Breather surface

In real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a generalization of the better-known dominated convergence theorem of Henri Lebesgue. It is a strong condition that depends on uniform integrability. It is useful when a dominating function cannot be found for the sequence of functions in question; when such a dominating function can be found, Lebesgue's theorem follows as a special case of Vitali's.

Statement of the theorem

Let $(X,\mathcal{ℱ},\mu )$ be a positive measure space. If

1. $\mu (X)<\mathrm{\infty }$
2. $\{f_n\}$ is uniformly integrable
3. $f_n(x)\to f(x)$ a.e. as $n\to \mathrm{\infty }$ and
4. $|f(x)|<\mathrm{\infty }$ a.e.

then the following hold:

1. $f\in {\mathcal{ℒ}}^1(\mu )$
2. $\underset{n\to \mathrm{\infty }}{\lim }\underset{X}{\int }|f_n-f|d\mu =0$.

Outline of Proof

For proving statement 1, we use Fatou's lemma: $\underset{X}{\int }\left|f\right|d\mu \le {\mathrm{lim\; inf}}_{n\to \mathrm{\infty }}\underset{X}{\int }\left|f_n\right|d\mu$

• Using uniform integrability, we have $\underset{E}{\int }\left|f_n\right|d\mu <1$ where $E$ is a set such that $\mu (E)<\delta$
• By Egorov's theorem, $f_n$ converges uniformly on the set $E^C$. $\underset{E^C}{\int }|f_n-f_p|d\mu <1$ for a large $p$ and $\mathrm{\forall }n>p$. Using triangle inequality, $\underset{E^C}{\int }\left|f_n\right|d\mu \le \underset{E^C}{\int }\left|f_p\right|d\mu +1=M$
• Plugging the above bounds on the RHS of Fatou's lemma gives us statement 1.

For statement 2, use $\underset{X}{\int }|f-f_n|d\mu \le \underset{E}{\int }\left|f\right|d\mu +\underset{E}{\int }\left|f_n\right|d\mu +\underset{E^C}{\int }|f-f_n|d\mu$, where $E\in X$ and $\mu (E)<\delta$.

• The terms in the RHS are bounded respectively using Statement 1, uniform integrability of $f_n$ and Egorov's theorem for all $n>N$.

Converse of the theorem

Let $(X,\mathcal{ℱ},\mu )$ be a positive measure space. If

1. $\mu (X)<\mathrm{\infty }$,
2. $f_n\in {\mathcal{ℒ}}^1(\mu )$ and
3. $\underset{n\to \mathrm{\infty }}{\lim }\underset{E}{\int }{f}_{n}d\mu$ exists for every $E\in \mathcal{ℱ}$

then $\{f_n\}$ is uniformly integrable.

Citations

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References

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Template:MathSciNet

External links

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