Abstract <p>A bijective mapping of the space of operator-valued functions to the set of complex-valued finite additive cylindrical measures on the trajectory space is constructed and studied. Conditions are established under which the Cauchy problem for a first-order equation with a variable operator generates a two-parameter evolution family of operators. A representation of the solution to the Cauchy problem with a variable perturbed generator is obtained using the path integral of a perturbation-defined functional on the trajectory space with respect to a cylindrical pseudomeasure defined by the unperturbed two-parameter evolution family of operators.</p>