  In this paper we extend the stability results of [4]}. Our utility
maximization problem is defined as an essential supremum of conditional
expectations of the terminal values of wealth processes, conditioned on the
filtration at the stopping time $\tau$. To establish our results, we extend the
classical results of convex analysis to maps from $L^0$ to $L^0$. The notion of
convex compactness introduced in [7] plays an important role in our analysis.
