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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Criticality.BlockIndependence
namespace OAI
section
noncomputable section
open Set MeasureTheory ProbabilityTheory
open scoped Classical symmDiff ENNReal
namespace CriticalPercolation.BernoulliProduct
variable {E : Type*}
def finiteCoordinateEvents : Set (Set (Set E)) :=
{A | ∃ S : Finset E, MeasurableSet[blockSigma (S : Set E)] A}
lemma finiteCoordinateEvents_ring : IsSetRing (finiteCoordinateEvents (E := E)) where
empty_mem := ⟨∅,@MeasurableSet.empty (Set E) (blockSigma ((∅ : Finset E) : Set E))⟩
union_mem := by
rintro A B ⟨S,hS⟩ ⟨T,hT⟩
refine ⟨S ∪ T,?_⟩
exact @MeasurableSet.union (Set E) (blockSigma ((S ∪ T : Finset E) : Set E)) _ _
(blockSigma_mono (by simp) _ hS) (blockSigma_mono (by simp) _ hT)
sdiff_mem := by
rintro A B ⟨S,hS⟩ ⟨T,hT⟩
refine ⟨S ∪ T,?_⟩
exact @MeasurableSet.diff (Set E) (blockSigma ((S ∪ T : Finset E) : Set E)) _ _
(blockSigma_mono (by simp) _ hS) (blockSigma_mono (by simp) _ hT)
lemma generateFrom_finiteCoordinateEvents :
MeasurableSpace.generateFrom (finiteCoordinateEvents (E := E)) =
(Set.instMeasurableSpace : MeasurableSpace (Set E)) := by
apply le_antisymm
· exact MeasurableSpace.generateFrom_le fun A ⟨S,hS⟩ => blockSigma_le _ _ hS
· change (⨆ e : E, coordinateSigma e) ≤ _
refine iSup_le fun e => ?_
intro A hA
apply MeasurableSpace.measurableSet_generateFrom
refine ⟨{e},?_⟩
exact (le_iSup_of_le e (le_iSup_of_le (Finset.mem_singleton_self e) le_rfl) :
coordinateSigma e ≤ blockSigma (({e} : Finset E) : Set E)) _ hA
lemma exists_finiteCoordinate_approx (q : E → unitInterval)
{A : Set (Set E)} (hA : MeasurableSet A) {ε : ℝ≥0∞} (hε : 0 < ε) :
∃ (S : Finset E) (B : Set (Set E)), MeasurableSet[blockSigma (S : Set E)] B ∧
law q (B ∆ A) < ε := by
obtain ⟨B,⟨S,hB⟩,h⟩ := exists_measure_symmDiff_lt_of_generateFrom_isSetRing
finiteCoordinateEvents_ring (μ := law q)
(show ∃ D : Set (Set (Set E)), D.Countable ∧ D ⊆ finiteCoordinateEvents ∧
law q (⋃₀ D)ᶜ = 0 from
⟨{Set.univ},by simp,by rintro A rfl; exact ⟨∅,@MeasurableSet.univ (Set E) (blockSigma ((∅ : Finset E) : Set E))⟩,by simp⟩)
generateFrom_finiteCoordinateEvents.symm hA hε
exact ⟨S,B,hB,h⟩
lemma measurable_image_equiv_blocks (e : E ≃ E) (S : Set E) :
@Measurable (Set E) (Set E) (blockSigma (e.symm '' S)) (blockSigma S)
(fun ω => e '' ω) := by
apply measurable_iff_comap_le.mpr
simp only [blockSigma, MeasurableSpace.comap_iSup]
refine iSup₂_le fun x hx => ?_
have hm : @Measurable (Set E) Prop (coordinateSigma (e.symm x)) inferInstance
(fun ω => x ∈ e '' ω) := by
have hh : (fun ω : Set E => x ∈ e '' ω) = (fun ω => e.symm x ∈ ω) := by
funext ω
exact propext (by simp)
rw [hh]
exact measurable_iff_comap_le.mpr le_rfl
have hc := hm.comap_le
rw [coordinateSigma,MeasurableSpace.comap_comp]
change MeasurableSpace.comap (fun ω : Set E => x ∈ e '' ω) inferInstance ≤ _
exact hc.trans (le_iSup_of_le (e.symm x) (le_iSup_of_le ⟨x,hx,rfl⟩ le_rfl))
end CriticalPercolation.BernoulliProduct
end
end
end OAI
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