uniformly integrable
Let be a positive measure on a measurable space. A collection of functions is uniformly integrable, if for every , there exists such that
(The absolute value sign outside of the integral above may appear under the integral sign instead without affecting the definition.)
The usefulness of this definition comes from the Vitali convergence theorem, which uses it to characterize the convergence of functions in .
Definition in probability theory
In probability , a different, and slightly stronger, definition of “uniform integrability”, is more commonly used:
A collection of functions is uniformly integrable, if for every , there exists such that
Assuming is a probability measure, this definition is equivalent to the previous one together with the condition that is uniformly bounded for all .
Properties
-
1.
If a finite number of collections are uniformly integrable, then so is their finite union.
-
2.
A single is always uniformly integrable.
To see this, observe that must be almost everywhere non-infinite. Thus goes to zero a.e. as , and it is bounded by . Then by the dominated convergence theorem. Choosing big enough so that , and letting , we have, when ,
Examples
-
1.
If is an integrable function, then the collection consisting of all measurable functions dominated by — that is, — is uniformly integrable.
-
2.
If is a random variable on a probability space , then the set of all of its conditional expectations,
is always uniformly integrable.
-
3.
If there is an unbounded increasing function such that
is uniformly bounded for all , then the collection is uniformly integrable.
References
- 1 Kai Lai Chung. A Course in Probability Theory, third ed. Academic Press, 2001.
- 2 Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications, second ed. Wiley-Interscience, 1999.
- 3 Jeffrey S. Rosenthal. A First Look at Rigorous Probability Theory. World Scientific, 2003.
Title | uniformly integrable |
---|---|
Canonical name | UniformlyIntegrable |
Date of creation | 2013-03-22 15:22:55 |
Last modified on | 2013-03-22 15:22:55 |
Owner | stevecheng (10074) |
Last modified by | stevecheng (10074) |
Numerical id | 23 |
Author | stevecheng (10074) |
Entry type | Definition |
Classification | msc 28A20 |
Synonym | uniform integrability |
Synonym | uniform absolute continuity |
Related topic | VitaliConvergenceTheorem |
Related topic | ConditionalExpectationsAreUniformlyIntegrable |