# Fatou's Lemma

/*not*always true that $$\int \lim_{n\to\infty}f_n = \lim_{n\to\infty}\int f_n.$$ (Take this sequence for example.)

**Fatou's Lemma**, the

**Monotone Convergence Theorem**(MCT), and the

**Dominated Convergence Theorem**(DCT) are three major results in the theory of Lebesgue integration which answer the question "When do $\displaystyle{ \lim_{n\to\infty} }$ and $\int$ commute?" The MCT and DCT tell us that

*if*you place certain restrictions on both the $f_n$ and $f$, then you can interchange the limit and integral. On the other hand, Fatou's Lemma says, "Here's the best you can do if you

*don't*put any restrictions on the functions."

*Real and Complex Analysis*(a.k.a "Big Rudin") which illustrates that the inequality in Fatou's Lemma can be a strict inequality.

**Fatou's Lemma:**Let $(X,\Sigma,\mu)$ be a measure space and $\{f_n:X\to[0,\infty]\}$ a sequence of nonnegative measurable functions. Then the function $\displaystyle{ \liminf_{n\to\infty} f_n}$ is measureable and $$\int_X \liminf_{n\to\infty} f_n \;d\mu \;\; \leq \;\; \liminf_{n\to\infty} \int_X f_n\;d\mu .$$

## Proof

**1st observation:** $\int g_k \leq \int f_n$ for all $n\geq k$. This follows easily from the fact that for a fixed $x\in X$, $\displaystyle{\inf_{n\geq k}\{f_n(x)\}}\leq f_n(x)$ whenever $n\geq k$ (by definition of infimum). Hence $\int \displaystyle{\inf_{n\geq k} f_n} \leq \int f_n$ for all $n\geq k$, as claimed. This allows us to write
\begin{align}
\int g_k\leq \inf_{n\geq k}\int f_n. \qquad \qquad (1)
\end{align}

**2nd observation:** $\{g_k\}$ is an increasing sequence and $\displaystyle{\lim_{k\to\infty} g_k}=h$ pointwise. Thus, by the Monotone Convergence Theorem,
\begin{align*}
\int\liminf_{n\to\infty} f_n =\int h = \lim_{k\to\infty} \int g_k \leq \lim_{k\to\infty} \inf_{n\geq k}\int f_n = \liminf_{n\to\infty} \int f_n
\end{align*}
where the inequality in the middle follows from (1).

Finally, $\liminf f_n$ is measurable as we've proved before in the footnotes here.

## Exercise from Big Rudin

*Real and Complex Analysis*.

*(Rudin, RCA, #1.8) Let $E\subset \mathbb{R}$ be Lebesgue measurable, and for $n\geq 0$ define*$$ f_n=\begin{cases} \chi_E &\text{if $n$ is even};\\ 1-\chi_E &\text{if $n$ is odd.} \end{cases} $$

*What is the relevance of this example to Fatou's Lemma?*