The Unapologetic Mathematician

Mathematics for the interested outsider

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