  The transformation of the path integral measure under the reduction procedure
in the dynamical systems with a symmetry is considered. The investigation is
carried out in the case of the Wiener--type path integrals that are used for
description of the diffusion on a smooth compact Riemannian manifold with the
given free isometric action of the compact semisimple unimodular Lie group. The
transformation of the path integral, which factorizes the path integral
measure, is based on the application of the optimal nonlinear filtering
equation from the stochastic theory. The integral relation between the kernels
of the original and reduced semigroup are obtained.
