| import Mathlib.Topology.Algebra.MulAction |
| import Mathlib.Topology.UniformSpace.HeineCantor |
| import Mathlib.Topology.MetricSpace.UniformConvergence |
|
|
| namespace OAI |
|
|
| /-! |
| # Compact observables under small group translations |
|
|
| A compactly supported continuous observable changes uniformly little |
| under a group element tending to the identity. This remains true along |
| arbitrary moving base points, including points leaving compact sets. |
| It supplies the topological contraction step in the Hopf argument on a |
| noncompact homogeneous space. |
| -/ |
|
|
| open Filter Set |
| open scoped Topology |
|
|
| namespace DukePrimeDegree |
|
|
| variable {G X : Type*} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] |
| [WeaklyLocallyCompactSpace G] [TopologicalSpace X] [MulAction G X] |
| [ContinuousSMul G X] |
|
|
| / |
| supported continuous real observable. -/ |
| theorem compact_observable_smul_tendstoUniformly |
| (f : X → ℝ) (hf : Continuous f) (hc : HasCompactSupport f) : |
| TendstoUniformly (fun a : G => fun x : X => f (a • x)) f (𝓝 1) := by |
| apply Metric.tendstoUniformly_iff.mpr |
| intro ε hε |
| obtain ⟨V, hVc, hV⟩ := exists_compact_mem_nhds (1 : G) |
| let C : Set X := tsupport f ∪ |
| (fun z : G × X => z.1⁻¹ • z.2) '' (V ×ˢ tsupport f) |
| have hCc : IsCompact C := hc.union |
| ((hVc.prod hc).image (continuous_fst.inv.smul continuous_snd)) |
| have hcont : Continuous (fun z : G × X => f (z.1 • z.2)) := by |
| fun_prop |
| obtain ⟨W, hW, hsmallW⟩ := hCc.mem_uniformity_of_prod |
| (f := fun a : G => fun x : X => f (a • x)) |
| (s := Set.univ) hcont.continuousOn (show (1 : G) ∈ Set.univ from Set.mem_univ _) |
| (Metric.dist_mem_uniformity hε) |
| have hsmall : ∀ᶠ a : G in 𝓝 1, |
| ∀ x : C, dist (f (a • (x : X))) (f x) < ε := by |
| have hW' : W ∈ 𝓝 (1 : G) := by simpa only [nhdsWithin_univ] using hW |
| filter_upwards [hW'] with a ha x |
| simpa only [one_smul, Set.mem_ofPred_eq] using hsmallW a ha x x.property |
| filter_upwards [hV, hsmall] with a ha hs x |
| by_cases hx : x ∈ C |
| · simpa only [dist_comm] using hs ⟨x, hx⟩ |
| · have hx0 : f x = 0 := by |
| by_contra hne |
| exact hx (Or.inl (subset_tsupport f hne)) |
| have hax0 : f (a • x) = 0 := by |
| by_contra hne |
| apply hx |
| exact Or.inr ⟨(a, a • x), ⟨ha, subset_tsupport f hne⟩, inv_smul_smul a x⟩ |
| simpa only [hx0, hax0, dist_self] using hε |
|
|
| / |
| tends to the identity. -/ |
| theorem compact_observable_moving_smul_difference |
| (f : X → ℝ) (hf : Continuous f) (hc : HasCompactSupport f) |
| {ι : Type*} {l : Filter ι} (x : ι → X) (a : ι → G) |
| (ha : Tendsto a l (𝓝 1)) : |
| Tendsto (fun n => |f (a n • x n) - f (x n)|) l (𝓝 0) := by |
| apply Metric.tendsto_nhds.mpr |
| intro ε hε |
| have hh := Metric.tendstoUniformly_iff.mp |
| (compact_observable_smul_tendstoUniformly (G := G) f hf hc) ε hε |
| filter_upwards [ha.eventually hh] with n hn |
| simpa only [Real.dist_eq, sub_zero, abs_abs, abs_sub_comm] using hn (x n) |
|
|
| end DukePrimeDegree |
|
|
| end OAI |
|
|
