Fatou’s Lemma

Fatou’s Lemma
Let (f_n) be a sequence of nonnegative measurable functions, then \displaystyle\int\liminf_{n\to\infty}f_n\,d\mu\leq\liminf_{n\to\infty}\int f_n\,d\mu.

A brilliant graphical way to remember Fatou’s Lemma (taken from the site http://math.stackexchange.com/questions/242920/what-are-some-tricks-to-remember-fatous-lemma).

The first two are f1 and f2 respectively, but even the smaller of these is larger than the area in the third picture, which is inf fn.



About mathtuition88

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2 Responses to Fatou’s Lemma

  1. nkpithwa says:

    Visual memory is the most powerful. Thanks for culling this for us. BTW, it reminded me of my first encounter with it in baby Rudin. 🙂 That further reminds me, ( a slight digress) have a look at “Proofs without Words” (Exercises in Visual thinking) by Roger B. Nelsen, published by MAA. In a nutshell, “draw as much as you can, wherever you can!”. (Richard Feynman advocated drawing even to understand/develop software). On the other hand, William B. Thurston, famous topologist (Fields Medallist), who passed away some time back, used to say that “thinking is seeing” !

    Liked by 1 person

    • Yeah, drawing pictures is a way to intuitively remember or understand results, that complements the usual rigorous proof. After viewing this picture, one can no longer worry about forgetting the direction of the inequality in Fatou’s Lemma!


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