Strong uniform distributions and ergodic theorems

Abstract:

Let $G$ and $H$ be locally compact $\sigma$-compact abelian groups, $\mathcal {A}$ a mapping from $G$ to $H$, and $\{ {\mu _n}\} _{n = 1}^\infty$ a sequence of measures on $G$. We define the notions: “$\mathcal {A}$ is a uniform distribution with respect ot $\{ {\mu _n}\}$” and “$\mathcal {A}$ is a strong uniform distribution". We give a number of examples of these notions and derive some general individual ergodic theorems for measure-preserving transformations with discrete spectrum.