 Concentrating spikes distinguish weak Lp and weak L1 compactness

The displayed spikes have $L^p$ norm one. Pairing with $g\in L^{p^{\prime }}$ and applying Holder inequality bounds the pairing by $({\int }_0^{1/j}|g{|}^{p^{\prime }}{)}^{1/p^{\prime }}$, which tends to zero. At exponent one, however, $j{1}_{(0,1/j)}$ cannot converge weakly in $L^1$: every interval away from zero forces any limit to vanish there, while the constant test function preserves integral one. Their failure of uniform integrability records exactly the obstruction in the Dunford-Pettis theorem.