A Class of Weak Bernoulli Transformations Associated with Representations of Real Numbers

A class W of transformations which are usually associated with representations of real numbers is considered. We prove that every T e W is weak Bernoulli. Our proof is based on Doeblin (1), Rudolfer and Wilkinson (5). Several authors (1), (2), (4), (5), (6), (7), (8), (9) have studied ergodic properties of many number- theoretical transformations but all assumed that proper atoms existed and that the improper ones could be 'approximated' by the proper ones. Our class W includes this case and not only that: it also includes the case when only improper atoms exist. Our only requirement is that T should have property (e). Both the transformation