The Lebesgue Convergence Theorems

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 2 April 2011

The Lebesgue Convergence Theorems

(Lebesgue's monotone convergence theorem)
Let $(X,ℳ)$ be a measurable space and let $\mu :ℳ\to [0,\infty ]$ be a positive measure on $(X,ℳ)$.
Let $f_n:X\to [0,\infty ]$, $n\in {\mathbb{Z}}_{>0}$, be a sequence of measurable functions such that

(a)   If $x\in X$ then $0\le f_1\left(x\right)\le f_2\left(x\right)\le \cdots \le \infty$, and

(b)   If $x\in X$ then ${\displaystyle \underset{n\to \infty }{\lim }{f}_n\left(x\right)}$ exists.

Let $f:X\to [0,\infty ]$ be given by $f\left(x\right)={\displaystyle \underset{n\to \infty }{\lim }{f}_n\left(x\right)}$.
Then $f$ is measurable and

${\displaystyle {\int }_X\left(\underset{n\to \infty }{\lim }{f}_n\right)d\mu }={\displaystyle \underset{n\to \infty }{\lim }\left({\int }_X{f}_{n}d\mu \right)}$.

(Fatou's lemma)
Let $(X,ℳ)$ be a measurable space and let $\mu :ℳ\to [0,\infty ]$ be a positive measure on $(X,ℳ)$.
Let $f_n:X\to [0,\infty ]$, $n\in {\mathbb{Z}}_{>0}$, be a sequence of measurable functions. Then

${\displaystyle {\int }_X\left(\underset{n\to \infty }{\mathrm{liminf}}{f}_n\right)d\mu }\le {\displaystyle \underset{n\to \infty }{\mathrm{liminf}}\left({\int }_X{f}_{n}d\mu \right)}$.

(Lebesgue's dominated convergence theorem)
Let $(X,ℳ)$ be a measurable space and let $\mu :ℳ\to [0,\infty ]$ be a positive measure on $(X,ℳ)$.
Let $f_n:X\to ℂ$, $n\in {\mathbb{Z}}_{>0}$, be a sequence of measurable functions such that

if $x\in X$ then ${\displaystyle \underset{n\to \infty }{\lim }{f}_n\left(x\right)}$ exists.

Assume that there exists $g\in L^1\left(\mu \right)$ such that

if $n\in {\mathbb{Z}}_{>0}$ and $x\in X$ then $\left|f_n\left(x\right)\right|\le g\left(x\right)$.

Then

(a)   $f\in L^1\left(\mu \right)$,

(b)   ${\displaystyle $ \underset{n\to \infty }{\lim }\left({\int }_X\right|f_n-f\left|d\mu \right)$}=0$,

(c)   ${\displaystyle {\int }_X\left(\underset{n\to \infty }{\lim }{f}_n\right)d\mu }={\displaystyle \underset{n\to \infty }{\lim }\left({\int }_X{f}_{n}d\mu \right)}$.

Notes and References

These notes were written for a course in "Measure Theory" at the Masters level at University of Melbourne. This presentation follows [Ru, Chapters 1-6].

References

[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR0924157.

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