Namespaces
Variants
Actions

Difference between revisions of "Fatou theorem (on Lebesgue integrals)"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (MR/ZBL numbers added)
Line 6: Line 6:
  
 
====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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Saks,  "Theory of the integral" , Hafner  (1952)  (Translated from French)</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)</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>
  
  
Line 18: Line 18:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Hewitt,  K.R. Stromberg,  "Real and abstract analysis" , Springer  (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.R. Halmos,  "Measure theory" , v. Nostrand  (1950)</TD></TR></table>
+
<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>

Revision as of 11:59, 27 September 2012

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 [1]. In the statement of it is often replaced by .

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 is a measure space, is -measurable for and for , then

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=12617
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article