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 converges almost-everywhere on a set to a function , then
It was first proved by P. Fatou . In the statement of it is often replaced by .
|||P. Fatou, "Séries trigonométriques et séries de Taylor" Acta Math. , 30 (1906) pp. 335–400 MR1555035 Zbl 37.0283.01|
|||S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French) MR0167578 Zbl 1196.28001 Zbl 0017.30004 Zbl 63.0183.05|
|||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|
This result is usually called Fatou's lemma. It holds in a more general form: If is a measure space, is -measurable for and for , then
It is not necessary that the sequence converges.
|[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|
Fatou theorem (on Lebesgue integrals). Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Fatou_theorem_(on_Lebesgue_integrals)&oldid=28189