I am reading Stein & Shakarchi. On page 62 we have the Monotone Convergence Theorem:
Suppose $\{f_n\}$ is a sequence of non-negative measurable functions with $f_n\nearrow f.$ Then $\displaystyle \lim_{n \to \infty} \int f_n = \int f$.
However, just before this theorem we have this much more powerful corrolary of Fatou's lemma:
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 $\displaystyle \lim_{n \to \infty} \int f_n = \int f$.
To me, this second corollary seems strictly better than the Monotone Convergence Theorem, yet it is the M.C.T. that has a name and is used often. Am I misunderstanding the theorems, or is there a reason why the M.C.T. is more popular?
Does this corollary have a name?
Best Answer
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:
Now, compare this with the result you give:
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.