Question about of Fatou's lemma in Rick Durrett's book. 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 $\mu$?

At the beginning of section 1.4 where Durrett defines the integral he assumes that the measure $\mu$ is $\sigma$-finite, so I guess it is a standing assumption about all integrals in this book that the underlying measure satisfies this. Remember that it is a probability book, so his main interest are finite measures.
