  We study maximal averages associated with singular measures on $\rr$. Our
main result is a construction of singular Cantor-type measures supported on
sets of Hausdorff dimension $1 - \epsilon$, $0 \leq \epsilon < {1/3}$ for which
the corresponding maximal operators are bounded on $L^p(\mathbb R)$ for $p > (1
+ \epsilon)/(1 - \epsilon)$. As a consequence, we are able to answer a question
of Aversa and Preiss on density and differentiation theorems in one dimension.
Our proof combines probabilistic techniques with the methods developed in
multidimensional Euclidean harmonic analysis, in particular there are strong
similarities to Bourgain's proof of the circular maximal theorem in two
dimensions.
  Updates: Andreas Seeger has provided an argument to the effect that our
global maximal operators are in fact bounded on L^p(R) for all p>1; in
particular, it follows that our differentiation theorems are also valid for all
p>1. Furthermore, David Preiss has proved that no such differentiation theorems
(let alone maximal estimates) can hold with p=1. These arguments are included
in the new version. We have also improved the exposition in a number of places.
