Uniform integrability

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

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.

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

  $$\int_{\{|f_n|>M\}} |f_n|\,d\lambda = \int_0^{1/n} n\,dx = 1,$$

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