The long-time behavior of the L2 energy for linear KdV-type equations with localized damping on the torus is studied. We show that exponential decay occurs provided the damping term is effective on measurable space-time sets of positive measure. To this end, we establish new unique continuation inequalities for observation sets that are measurable in space-time. This is based on joint work with Jingrui Niu and Shengquan Xiang.