Vitali convergence theorem

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In mathematics, the Vitali convergence theorem is a generalization of the more well-known dominated convergence theorem of Lebesgue. 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. The result is named after the Italian mathematician Giuseppe Vitali.

[edit] Statement of the theorem

Let (X, Σ, μ) be a measure space; let p ≥ 1 and let f_n : X → R be in the L^p space L^p(X, Σ, μ; R) for each natural number n ∈ N. Then f_n converges as n → ∞ to another measurable function f : X → R in L^p (i.e. in p^th mean) if and only if

f_n converges in measure to f;

the f_n are uniformly integrable in the sense that, for every ε > 0, there exists some t ≥ 0 such that, for all n ∈ N,

$\int_{[ | f_{n} | \geq t ]} | f_{n} (x) | \, \mathrm{d} \mu (x) < \varepsilon;$

and, for every ε > 0, there exists some set E ⊆ X with finite μ-measure such that, for all n ∈ N,

$\int_{X \setminus E} | f_{n} (x) |^{p} \, \mathrm{d} \mu (x) < \varepsilon.$

(If X has finite μ-measure, then this third condition is always satisfied, since one can take E = X in every case.)

[edit] References

Folland, Gerald B. (1999). Real analysis, Second edition, Pure and Applied Mathematics (New York), New York: John Wiley & Sons Inc., xvi+386. ISBN 0-471-31716-0. MR 1681462
Rosenthal, Jeffrey S. (2006). A first look at rigorous probability theory, Second edition, Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., xvi+219. ISBN 978-981-270-371-2. MR 2279622