9 — Fatou’s Lemma, MCT, DCT

Lecture 9 completes the “Big Three” convergence theorems:

Fatou’s Lemma,

Monotone Convergence Theorem (MCT) (proved again using Fatou),

Dominated Convergence Theorem (DCT)

We continue on a σ-finite measure space $(\Omega, \mathcal{F}, \mu)$.
Recall from last time:

If $\mu(\Omega) < \infty$, $f_n \to f$ almost everywhere, and $\vert f_n\vert \le M$, then by the Bounded Convergence Theorem: $\int f_n \to \int f.$

We now turn to the fundamental convergence theorems.

1. The liminf Construction

Given a sequence of real numbers $a_1,a_2,\dots$, define: $A_n = \inf_{k \ge n} a_k.$

Then:

$A_n \le A_{n+1}$ (infimum over a smaller tail),
$A_n \uparrow \liminf a_n$.

Thus: $\liminf_{n\to\infty} a_n = \lim_{n\to\infty} A_n.$

2. Fatou’s Lemma

Theorem (Fatou).
Assume $f_n \ge 0$ and $\mu$ is σ-finite. Then:

\[\int \liminf_{n\to\infty} f_n \;\; \le \;\; \liminf_{n\to\infty} \int f_n .\]

Proof

Define the pointwise infimum tails: $g_n(x) = \inf_{m \ge n} f_m(x).$

Properties visible in the notes (page 2):

$g_n \le f_n$,
$g_n \uparrow g$ pointwise, where
$g(x) = \lim_{n\to\infty} g_n(x) = \liminf_{n\to\infty} f_n(x).$

Since $f_n \ge g_n$, $\int f_n \ge \int g_n \qquad \forall n.$

Thus: $\liminf_{n\to\infty} \int f_n \;\ge\; \lim_{n\to\infty} \int g_n .$

Now apply the Monotone Convergence Theorem to $g_n\uparrow g$:

\[\lim_{n\to\infty} \int g_n = \int g = \int \liminf f_n.\]

Therefore: $\boxed{ \int \liminf f_n \le \liminf \int f_n. }$

3. Monotone Convergence Theorem (Beppo Levi), Reproved via Fatou

Theorem (MCT).
Let $f_n \ge 0$ with $f_n \uparrow f$. Then: $\int f_n \uparrow \int f.$

Proof (from page 3, “Fatou implies MCT”)

Fatou gives: $\int f = \int \lim f_n \le \liminf \int f_n.$

But since $f_n \le f$, $\int f_n \le \int f \quad\Longrightarrow\quad \limsup \int f_n \le \int f.$

Thus: $\int f \le \liminf\int f_n \le \limsup\int f_n \le \int f,$

forcing equality, hence: $\boxed{ \int f_n \to \int f. }$

4. Dominated Convergence Theorem (DCT)

Theorem (DCT).
Let $\mu$ be σ-finite.
Suppose:

$f_n \to f$ almost everywhere,
$\vert f_n\vert \le g$ for some integrable $g$ (i.e. $\int g < \infty$).

Then:

\[\int f_n \to \int f.\]

Proof Breakdown

Your notes (page 3–4) show the proof using:

Fatou on the sequences $g + f_n$ and $g - f_n$,
The positivity of integrals.

Let’s write it cleanly.

Step 1: Use Fatou on $g + f_n$

Note $g + f_n \ge 0$.
Apply Fatou:

\[\int \liminf (g + f_n) \le \liminf \int (g + f_n).\]

Since $f_n \to f$ a.e., $\liminf (g+f_n) = g+f.$

Thus: $\int (g+f) \le \liminf \left[\int g + \int f_n\right] = \int g + \liminf \int f_n.$

Cancel $\int g$: $\int f \le \liminf \int f_n.$

Step 2: Use Fatou on $g - f_n$

Similarly $g - f_n \ge 0$. Fatou gives:

\[\int (g-f) \le \liminf [\int g - \int f_n] = \int g - \limsup \int f_n.\]

Rearrange:

\[\limsup \int f_n \le \int f.\]

Step 3: Combine inequalities

We have: $\int f \le \liminf \int f_n \le \limsup \int f_n \le \int f.$

Thus:

\[\boxed{ \int f_n \to \int f. }\]

5. Summary of the Three Convergence Theorems

Fatou’s Lemma

$\int \liminf f_n \le \liminf \int f_n.$

Monotone Convergence (MCT)

If $f_n \uparrow f$,
$\int f_n \uparrow \int f.$

Dominated Convergence (DCT)

If $f_n \to f$ a.e. and $\vert f_n\vert \le g$ with $\int g < \infty$,
$\int f_n \to \int f.$

These results form the analytic backbone of measure-theoretic probability.

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