When reading a proof of the sub-additivity property of the outer measure on $\mathbb{R}$, that is:
If $\lbrace A_n \rbrace_{n=1}$ in $\mathscr{P}(\mathbb{R})$ then
$$ m^*\left( \bigcup_{n=1}^{\infty}\ A_n \right) \le \sum_{n=1}^{\infty}\ m^*(A_n) $$
I see the following sentence at the beginning of the proof:
If $\sum_{n=0}^{\infty}m^*(A_n)=\infty$, then the inequality is trivial.
This is trivial if the left-hand side of the inequality is a number. However, if it's also $\infty$ then how can we reason that the two infinities are equal or the $\infty$ on the right is bigger than the $\infty$ on the left?
There is only one $∞$ in $[0,∞]$, to which both sides of the inequality belong.
The extended real number line adds a symbol $\infty$ (and $-\infty$) and some rules for working with it that allow us to treat infinite cases more easily, but there is no distinction between different "sizes" of infinity, as this infinity is not denoting a cardinality but rather an "infinite number".