I am studying real analysis and I am being introduced to functions that take values on the extended real line, I have a fundamental doubt about this so I'll give an example to illustrate my confusion:
Consider the proposition "If $f$ is measurable then there exists a sequence $(\phi_n)_{n \in \mathbb N}$ of simple functions such that $\phi_n(x) \to f(x)$ for all $x$." So, take the case where there is $x$ with $f(x)=+\infty$, what it means $\phi_n(x) \to f(x)$ when $n \to \infty$?, I suppose the natural definition would be the same as for limits in $\mathbb R$, which is for any $M>0$, there is $n_0:$ for all $n \geq n_0$, $\phi_n(x)>M$. How can I define a metric here?, what would be $d(x,+\infty)$?, which are the open and closed sets?
Thanks in advance
The extended real line is not a metric space. You are correct in noting that $d(0,+\infty)$ must be larger than any $d(0,x)$ for any $x\in\Bbb R$, and therefore cannot be a real number.
However the extended real line is metrizable. This means that we can redefine the metric completely, while preserving the same topology (and therefore the same convergent sequences). To see this simply note that if $\Bbb R$ and $(0,1)$ are homeomorphic, then the extended real line is homeomorphic to $[0,1]$.
But more directly, how do define the open sets? The same way as before, using open intervals, only here you should note that $(x,+\infty]$ and $[-\infty,x)$ are considered an open interval as well, since the endpoints of the entire order are allowed to be included in open intervals.