In lecture we learned about Fatou's Lemma stated as follows:
Let $(X, \mathcal{S}, \mu)$ be a measure space and $(f_k : X \to [0,\infty])$ measurable and $f: X \to [0, \infty] $ a function such that: $ f = \liminf_{k \to \infty} f_k $ almost everywhere Then $f$ is measurable and $ \int_X f d\mu \leq \liminf_{k \to \infty} \int_X f_k d\mu. $
I am thinking that the statement written in this way is not correct. Since f need not be measurable. We can only guarantee this in complete measure spaces. Is there something I am missing or do we need to change the statement.