The divergence Borel–Cantelli Lemma revisited

TY - JOUR

T1 - The divergence Borel–Cantelli Lemma revisited

AU - Beresnevich, Victor

AU - Velani, Sanju

N1 - © 2022 The Author(s).

PY - 2023/3/1

Y1 - 2023/3/1

N2 - Let $(\Omega, \mathcal{A}, \mu)$ be a probability space. The classical Borel--Cantelli Lemma states that for any sequence of $\mu$-measurable sets $E_i$ ($i=1,2,3,\dots$), if the sum of their measures converges then the corresponding $\limsup$ set $E_\infty$ is of measure zero. In general the converse statement is false. However, it is well known that the divergence counterpart is true under various additional `independence' hypotheses. In this paper we revisit these hypotheses and establish both sufficient and necessary conditions for $E_\infty$ to have either positive or full measure.

AB - Let $(\Omega, \mathcal{A}, \mu)$ be a probability space. The classical Borel--Cantelli Lemma states that for any sequence of $\mu$-measurable sets $E_i$ ($i=1,2,3,\dots$), if the sum of their measures converges then the corresponding $\limsup$ set $E_\infty$ is of measure zero. In general the converse statement is false. However, it is well known that the divergence counterpart is true under various additional `independence' hypotheses. In this paper we revisit these hypotheses and establish both sufficient and necessary conditions for $E_\infty$ to have either positive or full measure.

U2 - 10.1016/j.jmaa.2022.126750

DO - 10.1016/j.jmaa.2022.126750

M3 - Article

SN - 0022-247X

VL - 519

JO - Journal of mathematical analysis and applications

JF - Journal of mathematical analysis and applications

IS - 1

M1 - 126750

ER -