An offhand remark in Morters and Peres' book on Brownian motion says that Brownian motion is a.s. absolutely continuous on compact intervals (see page 147, immediately preceding the statement of Theorem 6.1).
Does anyone know how to prove this result? I actually have some doubts as to whether this is correct, since elsewhere I have read that the sample paths of Brownian motion are not of bounded variation on compact intervals.
You are correct that Brownian motion is a.s. not absolutely continuous, since it does not even have finite variation on any interval. However, for the argument given in the book the authors need uniform continuity, not absolute continuity, so I suppose this is just a typo, and they meant to write "uniformly" instead of "absolutely". (And obviously this is true because Brownian motion is a.s. continuous, and $[0,t]$ is a compact interval.)