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 $\mu$?

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Assuming $m$ integers, such sets exists if and only if the measure is $\sigma$-finite. – Davide Giraudo Nov 23 '12 at 20:53
Correct, this would be $\sigma$-finiteness, but it is not needed for the proof of Fatou's lemma. I don't own Durrett's book, so I don't know exactly what he is doing there. – Lukas Geyer Nov 23 '12 at 21:03
I'm reading this copy of his book : stat-www.berkeley.edu/~aldous/205A/PTE4_Jan2010.pdf The proof is on page 22. – ablmf Nov 23 '12 at 21:08
At the very 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. I am not sure why he does this, but since it is a probability book, he mostly deals with finite measures anyway, so I guess it is not a big problem in the context. – Lukas Geyer Nov 23 '12 at 21:22
@LukasGeyer You can turn your comment in an answer. – Davide Giraudo Nov 26 '12 at 12:26

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.