We discuss measures on spaces of unparametrized paths related to the Wiener
measure. These measures arise naturally in the study of one-dimensional gr
avity coupled to scalar fields. Two kinds of discrete approximations are de
fined, the piecewise linear and the hypercubic approximations. The converge
nce of these approximations in the sense of weak convergence of measures is
proven. We describe a family of sets of unparametrized paths that are anal
ogous to cylinder sets of parametrized paths. Integrals over some of these
sets are evaluated in terms of Dirichlet propagators in bounded regions. (C
) 2001 American Institute of Physics.