# Vitali convergence theorem

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

1. ${\displaystyle \mu (X)<\infty }$
2. ${\displaystyle \{f_{n}\}}$ is uniformly integrable
3. ${\displaystyle f_{n}(x)\to f(x)}$ a.e. (or converges in measure) as ${\displaystyle n\to \infty }$ and
4. ${\displaystyle |f(x)|<\infty }$ a.e.

then the following hold:

1. ${\displaystyle f\in {\mathcal {L}}^{1}(\mu )}$
2. ${\displaystyle \lim _{n\to \infty }\int _{X}|f_{n}-f|d\mu =0}$.[1]

## Outline of Proof

For proving statement 1, we use Fatou's lemma: ${\displaystyle \int _{X}|f|d\mu \leq \liminf _{n\to \infty }\int _{X}|f_{n}|d\mu }$
• Using uniform integrability there exists ${\displaystyle \delta >0}$ such that we have ${\displaystyle \int _{E}|f_{n}|d\mu <1}$ for every set ${\displaystyle E}$ with ${\displaystyle \mu (E)<\delta }$
• By Egorov's theorem, ${\displaystyle {f_{n}}}$ converges uniformly on the set ${\displaystyle E^{C}}$. ${\displaystyle \int _{E^{C}}|f_{n}-f_{p}|d\mu <1}$ for a large ${\displaystyle p}$ and ${\displaystyle \forall n>p}$. Using triangle inequality, ${\displaystyle \int _{E^{C}}|f_{n}|d\mu \leq \int _{E^{C}}|f_{p}|d\mu +1=M}$
• Plugging the above bounds on the RHS of Fatou's lemma gives us statement 1.
For statement 2, use ${\displaystyle \int _{X}|f-f_{n}|d\mu \leq \int _{E}|f|d\mu +\int _{E}|f_{n}|d\mu +\int _{E^{C}}|f-f_{n}|d\mu }$, where ${\displaystyle E\in {\mathcal {F}}}$ and ${\displaystyle \mu (E)<\delta }$.
• The terms in the RHS are bounded respectively using Statement 1, uniform integrability of ${\displaystyle f_{n}}$ and Egorov's theorem for all ${\displaystyle n>N}$.

## Converse of the theorem

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

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

then ${\displaystyle \{f_{n}\}}$ is uniformly integrable.[1]

## Citations

1. ^ a b Rudin, Walter (1986). Real and Complex Analysis. p. 133. ISBN 978-0-07-054234-1.

## References

• Folland, Gerald B. (1999). Real analysis. Pure and Applied Mathematics (New York) (Second ed.). New York: John Wiley & Sons Inc. pp. xvi+386. ISBN 0-471-31716-0. MR1681462
• Rosenthal, Jeffrey S. (2006). A first look at rigorous probability theory (Second ed.). Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd. pp. xvi+219. ISBN 978-981-270-371-2. MR2279622