Uniform integrability - Imported Knowlpedia Content

A uniformly integrable family is a collection of integrable functions whose mass cannot escape to arbitrarily large values on sets of small measure. Let $(X,\mathcal{A},\mu)$ be a measure space and let $\mathcal{F}\subset L^1(X)$ (see L1 function). The family $\mathcal{F}$ is uniformly integrable if

\lim_{M\to\infty}\ \sup_{f\in\mathcal{F}} \int_{\{|f|>M\}} |f|\,d\mu = 0.

A sequence $(f_n)$ is uniformly integrable if the set $\{f_n:\ n\ge 1\}$ is uniformly integrable.

Uniform integrability is used to justify passing limits through the Lebesgue integral when one only has weaker convergence, such as convergence in measure; it rules out “spikes” that carry significant integral while living on very small sets.

so the supremum of the tail integrals does not go to $0$ as $M\to\infty$ .

Examples

If $(f_n)$ satisfies $|f_n|\le g$ almost everywhere for some $g\in L^1(X)$ , then $\{f_n\}$ is uniformly integrable; the integrable envelope $g$ forces the tail integrals over $\{|f_n|>M\}$ to be small uniformly in $n$ .
On $([0,1],\mathcal{B},\lambda)$ , the functions $f_n = n\,\mathbf{1}_{[0,1/n]}$ are not uniformly integrable: for any $M>0$ choose $n>M$ , then $\int_{\{|f_n|>M\}} |f_n|\,d\lambda = \int_0^{1/n} n\,dx = 1,$