# Characterizing the metric compactification of $L_{p}$ spaces by random measures

## TL;DR

This paper fully characterizes the metric compactification of $L_{p}$ spaces for $1 \\leq p < \\infty$, representing elements via random measures and revisiting key ergodic and isometry examples.

## Contribution

It provides a complete description of the metric compactification of $L_{p}$ spaces using random measures, extending understanding of their geometric structure.

## Key findings

- Elements of the compactification are represented by random measures.
- Revisits the $L_{p}$-mean ergodic theorem for $1 < p < \\infty$.
- Analyzes Alspach's isometry example with no fixed points.

## Abstract

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.