The DCT is stated as follows in Bartle's text:
Let $(f_n)$ be a sequence of integrable functions which converges almost everywhere to a real-valued measurable function $f$. If there exists an integrable function $g$ such that $|f_n| \le g $ for all $n$, then $f$ is integrable and $$ \int f \, d \mu = \lim \int f_n \, d \mu .$$
I was wondering if we need the condition that it converges a.e. to a "real-valued measurable function $f$". I believe we can replace it with conergence a.e. to a "real-valued function."
As each $f_n$ is integrable, by definition each $f_n$ is measurable. Hence, $f = \lim f_n = \lim \inf f_n$ is measurable. Am I missing something? Thanks.
The point is that $f$ should be measurable up to a null set since you only have almost everywhere convergence.
See for instance the version of DCT in Wikipedia, or the one in Folland's Real Analysis: