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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038260/f0382601.png" /> converges almost-everywhere on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038260/f0382602.png" /> to a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038260/f0382603.png" />, then
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038260/f0382604.png" /></td> </tr></table>
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It was first proved by P. Fatou [[#References|[1]]]. In the statement of it <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038260/f0382605.png" /> is often replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038260/f0382606.png" />.
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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====
 
====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>
 
<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====
 
====Comments====
This result is usually called Fatou's lemma. It holds in a more general form: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038260/f0382607.png" /> is a [[Measure space|measure space]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038260/f0382608.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038260/f0382609.png" />-measurable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038260/f03826010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038260/f03826011.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038260/f03826012.png" />, then
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038260/f03826013.png" /></td> </tr></table>
+
$$
 +
\int\limits f  d \mu  \leq  \lim\limits
 +
_ {n \rightarrow \infty }  \inf \int\limits f _ {n}  d \mu .
 +
$$
  
 
It is not necessary that the sequence converges.
 
It is not necessary that the sequence converges.

Latest revision as of 19:38, 5 June 2020


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
How to Cite This Entry:
Fatou theorem (on Lebesgue integrals). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fatou_theorem_(on_Lebesgue_integrals)&oldid=46905
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article