The Annals of Probability

Measure-Invariant Sets

Julius R. Blum and Pramod K. Pathak

Full-text: Open access


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.

Article information

Ann. Probab., Volume 1, Number 4 (1973), 590-602.

First available in Project Euclid: 19 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60B05: Probability measures on topological spaces
Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 62A05 43A05: Measures on groups and semigroups, etc.

Measure-invariant sets harmonic analysis completeness Fourier series translation parameter


Blum, Julius R.; Pathak, Pramod K. Measure-Invariant Sets. Ann. Probab. 1 (1973), no. 4, 590--602. doi:10.1214/aop/1176996888.

Export citation