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Vitali convergence theorem

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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.

Contents

Statement of the theorem

Let ( X , F , μ ) be a positive measure space. If

  1. μ ( X ) < ∞
  2. { f n } is uniformly integrable
  3. f n ( x ) → f ( x ) a.e. (or converges in measure) as n → ∞ and
  4. | f ( x ) | < ∞ a.e.

then the following hold:

  1. f ∈ L 1 ( μ )
  2. lim n → ∞ ∫ X | f n − f | d μ = 0 .

Outline of Proof

For proving statement 1, we use Fatou's lemma: ∫ X | f | d μ ≤ lim inf n → ∞ ∫ X | f n | d μ
  • Using uniform integrability there exists δ > 0 such that we have ∫ E | f n | d μ < 1 for every set E with μ ( E ) < δ
  • By Egorov's theorem, f n converges uniformly on the set E C . ∫ E C | f n − f p | d μ < 1 for a large p and ∀ n > p . Using triangle inequality, ∫ E C | f n | d μ ≤ ∫ E C | f p | d μ + 1 = M
  • Plugging the above bounds on the RHS of Fatou's lemma gives us statement 1.
    • For statement 2, use ∫ X | f − f n | d μ ≤ ∫ E | f | d μ + ∫ E | f n | d μ + ∫ E C | f − f n | d μ , where E ∈ F and μ ( E ) < δ .
  • 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 , F , μ ) be a positive measure space. If

    1. μ ( X ) < ∞ ,
    2. f n ∈ L 1 ( μ ) and
    3. lim n → ∞ ∫ E f n d μ exists for every E ∈ F

    then { f n } is uniformly integrable.

    References

    Vitali convergence theorem Wikipedia
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