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 characterization of the convergence in $L^{p}$ in terms of convergence in measure and a condition related to uniform integrability.

Statement of the theorem

Let $(f_{n})_{n\in \mathbb {N} }\subseteq L^{p}(X,\tau ,\mu ),f\in L^{p}(X,\tau ,\mu )$, with $1\leq p<\infty$. Then, $f_{n}\to f$ in $L^{p}$ if and only if we have

• (i) $f_{n}$ converge in measure to $f$.
• (ii) For every $\varepsilon >0$ there exists a measurable set $E_{\varepsilon }$ with $\mu (E_{\varepsilon })<\infty$ such that for every $G\in \tau$ disjoint from $E_{\varepsilon }$ we have, for every $n\in \mathbb {N}$

$\int _{G}|f_{n}|^{p}\,d\mu <\varepsilon ^{p}$

• (iii) For every $\varepsilon >0$ there exists $\delta (\varepsilon )>0$ such that, if $E\in \tau$ and $\mu (E)<\delta (\varepsilon )$ then, for every $n\in \mathbb {N}$ we have

$\int _{E}|f_{n}|^{p}\,d\mu <\varepsilon ^{p}$

Remark: If $\mu (X)$ is finite, then the second condition is trivially true (just pick a subset that covers all but a sufficiently small portion of the whole range). Also, (i) and (iii) implies the uniform integrability of $(|f_{n}|^{p})_{n\in \mathbb {N} }$, and the uniform integrability of $(|f_{n}|^{p})_{n\in \mathbb {N} }$ implies (iii).

Outline of Proof

For proving statement 1, we use Fatou's lemma: $\int _{X}|f|\,d\mu \leq \liminf _{n\to \infty }\int _{X}|f_{n}|\,d\mu$

• Using uniform integrability there exists $\delta >0$ such that we have $\int _{E}|f_{n}|\,d\mu <1$ for every set $E$ with $\mu (E)<\delta$
• 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, $\int _{E^{C}}|f_{n}|\,d\mu \leq \int _{E^{C}}|f_{p}|\,d\mu +1=M$
• Plugging the above bounds on the RHS of Fatou's lemma gives us statement 1.

For statement 2, use $\int _{X}|f-f_{n}|\,d\mu \leq \int _{E}|f|\,d\mu +\int _{E}|f_{n}|\,d\mu +\int _{E^{C}}|f-f_{n}|\,d\mu$, where $E\in {\mathcal {F}}$ 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

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. $\lim _{n\to \infty }\int _{E}f_{n}\,d\mu$ exists for every $E\in {\mathcal {F}}$

then $\{f_{n}\}$ is uniformly integrable.

References

• Modern methods in the calculus of variations. 2007. ISBN 9780387357843.
• 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

External links

• Vitali convergence theorem at PlanetMath.