This paper deals with approximation of invariant measures of stochastic evolution processes. Under certain conditions, we demonstrate that any limit point of invariant measures of the time discrete approximations, i.e., numerical scheme, must be an invariant measure of the underlying continuous stochastic evolution processes as the step size approaches zero. As an application, we study the invariant measures of Euler-Maruyama-type stochastic difference equations with Markovian switching and discuss their convergence as the step size tends to zero.