# History of Fatou's Lemma

This question concerns Fatou's Lemma:

Let $$(f_n)$$ be a sequence of nonnegative measurable functions on a measure space $$X$$. Define $$f: X\to [0,+\infty]$$ by $$f(x) :=\liminf_{n\to\infty} f_n(x)$$, for every $$x\in X$$. Then $$f$$ is measurable and $$\int_X f\,d\mu \le \liminf_{n\to\infty} \int_X f_n\,d\mu$$.

This result is, of course, very famous now, but I'm having a hard time tracing back its genesis. After his thesis on integration theory, Fatou has worked in several areas, including complex analysis and iterated functions; so a natural guess would be that this Lemma originates from his PhD thesis, but I couldn't find any precise quote. In fact, it seems that the first occurrence of the name "Fatou's Lemma" (par excellence) can be found in a 1932 article by Doob, although I'm not completely sure Doob was referring to precisely what today goes under the same name.

Does anyone have closer information?

• See pp. 375-376 of Séries trigonométriques et séries de Taylor, Acta Mathematica 30 (1906), 335-400 and this google search. Jan 18, 2021 at 19:04
• An interesting question to ask students when they learn Fatou's lemma: "What was it a lemma for?" You can also do that with Zorn's lemma. Or other standard results well-kown as "lemmas". Jan 19, 2021 at 19:46

Yes, Fatou formulated the lemma the modern way that Doob refers to. It appears in Fatou's paper Series trigonometriques et series de Taylor, p. 375 (Acta Math., 30 (1906) 335-400), which he presented as his doctoral thesis. It was a side result in proving that if $$f$$ is bounded and analytic for $$|z|<1$$, then $$\lim_{r\nearrow1} f(re^{it})$$ exists almost everywhere. This, in turn, was part of his proof of a general form of Parseval's identity for trigonometric series of summable functions. Fatou's research was personally encouraged and aided by Lebesgue himself.
"Theorem 6.6 is similar to the theorem of Beppo Levi referred to in 5.3. Fatou proved his theorem as follows: Let $$L$$ denote a positive number, and consider the set $$E$$ of all $$x$$ such that $$f(x) < L$$. A sequence of functions $$g_n$$ is constructed by setting $$g_n(x) = f_n(x)$$ if $$f_n(x) and $$g_n(x)=f(x)$$ otherwise. Then for $$x$$ in $$E$$, $$0, and $$g_n(x)$$ converges to $$f(x)$$. Lebesgue's Theorem 5.3 applies and shows that $$\int_E f= \lim \int_E g_n <\liminf \int_a^b f_n$$. The lemma then follows, since the right-hand side of the inequality is independent of $$L$$."
The theorem 5.3 referred to is Lebesgue's extension of the theorems of Arzela and Osgood to convergence of bounded measurable functions. The theorem of Levi was that if $$f_n$$ is a non-decreasing sequence of summable functions such that $$\lim \int f_n$$ is finite, then $$f(x) = \lim f_n (x)$$ is finite almost everywhere and summable with $$\int f_n= \lim \int f_n$$. It was published by Levi the same year in Sopra l'integrazione delle serie (Milano Ist. Lomb. Rend., (2) 39 (1906) 775-80) and used by Fubini to prove his theorem on exchanging the order of integration. Here is some broader context from Hawkins: