2013/11/18

Fatou's lemma and monotone convergence theorem

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. $\blacksquare$

0 comments:

Post a comment