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a44267b | 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 | 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]
/-- Small translations act uniformly continuously on a compactly
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
/-- The base point may vary arbitrarily while the translating element
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
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