## The Annals of Probability

### Dimensional Properties of One-Dimensional Brownian Motion

Robert Kaufman

#### Abstract

For each closed set $F \subseteq \lbrack 0, 1\rbrack, \dim X(F + t) = \min(1, 2 \dim F)$ for almost all $t > 0. (X$ is one-dimensional Brownian motion). For each closed set $F \subseteq \lbrack 0, 1 \rbrack$ of dimension greater than $1/2, m(X(F + t)) > 0$ for almost all $t > 0$. These statements are true outside a single null-set in the sample space.

#### Article information

Source
Ann. Probab., Volume 17, Number 1 (1989), 189-193.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176991503

Digital Object Identifier
doi:10.1214/aop/1176991503

Mathematical Reviews number (MathSciNet)
MR972780

Zentralblatt MATH identifier
0672.60077

JSTOR