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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 | import Mathlib
import OAI.Probability.CriticalPercolation.ProductMeasure.FiniteOpenWitness
namespace OAI
section
section
open scoped BigOperators
open scoped Classical ENNReal
open Set
open scoped Classical Topology
open Set MeasureTheory ProbabilityTheory
open scoped Classical
open Filter Topology
open Set MeasureTheory ProbabilityTheory Filter
open scoped Topology Classical
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
lemma hitEvent_isUpperSet (o : V) (A : Set V) : IsUpperSet (G.hitEvent o A) :=
fun _ _ h => G.hitEvent_mono_edges o A h
lemma hitEvent_hasFiniteOpenWitness (o : V) (A : Set V) :
BernoulliProduct.HasFiniteOpenWitness (G.hitEvent o A) := by
rintro ω ⟨v, hv, hvA⟩
obtain ⟨s, hs, hsω, hvs⟩ := G.mem_cluster_finite_witness hv
exact ⟨s, hs, hsω, v, hvs, hvA⟩
lemma lowerSemicontinuous_real_hitEvent [Countable E] (o : V) (A : Set V) :
LowerSemicontinuous (fun p : unitInterval => (G.law p).real (G.hitEvent o A)) := by
simpa [law, BernoulliProduct.law_const] using
BernoulliProduct.lowerSemicontinuous_real_finiteWitness (G.hitEvent o A)
(G.hitEvent_isUpperSet o A) (G.hitEvent_hasFiniteOpenWitness o A)
end CriticalPercolation.BondGraph
open Set MeasureTheory ProbabilityTheory Filter
open scoped Classical Topology
end
end
end OAI
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