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?
The event $\limsup A_n$ means $$ \bigcap_{m\le 1} \bigcup_{n\le m} A_n $$ that is, the event that infinitely many of the $A_i$ events happen.
The complement of this is of course that only finitely of them happen, so indeed $$ P(\limsup A_n)=0 $$ means that infinitely many of the $A_n$ happens almost never, which is the same as only finitely many $A_n$s happen almost surely.
(For comparison $\liminf A_n$ would be the event that all of the $A_n$s happen together, with at most finitely many exceptions).