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.

翻译：在此简短说明中,我们讨论巴恩多夫-尼尔森-林马,这是众所周知的博雷尔-坎特利-伦马的概括,虽然巴恩多夫-尼尔森-伦马所述结果是正确的,但并不从相应证据中提出的论点出发,在本说明中,我们展示这一点,并对这一列马提供另一种证据,我们还提议对博雷尔-坎特利-伦马进行新的概括。