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 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[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. ↑ 1.0 1.1 Rudin, Walter (1986). Real and Complex Analysis. p. 133. ISBN 978-0-07-054234-1.

References

External links