The confusion, if any, stems from not specifying the target space $E$ of the arrow $X:\Omega\to E$.
If $E$ is the real line, obviously $P(|X|\ge N)\to0$ when $N\to+\infty$, but if $E$ is the extended real ine, this need not be true anymore since $P(X=+\infty)$ and $P(X=-\infty)$ may be any nonnegative real numbers whose sum does not exceed $1$. In the second case, the sum $P(X=+\infty)+P(X=-\infty)$ might even be $1$, this would happen if P(|X| \mbox{finite})=0 and $P(|X|=+\infty)=1$.
Both topological spaces $E$ are endowed with a natural Borel sigma-algebra. Some people call random variable any random variable with values in the (non extended) real line, some consider the extended real line, and, although I prefer one choice over the other, both seem legitimate to me.
A common situation where the extended real line enters the picture is when a sequence $(\xi_t)$ of random variables indexed by $t\in T$ is given, for example with $T=\mathbb{N}$, and one defines $X$ as the smallest index $t$ such that an event like $[\xi_t\in B]$ happens, for a given measurable subset $B$ of the common target space of the random variables $\xi_t$. Then the natural target space of $X$ is the extended line because $X=+\infty$ on the event $[\forall t\in T, \xi_t\notin B]$, and it may well happen that $P(X=+\infty)>0$.
However, I can think of no natural situation where a random variable $X$ such that $P(|X|=+\infty)=1$ is involved.
The relevant wikipedia paragraph explains this concisely and competently.