  In prior work \cite{AD} of Lars Andersson and Bruce K. Driver, the path space
with finite interval over a compact Riemannian manifold is approximated by
finite dimensional manifolds $H_{x,\P} (M)$ consisting of piecewise geodesic
paths adapted to partitions $\P$ of $[0,T]$, and the associated Wiener measure
is also approximated by a sequence of probability measures on finite
dimensional manifolds. In this article, we will extend their results to the
general path space(possibly with infinite interval) over a non-compact
Riemannian manifold by using the cutoff method of compact Riemannian manifolds.
Extension to the free path space. As applications, we obtain integration by
parts formulas in the path space $W^T_x(M)$ and the free path space $W^T(M)$
respectively.
