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.
- Using uniform integrability, we have
-
- 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
.
- The terms in the RHS are bounded respectively using Statement 1, uniform integrability of
-
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