De la Vallée-Poussin criterion (${\sup }_j\int \Psi (|v_j|)<\infty ,\Psi (s)/s\to \infty$)

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 $\Psi (s)\ge As$ for $s>R_A$; then ${\int }_E|v_j|\le R_A\mu (E)+A^{-1}{\sup }_j\int \Psi (|v_j|)$ gives uniform integrability. Conversely choose increasing thresholds $R_m\to \infty$ so ${\sup }_j{\int }_{|v_j|>R_m}|v_j|\le 2^{-m}$. The convex function $\Psi (s)={\sum }_{m\ge 1}(s-R_m{)}_{+}$ is locally a finite sum, has uniformly bounded integrals by summing these tail estimates, and is superlinear because for $s\ge 2R_N$ its first $N$ terms sum to at least $Ns/2$. This proves the criterion in both directions.

Choose increasing $R_m\to \infty$ so $H(t)\ge m^2|t|$ whenever $|t|\ge R_m$. Put $\Psi (s)={\sum }_{m\ge 1}(s-R_m{)}_{+}$ for $s\ge 0$. This is nonnegative, increasing and convex. If $m$ terms are active, then $\Psi (s)\le ms\le H(t)$ for $s=|t|$. For every fixed $N$, $s\ge 2R_N$ gives $\Psi (s)\ge Ns/2$, proving superlinearity. Thus a nonconvex superlinear energy lower bound still supplies the convex function needed in the De la Vallée-Poussin criterion. A continuous superlinear function not initially nonnegative can first have its finite global minimum subtracted.