 Uniform integrability gives uniform compactness of primitives

For a uniformly integrable bounded sequence of derivatives on a compact interval and a fixed initial value, its primitives are uniformly bounded and equicontinuous. The increment over any short interval is bounded by the integral of the absolute derivative on that interval. The Arzelà-Ascoli theorem gives relative uniform compactness. If the derivatives converge weakly in $L^1$, tests by interval indicators identify every pointwise limit with $u(x)=\alpha +{\int }_a^{x}v$. Hence every convergent subsequence has this same limit and the primitives converge uniformly. The limit belongs to $W^{1,1}$ and retains any common terminal value. Uniform integrability alone gives no essential-supremum bound on its derivative, so this compactness statement does not assert a Lipschitz limit.