# The Unapologetic Mathematician

## The Fatou-Lebesgue Theorem

Now we turn to the Fatou-Lebesgue theorem. Let $\{f_n\}$ be a sequence of integrable functions (this time we do not assume they are non-negative) and $g$ be some other function which dominates this sequence in absolute value. That is, we have $\lvert f_n(x)\rvert\leq g(x)$ a.e. for all $n$. We define the functions

\displaystyle\begin{aligned}f_*(x)&=\liminf\limits_{n\to\infty} f_n(x)\\f^*(x)&=\limsup\limits_{n\to\infty} f_n(x)\end{aligned}

These two functions are integrable, and we have the sequence of inequalities

$\displaystyle\int f_*\,d\mu\leq\liminf\limits_{n\to\infty}\int f_n\,d\mu\leq\limsup\limits_{n\to\infty}\int f_n\,d\mu\leq\int f^*\,d\mu$

Again, this is often stated for a sequence of measurable functions, but the dominated convergence theorem allows us to immediately move to the integrable case. In fact, if the sequence $\{f_n\}$ converges pointwise a.e., then $f_*=f^*$ a.e. and the inequality collapses and gives us exactly the dominated convergence theorem back again.

Since $g$ dominates the sequence $\{f_n\}$, the sequence $\{g+f_n\}$ will be non-negative. Fatou’s lemma then tells us that

\displaystyle\begin{aligned}\int g\,d\mu+\int f_*\,d\mu&=\int g+f_*\,d\mu\\&=\int\liminf\limits_{n\to\infty}(g+f_n)\,d\mu\\&\leq\liminf\limits_{n\to\infty}\int g+f_n\,d\mu\\&=\int g\,d\mu+\liminf\limits_{n\to\infty}\int f_n\,d\mu\end{aligned}

Cancelling the integral of $g$ we find the first asserted inequality. The second one is true by the definition of limits inferior and superior. The third one is essentially the same as the first, only using the non-negative sequence $\{g-f_n\}$.