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We discuss measures on spaces of unparametrized paths related to the Wienermeasure. These measures arise naturally in the study of one-dimensional gravitycoupled to scalar fields. Two kinds of discrete approximations are defined, thepiecewise linear and the hypercubic approximations. The convergence of theseapproximations in the sense of weak convergence of measures is proven. Wedescribe a family of sets of unparametrized paths that are analogous tocylinder sets of parametrized paths. Integrals over some of these sets areevaluated in terms of Dirichlet propagators in bounded regions.Comment: 29 pages, 2 figure

Discrete approximations to integrals over unparametrized paths