On Ergodic Sequences of Measures

Abstract

Let $Z$ be the group of integers and $\bar Z$ its Bohr compactification. A sequence of probability measures $\{ {\mu _n},n = 1,2, \ldots \}$ defined on $Z$ is said to be ergodic provided ${\mu _n}$ converges weakly to $\bar \mu$, the Haar measure on $\bar Z$. Let ${A_n} \subset Z,n = 1,2, \ldots$ and define ${\mu _n}$ by ${\mu _n}(B) = |{A_n} \cap B|/|{A_n}|$ where $|B|$ is the cardinality of $B$. Then it is easy to show that if $|{A_n} \cap {A_n} + k|/|{A_n}| \to 1$ for every $k \in Z$, then ${\mu _n}$ is ergodic. Let $0 \leq {p_k} \leq 1$. In this paper we construct (random) sequences $\{ {\mu _n}\}$ which are ergodic, and such that $\lim (|{A_n} \cap {A_n} + k|/|{A_n}|) = {p_k}$, for every $k \in Z$.