A.toLinearMap.range := by
rw [ae_iff]
simpa only [not_not,Set.ofPred_mem_eq] using hn _ hr
obtain ⟨q,hq,hq'⟩ := (hae.and hnot).exists
exact hq' hq
exact LinearMap.injective_iff_surjective.mpr (LinearMap.range_eq_top.mp hr)
end CriticalPercolation.LocalProjective
end
end
end OAI
|