File size: 3,036 Bytes
67aee1d | 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 | import OAI.Analysis.FiniteFactor.OrbitSampling.CylinderSystem
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter Finset
open scoped Classical ENNReal Topology
noncomputable section
theorem rational_Icc_nhds {x : ℝ} {u : Set ℝ} (hu : u ∈ 𝓝 x) :
∃ a b : ℚ, Set.Icc (a : ℝ) b ∈ 𝓝 x ∧ Set.Icc (a : ℝ) b ⊆ u := by
obtain ⟨l,r,hlr,hsub⟩ := mem_nhds_iff_exists_Ioo_subset.mp hu
obtain ⟨a,hla,hax⟩ := exists_rat_btwn hlr.1
obtain ⟨b,hxb,hbr⟩ := exists_rat_btwn hlr.2
exact ⟨a,b,Icc_mem_nhds hax hxb,fun y hy => hsub ⟨hla.trans_le hy.1,hy.2.trans_lt hbr⟩⟩
/-- The finite rational cylinders contain arbitrarily small neighborhoods
in the full real product, including points outside the unit cube. -/
theorem sourceSamplingCylinder_basis (u : Set SourceSamplingPair) (hu : IsOpen u)
(p : SourceSamplingPair) (hp : p ∈ u) :
∃ s ∈ Set.range sourceSamplingCylinder, s ∈ 𝓝 p ∧ s ⊆ u := by
obtain ⟨ux,hux,uy,huy,hxy⟩ := mem_nhds_prod_iff.mp (hu.mem_nhds hp)
rw [nhds_pi,Filter.mem_pi'] at hux huy
obtain ⟨Kx,tx,htx,hsubx⟩ := hux
obtain ⟨Ky,ty,hty,hsuby⟩ := huy
choose ax bx hnx hsx using fun i => rational_Icc_nhds (htx i)
choose ay by' hny hsy using fun i => rational_Icc_nhds (hty i)
let T : SourceSamplingRationalTest := ⟨Kx ∪ Ky,fun i => ((ax i,bx i),(ay i,by' i))⟩
have hTx (i : SourceCoordinate) (hi : i ∈ T.1) :
sourceSamplingRationalX T i = Set.Icc (ax i : ℝ) (bx i) := by
simp [sourceSamplingRationalX,sourceSamplingRationalEndpoints,hi,T]
have hTy (i : SourceCoordinate) (hi : i ∈ T.1) :
sourceSamplingRationalY T i = Set.Icc (ay i : ℝ) (by' i) := by
simp [sourceSamplingRationalY,sourceSamplingRationalEndpoints,hi,T]
refine ⟨sourceSamplingCylinder T,⟨T,rfl⟩,?_,?_⟩
· have he : sourceSamplingCylinder T = ⋂ i ∈ T.1,
{q : SourceSamplingPair | q.1 i ∈ sourceSamplingRationalX T i ∧
q.2 i ∈ sourceSamplingRationalY T i} := by ext q; simp [sourceSamplingCylinder]
rw [he]
apply (Filter.biInter_mem (Set.toFinite (T.1 : Set SourceCoordinate))).mpr
intro i hi
have hx := ((continuous_apply i).comp continuous_fst).continuousAt (hnx i)
have hy := ((continuous_apply i).comp continuous_snd).continuousAt (hny i)
change {q : SourceSamplingPair | q.1 i ∈ Set.Icc (ax i : ℝ) (bx i)} ∈ 𝓝 p at hx
change {q : SourceSamplingPair | q.2 i ∈ Set.Icc (ay i : ℝ) (by' i)} ∈ 𝓝 p at hy
filter_upwards [hx,hy] with q hq1 hq2
exact ⟨by simpa only [hTx i hi] using hq1,by simpa only [hTy i hi] using hq2⟩
· intro q hq
apply hxy
constructor
· apply hsubx
intro i hi
have hmem : i ∈ T.1 := Finset.mem_union_left _ hi
exact hsx i (by simpa only [hTx i hmem] using (hq i hmem).1)
· apply hsuby
intro i hi
have hmem : i ∈ T.1 := Finset.mem_union_right _ hi
exact hsy i (by simpa only [hTy i hmem] using (hq i hmem).2)
end
end FiniteFactor
end OAI
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