Why is the Monotone Convergence Theorem more famous than it's stronger cousin?

Some people do call this corollary monotone convergence theorem. Moreover, some instructora introduce this corollary as M.C.T., rather than the origianl version of M.C.T., for instance, Professor Mark J. Schervish and Professor Alessandro Rinaldo at CMU (Lecture notes). It's fine because this corollary is stronger and more practical.

However, from the perspective of mathematics, I think it's better to be traditional and use the original version of M.C.T. and just regard that corollary as a useful corollary.

Let's take a look at the Lebesgue's dominant convergence theorem:

Suppose $\{f_n\}$ and $f$ are measurable functions. If there exists a non-negative function $g$ such that $\int g <\infty$ and $|f_n|\leq g$ a.e., for all $n$. Then $f_n\overset{a.e.}{\rightarrow} f$ or $f_n\overset{\mu}{\rightarrow} f$ implies that $$\lim_{n \to \infty} \int f_n = \int f. $$

Now, compare this with the result you give:

Suppose $f$ is a non-negative measurable function, and $\{f_n\}$ a sequence of non-negative measurable functions with $f_n(x) \le f(x)$ and $f_n(x) \to f(x)$ for a.e. $x$. Then $\displaystyle \lim_{n \to \infty} \int f_n = \int f$.

We can find that when $\int f <\infty$, the result is implied by the Lebesgue's D.C.T. When $\int f =\infty$, by Fatou's lemma we have

$$\infty = \int f = \int \liminf_{n\rightarrow \infty} f_n \leq \liminf_{n\rightarrow \infty} \int f_n\Rightarrow \int f_n\rightarrow \infty.$$

In summary, this result seems more similar to D.C.T than M.C.T. This is one reason why it cannot take the place of M.C.T.

The other reason is more fundamental. If we take a look at the proof of the Fatou's lemma, we may find that it relies on the monotonicity of $g_{k} = \inf_{n\geq k}f_n$ and $g_k\nearrow \liminf_{n\rightarrow \infty} f_n$. That is to say, the Fatou's lemma is actually implied by the M.C.T.

In other words, M.C.T. is the most fundamental theorem, then the Fatou's lemma and finally the D.C.T. Therefore from the perspective of mathematician, it's definitely better to use the "naive version" (actually the only correct version) of M.C.T., but not the seemingly stronger result you give, which is implied by D.C.T. and Fatou's lemma.

Nevertheless, it is true that the result you give here is much more convenient to use than either M.C.T. or D.C.T. That's why some instructors prefer to introduce this result as a substitution for M.C.T.