Difference between revisions of "Fatou theorem (on Lebesgue integrals)"
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− | It was first proved by P. Fatou [[#References|[1]]]. In the statement of it | + | 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 | + | 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. | 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 |
Fatou theorem (on Lebesgue integrals). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fatou_theorem_(on_Lebesgue_integrals)&oldid=46905