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 be a positive measure space. If
- is uniformly integrable
- a.e. as and
- a.e.
then the following hold:
- .[1]
Outline of Proof
- For proving statement 1, we use Fatou's lemma:
-
- Using uniform integrability, we have where is a set such that
- By Egorov's theorem, converges uniformly on the set . for a large and . Using triangle inequality,
- Plugging the above bounds on the RHS of Fatou's lemma gives us statement 1.
-
- For statement 2, use , where and .
-
- The terms in the RHS are bounded respectively using Statement 1, uniform integrability of and Egorov's theorem for all .
-
Converse of the theorem
Let be a positive measure space. If
- ,
- and
- exists for every
then is uniformly integrable.[1]
Citations
References
- Lua error in package.lua at line 80: module 'strict' not found. MR 1681462
- Lua error in package.lua at line 80: module 'strict' not found. MR 2279622