Measure-Invariant Sets

Abstract

Let $\mu$ be a probability measure on $(\mathscr{R}, \mathscr{B})$, where $\mathscr{R}$ is the real line and $\mathscr{B}$ the family of Borel sets on $\mathscr{R}$. A measurable set `$A$' is called $\mu$-invariant if $\mu(A + \theta) = \mu(A) \mathbf{\forall} \theta, -\infty < \theta < \infty$. Let $\mathscr{A}(\mu)$ denote the family of all $\mu$-invariant sets. Let $S(\mu)$ denote the set where the characteristic function of $\mu$ vanishes. In this paper we establish the following results concerning $\mu$-invariant sets. (i) Suppose $S(\mu) \cap \overline{\lbrack S(\mu) \oplus S(\mu) \rbrack}$ is compact. Then $A$ is $\mu$-invariant implies $\mu(A) = 0, \frac{1}{2}, 1$. (ii) Fourier series representations are developed to study $\mu$-invariant sets. (iii) Dependence of $\mathscr{A}(\mu \ast \nu)$ on $\mathscr{A}(\mu)$ and $\mathscr{A}(\nu)$ is examined and representations for $\mu \ast \nu$-invariant sets are derived. (iv) Dependence of $\mathscr{A}(\mu)$ on $S(\mu)$ is carefully examined. (v) $A$ conjecture that $\mathscr{A}(\mu) \subset \mathscr{A}(\nu)$ implies that $\mu$ is a factor of $\nu$ is shown to be false.