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^[1]

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

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

then the following hold:

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

Outline of Proof

For proving statement 1, we use Fatou's lemma: $\int_X|f|d\mu\le \liminf_{n\to\infty} \int_X|f_n|d\mu$

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

For statement 2, use $\int_{X}|f-f_n|d\mu\le \int_{E}|f|d\mu+\int_{E}|f_n|d\mu+\int_{E^C}|f-f_n|d\mu$, where $E\in \mathcal{F}$ 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^[1]

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

1. $\mu(X)<\infty$,
2. $f_n\in \mathcal{L}^1(\mu)$ and
3. $\lim_{n\to\infty}\int_E f_nd\mu$ exists for every $E\in\mathcal{F}$

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

Citations

1. 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. MR 1681462
• 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. MR 2279622

External links

• Vitali convergence theorem at PlanetMath.org.