Vitali convergence theorem
- Vitali convergence theorem
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.
tatement 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,
::
* and, for every "ε" > 0, there exists some set "E" ⊆ "X" with finite "μ"-measure such that, for all "n" ∈ N,
::
: (If "X" has finite "μ"-measure, then this third condition is always satisfied, since one can take "E" = "X" in every case.)
References
* cite book
last = Folland
first = Gerald B.
title = Real analysis
series = Pure and Applied Mathematics (New York)
edition = Second edition
publisher = John Wiley & Sons Inc.
location = New York
year = 1999
pages = xvi+386
isbn = 0-471-31716-0 MathSciNet|id=1681462
* cite book
last = Rosenthal
first = Jeffrey S.
title = A first look at rigorous probability theory
edition = Second edition
publisher = World Scientific Publishing Co. Pte. Ltd.
location = Hackensack, NJ
year = 2006
pages = xvi+219
isbn = 978-981-270-371-2 MathSciNet|id=2279622
External links
*
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