## The Annals of Probability

- Ann. Probab.
- Volume 12, Number 2 (1984), 615-630.

### On the Cadlaguity of Random Measures

Robert J. Adler and Paul D. Feigin

#### Abstract

We consider finitely additive random measures taking independent values on disjoint Borel sets in $R^k$, and ask when such measures, restricted to some subclass $\mathscr{A}$ of closed Borel sets, possess versions which are "right continuous with left limits", in an appropriate sense. The answer involves a delicate relationship between the "Levy measure" of the random measure and the size of $\mathscr{A}$, as measured via an entropy condition. Examples involving stable measures, Dudley's class $I(k, \alpha, M)$ of sets in $R^k$ with $\alpha$-times differentiable boundaries, and convex sets are considered as special cases, and an example given to show what can go wrong when the entropy of $\mathscr{A}$ is too large.

#### Article information

**Source**

Ann. Probab. Volume 12, Number 2 (1984), 615-630.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993309

**Digital Object Identifier**

doi:10.1214/aop/1176993309

**Mathematical Reviews number (MathSciNet)**

MR735857

**Zentralblatt MATH identifier**

0542.60050

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G17: Sample path properties

Secondary: 60J30 60G15: Gaussian processes

**Keywords**

Random measures independent increments cadlag entropy convex sets

#### Citation

Adler, Robert J.; Feigin, Paul D. On the Cadlaguity of Random Measures. Ann. Probab. 12 (1984), no. 2, 615--630. doi:10.1214/aop/1176993309. https://projecteuclid.org/euclid.aop/1176993309