A large and central part of ergodic theory deals with abstract mathematical objects called measure-preserving transformations of a measure space. By a measure space (or probability space) we mean a set of points (like the points on the unit interval), together with a collection of subsets called measurable sets (like the Lebesgue measurable subsets of the unit interval) and a measure or probability assigned to each of these subsets (like Lebesgue measure). By a measure-preserving transformation on a measure space we mean a mapping which assigns to each point in the measure space another point in a one-to-one, onto way, and so that each measurable set is transformed onto a measurable set of the same measure.