Uniform integrability

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Uniform integrability is an important concept in real analysis, functional analysis and measure theory, and plays a vital role in the theory of martingales.


Let be a positive measure space. A set is called uniformly integrable if to each there corresponds a such that

whenever and

Formal definition

The following definition applies.[1]

Related corollaries

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  • Definition 1 could be rewritten by taking the limits as
Clearly , and indeed for all n. However,
and comparing with definition 1, it is seen that the sequence is not uniformly integrable.
Non-UI sequence of RVs. The area under the strip is always equal to 1, but pointwise.
and bounding each of the two, it can be seen that a uniformly integrable random variable is always bounded in .
then the class of random variables is uniformly integrable.

Relevant theorems

A class of random variables is uniformly integrable if and only if it is relatively compact for the weak topology .
The family is uniformly integrable if and only if there exists a non-negative increasing convex function such that

Relation to convergence of random variables



  1. {{#invoke:citation/CS1|citation |CitationClass=book }}
  2. Dellacherie, C. and Meyer, P.A. (1978). Probabilities and Potential, North-Holland Pub. Co, N. Y. (Chapter II, Theorem T25).
  3. Meyer, P.A. (1966). Probability and Potentials, Blaisdell Publishing Co, N. Y. (p.19, Theorem T22).
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  • J. Diestel and J. Uhl (1977). Vector measures, Mathematical Surveys 15, American Mathematical Society, Providence, RI ISBN 978-0-8218-1515-1