View source for Fatou theorem (on Lebesgue integrals)

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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 [[#References|[1]]]. In the statement of it  $  \lim\limits _ {n \rightarrow \infty }   \inf $
is often replaced by  $  \sup _ {n} $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Fatou,   "Séries trigonométriques et séries de Taylor"  ''Acta Math.'' , '''30'''  (1906)  pp. 335–400  {{MR|1555035}}  {{ZBL|37.0283.01}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Saks,   "Theory of the integral" , Hafner  (1952)  (Translated from French)  {{MR|0167578}} {{ZBL|1196.28001}} {{ZBL|0017.30004}}  {{ZBL|63.0183.05}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.P. Natanson,   "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M.  (1961)  (Translated from Russian)  {{MR|0640867}} {{MR|0409747}} {{MR|0259033}} {{MR|0063424}} {{ZBL|0097.26601}} </TD></TR></table>

====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|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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Hewitt,   K.R. Stromberg,   "Real and abstract analysis" , Springer  (1965)  {{MR|0188387}} {{ZBL|0137.03202}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.R. Halmos,   "Measure theory" , v. Nostrand  (1950)  {{MR|0033869}} {{ZBL|0040.16802}} </TD></TR></table>