# uniformly integrable

Let $\mu$ be a positive measure  on a measurable space   . A collection  of functions $\{f_{\alpha}\}\subset\mathbf{L}^{1}(\mu)$ is uniformly integrable, if for every $\epsilon>0$, there exists $\delta>0$ such that

 $\displaystyle\Bigl{\lvert}\int_{E}f_{\alpha}\,d\mu\Bigr{\rvert}<\epsilon\quad% \textrm{whenever \mu(E)<\delta, for any \alpha.}$

The usefulness of this definition comes from the Vitali convergence theorem, which uses it to characterize the convergence of functions in $\mathbf{L}^{1}(\mu)$.

## Definition in probability theory

In probability , a different, and slightly stronger, definition of “uniform integrability”, is more commonly used:

A collection of functions $\{f_{\alpha}\}\subset\mathbf{L}^{1}(\mu)$ is uniformly integrable, if for every $\epsilon>0$, there exists $t\geq 0$ such that

 $\displaystyle\int_{[\lvert f_{\alpha}\rvert\geq t]}\lvert f_{\alpha}\rvert\,d% \mu<\epsilon\quad\textrm{for every \alpha.}$

## Properties

1. 1.

If a finite number of collections are uniformly integrable, then so is their finite union.

2. 2.

A single $f\in\mathbf{L}^{1}(\mu)$ is always uniformly integrable.

To see this, observe that $f$ must be almost everywhere non-infinite. Thus $f\cdot 1_{[\lvert f\rvert>k]}$ goes to zero a.e. as $k\to\infty$, and it is bounded   by $\lvert f\rvert$. Then $\int_{[\lvert f\rvert>k]}\lvert f\rvert d\mu\to 0$ by the dominated convergence theorem. Choosing $k$ big enough so that $\int_{[\lvert f\rvert>k]}\lvert f\rvert d\mu<\epsilon$, and letting $\delta=\epsilon/k$, we have, when $\mu(E)<\delta$,

 $\displaystyle\int_{E}\lvert f\rvert d\mu=\int_{E\cap[\lvert f\rvert\leq k]}% \lvert f\rvert d\mu+\int_{E\cap[\lvert f\rvert>k]}\lvert f\rvert d\mu\leq k\mu% (E)+\epsilon=2\epsilon\,.$

## Examples

1. 1.

If $g$ is an integrable function, then the collection consisting of all measurable functions  $f$ dominated by $g$ — that is, $\lvert f\rvert\leq g$ — is uniformly integrable.

2. 2.

If $X$ is a $\mathbf{L}^{1}$ random variable  on a probability space $\Omega$, then the set of all of its conditional expectations,

 $\{\mathbb{E}[X\mid\mathcal{G}]\colon\mathcal{G}\text{ is a \sigma-algebra of% \Omega}\}\,,$

is always uniformly integrable.

3. 3.

If there is an unbounded  increasing function $\phi\colon[0,\infty)\to[0,\infty)$ such that

 $\int\lvert f_{\alpha}\rvert\phi(\lvert f_{\alpha}\rvert)\,d\mu$

is uniformly bounded for all $\alpha$, then the collection $\{f_{\alpha}\}$ 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 UniformlyIntegrable 2013-03-22 15:22:55 2013-03-22 15:22:55 stevecheng (10074) stevecheng (10074) 23 stevecheng (10074) Definition msc 28A20 uniform integrability uniform absolute continuity VitaliConvergenceTheorem ConditionalExpectationsAreUniformlyIntegrable