History of Fatou's Lemma

Question

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?

Answer 1

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.

The details are described in Lebesgue's Theory of Integration: Its Origins and Development by Hawkins, pp. 168-172. Theorem 6.6 in the quote below is what we now call the Fatou's lemma:

"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 $f_n(x) <L$ and $g_n(x)=f(x)$ otherwise. Then for $x$ in $E$, $0<g_n(x)<L$, 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:

"Parseval's equality for any summable function, bounded or not, was obtained shortly thereafter by Pierre Fatou (1878-1929). As his doctoral thesis Fatou [1906] presented a paper on "Series trigonometriques et series de Taylor" with the principal aim "to show the advantage that can be obtained in these questions from the new notions of the measure of sets and the generalized definite integral" [1906: 335]. He was particularly encouraged in this undertaking by his "friend H. Lebesgue who has not ceased to interest himself with my researches and whose advice has been very useful to me" [1906: 338]."