Path integrals and ordering of operators

A method, not based on finite-multiplicity approximations, is proposed for constructing the Feynman path integral for a particle in a curved space whose geometry is defined by the kinetic energy. For the example of a system with the Hamiltonian $H=f^2(x)p^2$ (and some other systems) it is shown that the path integral can be obtained by a change of the variables of integration from a Gaussian functional integral, and this then makes it possible to associate the function $H$ uniquely with an operator. The procedure for constructing the operator corresponding to a classical function of the coordinates and the momenta, for given form of the Hamiltonian, is also considered.