Fatou theorem (on Lebesgue integrals)

A theorem on passing to the limit under a Lebesgue integral: If a sequence of measurable (real-valued) non-negative functions $ f _ {1} , f _ {2} \dots $ converges almost-everywhere on a set $ E $ to a function $ f $, then

$$ \int\limits _ { E } f ( x) dx \leq \ \lim\limits _ {n \rightarrow \infty } \inf \ \int\limits _ { E } f _ {n} ( x) dx. $$

It was first proved by P. Fatou [1]. In the statement of it $ \lim\limits _ {n \rightarrow \infty } \inf $ is often replaced by $ \sup _ {n} $.

References

[1]	P. Fatou, "Séries trigonométriques et séries de Taylor" Acta Math. , 30 (1906) pp. 335–400 MR1555035 Zbl 37.0283.01
[2]	S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French) MR0167578 Zbl 1196.28001 Zbl 0017.30004 Zbl 63.0183.05
[3]	I.P. Natanson, "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M. (1961) (Translated from Russian) MR0640867 MR0409747 MR0259033 MR0063424 Zbl 0097.26601

Comments

This result is usually called Fatou's lemma. It holds in a more general form: If $ ( \mathfrak X , {\mathcal A} , \mu ) $ is a measure space, $ f _ {n} : \mathfrak X \rightarrow [ 0 , \infty ] $ is $ {\mathcal A} $- measurable for $ n = 1 , 2 \dots $ and $ f ( x) = \lim\limits _ {n \rightarrow \infty } \inf f _ {n} ( x) $ for $ x \in \mathfrak X $, then

$$ \int\limits f d \mu \leq \lim\limits _ {n \rightarrow \infty } \inf \int\limits f _ {n} d \mu . $$

It is not necessary that the sequence converges.

References

[a1]	E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) MR0188387 Zbl 0137.03202
[a2]	P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802