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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Criticality.BlockIndependence
namespace OAI
section
section
noncomputable section
open Set MeasureTheory ProbabilityTheory Filter
open scoped Classical ENNReal
namespace CriticalPercolation.BernoulliProduct
variable {E : Type*}
lemma bitMeasure_false_ne_zero (p : unitInterval) (hp : (p:ℝ)<1) :
bitMeasure p {False} ≠ 0 := by
have hh := bitMeasure_real_singleton p False
simp only [bitWeight,ite_false] at hh
intro hz
have : (bitMeasure p).real {False}=0 := by simp only [measureReal_def,hz,ENNReal.toReal_zero]
linarith
theorem frequently_closedBlock_ae (p : unitInterval) (hp : (p:ℝ)<1)
(S : ℕ → Set E) (hS : ∀ n, (S n).Finite)
(hdis : Pairwise (fun n m => Disjoint (S n) (S m))) (N : ℕ)
(hN : ∀ n, (S n).ncard=N) :
∀ᵐ ω ∂law (fun _ : E => p), ∃ᶠ n in atTop, ω ∈ closedBlock (S n) := by
let A := fun n => closedBlock (S n)
have hm' : ∀ n, MeasurableSet[blockSigma (S n)] (A n) :=
fun n => measurableSet_closedBlock (S n) (hS n).countable
have hm : ∀ n, MeasurableSet (A n) := fun n => blockSigma_le (S n) _ (hm' n)
have hi := independent_block_events (fun _ : E => p) S hdis A hm'
have hprob : ∀ n, law (fun _ : E => p) (A n) = (bitMeasure p {False})^N := by
intro n
rw [law_closedBlock_const p (S n) (hS n),hN]
have hsum : (∑' n, law (fun _ : E => p) (A n)) = ∞ := by
simp_rw [hprob]
exact ENNReal.tsum_const_eq_top_of_ne_zero (pow_ne_zero _ (bitMeasure_false_ne_zero p hp))
have hlim := measure_limsup_eq_one hm hi hsum
have hl : ∀ᵐ ω ∂law (fun _ : E => p), ω ∈ limsup A atTop :=
(mem_ae_iff_prob_eq_one (MeasurableSet.measurableSet_limsup hm)).mpr hlim
exact hl.mono (fun _ h => Filter.mem_limsup_iff_frequently_mem.mp h)
end CriticalPercolation.BernoulliProduct
end
end
end
end OAI
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