Distal systems in topological dynamics and ergodic theory

We generalize a result of Lindenstrauss on the interplay between measurable and topological dynamics which shows that every separable ergodic measurably distal dynamical system has a minimal distal model. We show that such a model can, in fact, be chosen completely canonically. The construction is performed by going through the Furstenberg–Zimmer tower of a measurably distal system and showing that at each step, there is a simple and canonical distal minimal model. This hinges on a new characterization of isometric extensions in topological dynamics.