= De la Vallée-Poussin criterion
{c}
{title2=$\sup_j\int\Psi(|v_j|)<\infty,\quad\Psi(s)/s\to\infty$}
On a finite measure space, a family is <uniformly integrable> exactly when it admits the displayed bound for some nonnegative increasing convex superlinear <function> $\Psi$. By the <Dunford-Pettis theorem> this ensures relative weak <compactness> in $L^1$. For the sufficient direction, choose $R_A$ so $\Psi(s)\geq As$ for $s>R_A$; then $\int_E|v_j|\leq R_A\mu(E)+A^{-1}\sup_j\int\Psi(|v_j|)$ gives <uniform integrability>. Conversely choose increasing thresholds $R_m\to\infty$ so $\sup_j\int_{|v_j|>R_m}|v_j|\leq2^{-m}$. The <convex function> $\Psi(s)=\sum_{m\geq1}(s-R_m)_+$ is locally a finite sum, has uniformly bounded integrals by summing these tail estimates, and is superlinear because for $s\geq2R_N$ its first $N$ terms sum to at least $Ns/2$. This proves the criterion in both directions.
Back to article page
