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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 | import Mathlib
import OAI.Probability.CriticalPercolation.LinearGroups.LocalProjectiveAction
namespace OAI
section
noncomputable section
open Set Function Filter MeasureTheory NormedSpace Module
open scoped Classical Topology ENNReal
namespace CriticalPercolation.LocalProjective
variable {K V : Type*} [NontriviallyNormedField K] [NormedAddCommGroup V] [NormedSpace K V]
variable [CompleteSpace K] [FiniteDimensional K V]
omit [CompleteSpace K] in
lemma small_intersection_rank (U W : Submodule K V)
(he : finrank K U=finrank K W) (hne : U≠W) :
finrank K ↥(U⊓W)<finrank K U := by
apply Submodule.finrank_lt_finrank_of_lt
refine lt_of_le_of_ne inf_le_left ?_
intro hi
have hle : U≤W := by rw [←hi]; exact inf_le_right
exact hne (Submodule.eq_of_le_of_finrank_eq hle he)
lemma finite_positive_subspaces (μ : Measure (Space K V)) [IsFiniteMeasure μ]
(d : ℕ) (ε : ℝ≥0∞) (hε : ε≠0)
(hnull : ∀ W : Submodule K V, finrank K W<d → μ (subspace W)=0) :
{W : Submodule K V | finrank K W=d ∧ ε≤μ (subspace W)}.Finite := by
let T := {W : Submodule K V | finrank K W=d ∧ ε≤μ (subspace W)}
by_contra hfin
have hinf : T.Infinite := hfin
let : Infinite T := hinf.to_subtype
have hdis : Pairwise (AEDisjoint μ on (fun W:T => subspace W.val)) := by
intro U W hne
change μ (subspace U.val∩subspace W.val)=0
rw [←subspace_inf]
apply hnull
rw [←U.property.1]
apply small_intersection_rank _ _ (U.property.1.trans W.property.1.symm)
exact fun he => hne (Subtype.ext he)
have hs := tsum_measure_le_measure_univ
(fun W:T => (measurable_subspace W.val).nullMeasurableSet) hdis
have hc : (∑' _ : T, ε)≤∑' W:T, μ (subspace W.val) :=
ENNReal.tsum_le_tsum (fun W => W.property.2)
rw [ENNReal.tsum_const_eq_top_of_ne_zero hε] at hc
exact (measure_ne_top μ Set.univ) (top_unique (hc.trans hs))
omit [CompleteSpace K] in
lemma exists_minimal_positive_subspace (μ : Measure (Space K V)) [IsProbabilityMeasure μ]
(h : ∃ W : Submodule K V, W≠⊤ ∧ μ (subspace W)≠0) :
∃ W : Submodule K V, W≠⊥ ∧ W≠⊤ ∧ μ (subspace W)≠0 ∧
∀ U : Submodule K V, finrank K U<finrank K W → μ (subspace U)=0 := by
have hex : ∃ d:ℕ, ∃ W : Submodule K V,
finrank K W=d ∧ W≠⊤ ∧ μ (subspace W)≠0 := by
obtain ⟨W,hW,hμ⟩ := h
exact ⟨finrank K W,W,rfl,hW,hμ⟩
obtain ⟨W,hWd,hWt,hWμ⟩ := Nat.find_spec hex
refine ⟨W,?_,hWt,hWμ,?_⟩
· intro he
exact hWμ (by simp [he])
· intro U hU
by_contra hUμ
have hut : U≠⊤ := by
intro he
have hh := Submodule.finrank_le W
rw [he,finrank_top] at hU
omega
have hi := Nat.find_min' hex ⟨U,rfl,hut,hUμ⟩
omega
lemma measure_subspace_map_eq (μ : Measure (Space K V))
(a : V ≃L[K] V) (hμ : Measure.map (action a) μ=μ)
(W : Submodule K V) :
μ (subspace (W.map a.toLinearEquiv.toLinearMap))=μ (subspace W) := by
conv_lhs => rw [←hμ]
rw [Measure.map_apply (continuous_action a).measurable
(measurable_subspace _),preimage_action_subspace]
lemma exists_finite_positive_subspace_orbit {G : Type*} [Group G]
(ρ : G →* (V ≃L[K] V)) (μ : Measure (Space K V)) [IsProbabilityMeasure μ]
(hμ : ∀ g, Measure.map (action (ρ g)) μ=μ)
(h : ∃ W : Submodule K V, W≠⊤ ∧ μ (subspace W)≠0) :
∃ W : Submodule K V, W≠⊥ ∧ W≠⊤ ∧
(Set.range (fun g => W.map (ρ g).toLinearEquiv.toLinearMap)).Finite := by
obtain ⟨W,hWb,hWt,hWμ,hnull⟩ := exists_minimal_positive_subspace μ h
refine ⟨W,hWb,hWt,?_⟩
apply (finite_positive_subspaces μ (finrank K W) _ hWμ hnull).subset
rintro U ⟨g,rfl⟩
refine ⟨(ρ g).toLinearEquiv.finrank_map_eq W,?_⟩
exact (measure_subspace_map_eq μ (ρ g) (hμ g) W).ge
end CriticalPercolation.LocalProjective
end
end
end OAI
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