In this short note, we discuss the Barndorff-Nielsen lemma, which is a
generalization of well-known Borel-Cantelli lemma. Although the result stated
in the Barndorff-Nielsen lemma is correct, it does not follow from the argument
proposed in the corresponding proof. In this note, we show this and offer an
alternative proof of this lemma. We also propose a new generalization of
Borel-Cantelli lemma.