  We study the non-uniformity of probability measures on the interval and the
circle. On the interval, we identify the Wasserstein-$p$ distance with the
classical $L^p$-discrepancy. We thereby derive sharp estimates in Wasserstein
distances for the irregularity of distribution of sequences on the interval and
the circle. Furthermore, we prove an $L^p$-adapted Erd\H{o}s$-$Tur\'{a}n
inequality.
