# 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]

then the following hold:

## Outline of Proof

For proving statement 1, we use Fatou's lemma: ${\displaystyle \int _{X}|f|d\mu \leq \liminf _{n\to \infty }\int _{X}|f_{n}|d\mu }$
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 ${\displaystyle E\in X}$ and ${\displaystyle \mu (E)<\delta }$.

## Converse of the theorem[1]

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

## Citations

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## References

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