| import OAI.GroupTheory.Thompson.InvariantMean | |
| import OAI.GroupTheory.Thompson.ProbabilityDefects | |
| import Mathlib.Analysis.LocallyConvex.Separation | |
| import Mathlib.Analysis.LocallyConvex.WithSeminorms | |
| import Mathlib.Analysis.Convex.Topology | |
| namespace OAI | |
| /-! | |
| # Finite probabilities approximately invariant under a finite set | |
| The usual positive normalized invariant mean forces zero to lie in the | |
| closure of the actual finite-probability defect set. The separation argument | |
| uses bounded coordinate functions of a continuous linear functional on ℓ¹; | |
| no identification of the full dual of ℓ¹ is assumed. | |
| -/ | |
| noncomputable section | |
| open scoped ENNReal BigOperators | |
| universe uG | |
| namespace ThompsonNonamenability | |
| variable {G : Type uG} [Group G] [DecidableEq G] | |
| local instance : Fact (1 ≤ (1 : ℝ≥0∞)) := ⟨le_rfl⟩ | |
| /-- A separator gives one bounded real function for each translation. -/ | |
| def separatorCoordinate {S : Finset G} (ell : DefectSpace S →L[ℝ] ℝ) | |
| (h : {h : G // h ∈ S}) : BoundedReal G := | |
| ⟨fun g => ell (lp.single 1 (h, g) 1), memℓp_infty ⟨‖ell‖, by | |
| rintro y ⟨g, rfl⟩ | |
| have hb := ell.le_opNorm (lp.single 1 (h, g) 1) | |
| simpa only [lp.norm_single (by norm_num : 0 < (1 : ℝ≥0∞)), | |
| norm_one, mul_one] using hb⟩⟩ | |
| omit [Group G] in | |
| @[simp] | |
| theorem separatorCoordinate_apply [Group G] {S : Finset G} | |
| (ell : DefectSpace S →L[ℝ] ℝ) (h : {h : G // h ∈ S}) (g : G) : | |
| separatorCoordinate ell h g = ell (lp.single 1 (h, g) 1) := rfl | |
| /-- The finite sum whose mean vanishes by left invariance. -/ | |
| def separatorDefect {S : Finset G} (ell : DefectSpace S →L[ℝ] ℝ) : BoundedReal G := | |
| ∑ h : {h : G // h ∈ S}, | |
| (leftPull h.val (separatorCoordinate ell h) - separatorCoordinate ell h) | |
| theorem separatorDefect_apply {S : Finset G} (ell : DefectSpace S →L[ℝ] ℝ) | |
| (g : G) : separatorDefect ell g = ell (diracDefect S g) := by | |
| classical | |
| cases Subsingleton.elim (inferInstance : DecidableEq G) (Classical.decEq G) | |
| simp only [separatorDefect, lp.coeFn_sum, Finset.sum_apply, | |
| lp.coeFn_sub, Pi.sub_apply, leftPull_apply, separatorCoordinate_apply, | |
| diracDefect, map_sum, map_sub] | |
| theorem InvariantMean.separatorDefect_eq_zero (M : InvariantMean G) | |
| {S : Finset G} (ell : DefectSpace S →L[ℝ] ℝ) : M (separatorDefect ell) = 0 := by | |
| classical | |
| change M.toLinearMap (∑ h : {h : G // h ∈ S}, | |
| (leftPull h.val (separatorCoordinate ell h) - separatorCoordinate ell h)) = 0 | |
| simp only [map_sum, map_sub, M.left_invariant, sub_self, Finset.sum_const_zero] | |
| /-- Separation contradicts the existence of a positive normalized invariant mean. -/ | |
| theorem InvariantMean.zero_mem_closure_defectSet (M : InvariantMean G) | |
| (S : Finset G) : (0 : DefectSpace S) ∈ closure (defectSet S) := by | |
| classical | |
| by_contra hout | |
| obtain ⟨ell, c, hc, hsep⟩ := geometric_hahn_banach_point_closed | |
| (convex_defectSet S).closure isClosed_closure hout | |
| have hcpos : 0 < c := by simpa only [ell.map_zero] using hc | |
| have hpoint (g : G) : c ≤ separatorDefect ell g := by | |
| rw [separatorDefect_apply] | |
| apply le_of_lt | |
| apply hsep | |
| apply subset_closure | |
| exact diracDefect_mem_defectSet S g | |
| have hmean : c ≤ M (separatorDefect ell) := by | |
| simpa only [InvariantMean.map_const] using | |
| M.mono (f := c • (1 : BoundedReal G)) (g := separatorDefect ell) | |
| (fun g => by simpa only [BoundedReal.smul_one_apply] using hpoint g) | |
| rw [M.separatorDefect_eq_zero] at hmean | |
| exact (not_le_of_gt hcpos) hmean | |
| /-- Actual nonnegative finite probabilities with arbitrarily small total defect. -/ | |
| theorem InvariantMean.exists_probability_small_defect (M : InvariantMean G) | |
| (S : Finset G) {ε : ℝ} (hε : 0 < ε) : | |
| ∃ p : G →₀ ℝ, IsProbabilityWeights p ∧ ‖defectMap S p‖ < ε := by | |
| classical | |
| obtain ⟨v, hv, hvsmall⟩ := Metric.mem_closure_iff.mp | |
| (M.zero_mem_closure_defectSet S) ε hε | |
| obtain ⟨p, hp, rfl⟩ := hv | |
| refine ⟨p, hp, ?_⟩ | |
| simpa only [dist_zero_left] using hvsmall | |
| end ThompsonNonamenability | |
| end | |
| end OAI | |