In this post, we deduce Fatou's lemma and monotone convergence theorem (MCT) from each other.
Fix a measure space $(\Omega,\cF,\mu)$.
Fatou's lemma. Let $\{f_n\}_{n=1}^\infty$ be a collection of non-negative integrable functions on $(\Omega,\cF,\mu)$. Then,
\begin{align}\label{Fatous}
\int \liminf_{n\to\infty} f_n d\mu \leq \liminf_{n\to\infty} \int f_n d\mu
\end{align}
Monotone convergence theorem. Let $\{f_n\}_{n=1}^\infty$ be a sequence of nonnegative integrable functions on $(\Omega,\cF,\mu)$ such that $f_n \leq f_j$ with $j \geq n$, i.e., $f_n \leq f_{n+1}$ for all $n \geq 1$ and $x \in \Omega$. If $f_n \to f$ pointwise, then,
\begin{align}\label{MCT}
\lim_{n\to\infty} \int f_n d\mu = \int f d\mu
\end{align}
Proof of Fatou's lemma. Let $g_n = \inf_{j\geq n} f_j$. And notice that $g_n \leq g_{n+1}$ for all $n\geq 1$. Also, define $\lim_n g_n = g = \liminf_n f_n$. And the last thing to note is, $\int g_n d\mu \leq \inf_j \int f_j d\mu$ for $j \geq n$, i.e., integral of the infimum is less than the infimum of integrals. By the monotone convergence theorem,
\begin{align*}
\lim_{n \to \infty} \int g_n d\mu = \int g d\mu = \int \liminf_{n\to\infty} f_n d\mu
\end{align*}
Also notice, since $\int g_n d\mu \leq \inf_{j\geq n} \int f_j d\mu$,
\begin{align*}
\lim_{n\to\infty} g_n d\mu \leq \lim_{n\to\infty} \inf_{j\geq n} \int f_j d\mu
\end{align*}
By combining these, we have the result,
\begin{align}
\int \liminf_{n\to\infty} f_n d\mu \leq \liminf_{n\to\infty} \int f_n d\mu
\end{align}$\blacksquare$
Proof of monotone convergence theorem. Since $f_n \leq f$ for every $n \geq 1$ and integrals preserve monotonicity, then $\int f_n d\mu \leq \int f d\mu$ for all $n \geq 1$. Then we have,
\begin{align}\label{lft}
\lim_{n\to\infty} \int f_n d\mu \leq \int f d\mu
\end{align}
On the other hand, for the converse, apply Fatou's lemma. We have,
\begin{align*}
\lim_{n\to\infty} f_n = f
\end{align*}
by assumption. Since the limit exists, we write, $\liminf_{n\to\infty} f_n = \lim_{n\to\infty} f_n$. Write Fatou's,
\begin{align*}
\int \liminf_{n\to\infty} f_n d\mu &= \int \lim_{n\to\infty} f_n d\mu \\
&\leq \liminf_{n\to\infty} \int f_n d\mu \\
&= \lim_{n\to\infty} \int f_n d\mu
\end{align*}
then we have,
\begin{align}\label{rght}
\int f d\mu \leq \lim_{n\to\infty} \int f_n d\mu
\end{align}
By combining \eqref{lft} and \eqref{rght}, the result follows. $\blacksquare$
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