Generalization of the Borel-Cantelli Lemma

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In the present note a generalization of Borel-Cantelli Lemma is derived.

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Conditional independence, conditional mixing and conditional association (2009, methodology): “We will now obtain a conditional version of the Borel-Cantelli lemma due to Petrov (2004). For simplicity, we write P(A|F) for EF (IA) in the following discussion.”
Measures of pseudorandomness for finite sequences: typical values (2007, result): “More recently, Petrov used this result to generalize the Borel–Cantelli lemma [13, 14].”