What are Some Tricks to Remember Fatou’s Lemma?

For a sequence of non-negative measurable functions f_n, Fatou’s lemma is a statement about the inequality

\int \liminf_{n\rightarrow \infty} f_n \mathrm{d}\mu \leq \liminf_{n\rightarrow \infty}(\int f_n \mathrm{d} \mu)

or alternatively (for sequences of real functions dominated by some integrable function)

\limsup_{n\rightarrow \infty}(\int f_n \mathrm{d} \mu) \leq \int \limsup_{n\rightarrow \infty} f_n \mathrm{d}\mu

I keep forgetting the direction of these two inequalities. I know that using the concepts repeatedly is the best way to remember them.

But I am interested about learning intuitive tricks that people use to quickly remember them.

(For instance, to remember the direction of Jensen’s inequality, I just picture a convex function and a line intersecting it.)

Answer

I like to think of the following pictures. The first two are \int f_1 and \int f_2 respectively, but even the smaller of these is larger than the area in the third picture, which is \int \inf f_n. Of course, Fatou’s lemma is more subtle since we’re talking about the limit infimum rather than just the minimum, but for the purpose of intuition this helps to make sure the inequalities go the right way.

Attribution
Source : Link , Question Author : Learner , Answer Author : Elchanan Solomon

Leave a Comment