It is my understanding that in probability, and especially as a consequence from the Borel-Cantelli Lemma, "almost surely infinitely often" corresponds to some event $A_n$ whereby:
$$ P(\limsup A_n) =1 $$
However, I am not sure what "almost surely finitely often" means. Does this mean that:
$$ P(\limsup A_n) =0 $$ ? If so, why?