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