Below is the common statement of the Monotone Convergence Theorem.
Suppose $\{f_n\}$ is a sequence of non-negative measurable functions with $f_n(x) \leq f_{n+1}(x)$ a.e. and $\lim \limits_{n\rightarrow\infty} f_n(x) = f(x)$. Then $$\lim \limits_{n\rightarrow\infty} \int f_n = \int f$$
But proof requires only $f_n(x) \leq f(x)$ since it is sufficient to conclude that $\int f_n \leq \int f$ for all $n$ by monotonicity. So what is the value of the more restrictive statement than it could be (except that it leads to a nice name for the theorem).
Edit: Proof.
By Fatou's lemma $\int f \leq \lim \limits_{n\rightarrow\infty} \inf \int f_n$. Now, given $f_n(x) \leq f(x)$ it follows that $\int f_n \leq \int f$ for all $n$ by monotonicity. Hence $\lim \limits_{n\rightarrow\infty} \sup \int f_n \leq \int f$. Which combined with Fatou's lemma gives the desired result.
I guess that relaxing the hipotesis about monotonicity you get a weakening of the dominated convergence theorem (for nonnegative functions dominated by your $f(x)$). Here you construct the limit (by monotonicity) and you prove it's a measurable function. Hope this was useful.