Let $f_n$ be a sequence of nonnegative Lebesgue measurable functions on $X \subset \mathbb{R}^n.$ Suppose $f_n$ converes to $f$ almost everywhere, and $f_n \leq f.$ I want show that $\int f =$ lim $\int f_n.$
Partial Proof:
If $f \in L^{1},$ then the equality follows from DCT. This gave me the motivation to believe the result might be true.
Proof: [f is just measurable]
$\int f = \int \lim f_n = \int \underline{\lim} f_n \leq \underline{\lim} \int f_n \leq \overline{\lim} \int f_n \leq \int f.$ The first inequality is the standard Fatou's Lemma, the second one is true for any sequence, and the third one is due to the given hypothesis. Therefore, the result is true. Is there any issue with this argument? Is there any other way we can argue to prove to show the equality? Is this result true for any general measure space? Thanks so much.