= Solution
Set $Y_n=\sup_{m\ge n}|f_m|$. It is an extended-real <measurable function> and decreases to $Y=\limsup_m|f_m|$. If $f_n\to0$ <almost surely>, then $Y_n\to0$ <almost surely>. For any $\epsilon>0$, the decreasing sets $\{Y_n>\epsilon\}$ have intersection contained, up to a <null set>, in the exceptional set for convergence. Continuity from above of the <probability measure> gives $P(Y_n>\epsilon)\to0$.
Conversely, if $Y_n\to0$ in <probability>, then $P(Y>\epsilon)\le P(Y_n>\epsilon)\to0$ for every positive $\epsilon$. Taking rational $\epsilon>0$ gives $Y=0$ <almost surely>, so $|f_n|\to0$ <almost surely>. This proves both directions, even when some individual $Y_n$ take the value infinity.
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