In Probability Theory and Examples, Theorem $1.5.4$, Fatou's Lemma, says
If $f_n \ge 0$ then $\liminf_{n \to \infty} \int f_n d\mu \ge \int \left(\liminf_{n \to \infty} f_n \right) d\mu. $
In the proof, the author says
Let $E_m \uparrow \Omega$ be sets of finite measure.
I'm confused, as without any information on the measure $\mu$, how can we guarantee this kind of sequence of events must exist? Has the author missed some additional condition on $\mu$?