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Atlas.TheoryOfProbability.code.BorelCantelliSecond

theorem second_Borel_Cantelli {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {A : Set Ω} (hm : ∀ (n : ), MeasurableSet (A n)) (hindep : ProbabilityTheory.iIndepSet A μ) (hsum : ∑' (n : ), μ (A n) = ) :

Second Borel–Cantelli lemma. If the events Aₙ are (mutually) independent and ∑ₙ μ(Aₙ) = ∞, then almost every point belongs to infinitely many of the Aₙ, i.e. μ(limsup Aₙ) = 1.