A generic transformation is invertible

Abstract

We show that, on a standard non-atomic probability space, invertible measure-preserving transformations form a dense $G_\delta$ subset of the space of all measure-preserving transformations endowed with the strong (=weak) operator topology. This implies that all properties which are generic for invertible transformations are also generic for general ones. We further show that invertible Koopman operators form a dense $G_\delta$ subset of all bi-stochastic operators for the weak operator topology, and the same holds for general Koopman operators.