  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.
