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 $(X,{\mathcal {F}},\mu )$ 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: $\int _{X}|f|d\mu \leq \liminf _{n\to \infty }\int _{X}|f_{n}|d\mu$

• 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, $\int _{E^{C}}|f_{n}|d\mu \leq \int _{E^{C}}|f_{p}|d\mu +1=M$
• 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 $E\in X$ 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. $\mu (X)<\infty$,
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

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External links

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