## The Lebesgue Convergence Theorems

Last updates: 2 April 2011

## The Lebesgue Convergence Theorems

(Lebesgue's monotone convergence theorem)
Let $\left(X,ℳ\right)$ be a measurable space and let $\mu :ℳ\to \left[0,\infty \right]$ be a positive measure on $\left(X,ℳ\right)$.
Let ${f}_{n}:X\to \left[0,\infty \right]$, $n\in {ℤ}_{>0}$, 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 $\underset{n\to \infty }{\mathrm{lim}}{f}_{n}\left(x\right)$ exists.
Let $f:X\to \left[0,\infty \right]$ be given by $f\left(x\right)=\underset{n\to \infty }{\mathrm{lim}}{f}_{n}\left(x\right)$.
Then $f$ is measurable and
 ${\int }_{X}\left(\underset{n\to \infty }{\mathrm{lim}}{f}_{n}\right)\phantom{\rule{0.2em}{0ex}}d\mu =\underset{n\to \infty }{\mathrm{lim}}\left({\int }_{X}{f}_{n}\phantom{\rule{0.2em}{0ex}}d\mu \right)$.

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

 ${\int }_{X}\left(\underset{n\to \infty }{\mathrm{liminf}}{f}_{n}\right)\phantom{\rule{0.2em}{0ex}}d\mu \le \underset{n\to \infty }{\mathrm{liminf}}\left({\int }_{X}{f}_{n}\phantom{\rule{0.2em}{0ex}}d\mu \right)$.

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

 if $x\in X$ then $\underset{n\to \infty }{\mathrm{lim}}{f}_{n}\left(x\right)$ exists.
Assume that there exists $g\in {L}^{1}\left(\mu \right)$ such that
 if $n\in {ℤ}_{>0}$ and $x\in X$ then $|{f}_{n}\left(x\right)|\le g\left(x\right)$.
Then
(a)   $f\in {L}^{1}\left(\mu \right)$,
(b)   $\underset{n\to \infty }{\mathrm{lim}}\left({\int }_{X}|{f}_{n}-f|\phantom{\rule{0.2em}{0ex}}d\mu \right)=0$,
(c)   ${\int }_{X}\left(\underset{n\to \infty }{\mathrm{lim}}{f}_{n}\right)\phantom{\rule{0.2em}{0ex}}d\mu =\underset{n\to \infty }{\mathrm{lim}}\left({\int }_{X}{f}_{n}\phantom{\rule{0.2em}{0ex}}d\mu \right)$.

## 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.