= Uniform integrability gives uniform compactness of primitives
{title2=$v_j\rightharpoonup v\text{ in }L^1,\quad u_j(x)=\alpha+\int_a^xv_j\implies u_j\to u\text{ uniformly}$}
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 $u(x)=\alpha+\int_a^xv$. 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.
Back to article page
