Comparison of the Bounded Convergence Theorem (BCT), Monotone Convergence Theorem (MCT), and Dominated Convergence Theorem (DCT)

Question

I'm trying to understand the relationship between the following theorems:

Bounded Convergence Theorem (BCT):

For a uniformly bounded sequence $f_n \to f$ a.e. on a set $E$ of finite measure, we have

$$ \lim_{n \to \infty} \int_E f_n = \int_E \lim_{n \to \infty} f_n $$

Monotone Convergence Theorem (MCT):

If $f_n \ge 0$ and $f_n \uparrow f$ a.e., then

$$ \int f_n \to \int f $$

Dominated Convergence Theorem (DCT):

If $f_n \to f$ a.e., $|f_n| \le g$, $\int g < \infty$, then

$$ \int \lim f_n = \lim \int f_n $$

I'm trying to understand how these convergence theorems are induced from uniform convergence, Egorov's theorem, and Fatou's lemma.

Another topic I want to explore is if these convergence theorems have analogous relationships within sequence continuities (without the integrals), as well as if these convergence theorems involving integrals are subsets/supersets of each other. I would like to find out which cases are more general, and what types of specific cases make one convergence theorem equivalent to another.

Answer 1

Once you have the MCT, everything else follows.

First, we can show that Fatou's lemma follows from MCT.

Proof: Suppose $f_n \geqslant 0$ and define $g_m = \inf_{k \geqslant m} f_k$. It follows that $g_m \leqslant f_n$ and $\int g_m \leqslant \int f_n$ for all $n \geqslant m$. Thus, $\int g_m \leqslant \liminf_{n \to \infty} \int f_n$. The sequence $(g_m)$ is increasing and by definition $\lim_{m \to \infty} g_m = \liminf_{n \to \infty} f_n$. By the MCT, it follows that

$$\int \liminf_{n \to \infty} f_n = \int\lim_{m \to \infty} g_m = \lim_{m \to \infty}\int g_m \leqslant \liminf_{n \to \infty} \int f_n\quad \text{(Fatou's lemma)}$$

Then we can show that DCT follows from Fatou's lemma.

Proof: We can assume WLOG that $f_n \to f$. (otherwise redefine appropriately on the measure zero set where $f_n \not\to f$). Since $|f_n| \leqslant g$, we have $g+f_n \geqslant 0$. Using Fatou's lemma, it follows that

$$\int g + \int f = \int(f+g) \leqslant \liminf_{n \to \infty}\int(g + f_n) = \int g + \liminf_{n \to \infty}\int f_n,$$

and, hence,

$$\tag{*} \int f \leqslant \liminf_{n \to \infty}\int f_n$$

Similarly, applying Fatou's lemma to $g- f_n \geqslant 0$, we get

$$\tag{**} \limsup_{n \to \infty} \int f_n \leqslant \int f$$

Together (*) and (**) imply that

$$\lim_{n \to \infty} \int f_n = \int f \quad \text{(DCT)}$$