The Unapologetic Mathematician

Fatou’s Lemma

Today we prove Fatou’s Lemma, which is a precursor to the Fatou-Lebesgue theorem, and an important result in its own right.

If $\{f_n\}$ is a sequence of non-negative integrable functions then the function defined pointwise as

$\displaystyle f_*(x)=\liminf\limits_{n\to\infty} f_n(x)$

is also integrable, and we have the inequality

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

In fact, the lemma is often stated for a sequence of measurable functions and concludes that $f_*$ is measurable (along with the inequality), but we already know that the limit inferior of a sequence of measurable functions is measurable, and so the integrable case is the most interesting part for us.

So, we define the functions

$\displaystyle g_n(x)=\inf\limits_{i\geq n}f_i(x)$

so that each $g_n$ is integrable, each $g_n\leq f_n$ and the sequence $\{g_n\}$ is pointwise increasing. Monotonicity tells us that for each $n$ we have

$\displaystyle\int g_n\,d\mu\leq\int f_n\,d\mu$

and it follows that

$\displaystyle\lim\limits_{n\to\infty}\int g_n\,d\mu\leq\liminf\limits_{n\to\infty}\int f_n\,d\mu<\infty$

We also know that

$\displaystyle\lim\limits_{n\to\infty}g_n(x)=\liminf\limits_{n\to\infty}f_n(x)=f_*(x)$

which means we can bring the monotone convergence theorem to bear. This tells us that

$\displaystyle\int f_*\,d\mu=\lim\limits_{n\to\infty}\int g_n\,d\mu\leq\liminf\limits_{n\to\infty}\int f_n\,d\mu$

as asserted.

If it happened that $f_*$ were not integrable, then some of the $f_n$ would have to be only measurable — not integrable — themselves. And it couldn’t just be a finite number of them, or we could just drop them from the sequence. No, there would have to be an infinite subsequence of non-integrable $f_n$, which would mean an infinite subsequence of their integrals would diverge to $\infty$. Thus when we take the limit inferior of the integrals we get $\infty$, as we do for the integral of $f$ itself, and the inequality still holds.

June 16, 2010 Posted by | Analysis, Measure Theory | 6 Comments