Characterizing the metric compactification of $L_p$ spaces by random measures

We present a complete characterization of the metric compactification of $L_p$ spaces for $1 \leq p < \infty$ . Each element of the metric compactification of $L_p$ is represented by a random measure on a certain Polish space. By way of illustration, we revisit the $L_p$ -mean ergodic theorem for $1 < p < \infty$ , and Alspach’s example of an isometry on a weakly compact convex subset of $L_1$ with no fixed points.