The divergence Borel–Cantelli Lemma revisited

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.