Compact measure-preserving system
= Compact measure-preserving system
A probability <measure-preserving system> is compact if every $L^2$ observable has a relatively compact <Koopman operator> orbit in the global $L^2$ norm. For an invertible system one may use all integer times; for a general transformation use nonnegative times. This is a measure-theoretic definition, stronger than compactness of the underlying topological space.