A dynamical system generated by a set of maps and corresponding probabilities such that at each iteration a map is selected with corresponding probability is called a random dynamical system. The paper studies position dependent random dynamical systems where probabilities are functions of the position in connection with the existence of absolutely continuous invariant measures and their stability with respect to small random perturbations. A stochastic perturbation of a random map \(T\) is defined as a family of Markov processes \(\mathcal T _\varepsilon , \varepsilon >0 \), with certain properties. The invariant measures associated with \(\mathcal T_\varepsilon\) are considered and their weak convergence to the invariant measure of the map \(T\) as \(\varepsilon \to 0\) is shown.