For a simple point process $\Xi$ on a suitable topological space,
the associated Palm distribution at a point s may be approximated by the
conditional distribution, given that $\Xi$ hits a small neighborhood of $s$. To
study the corresponding approximation problem for more general random sets, we
develop a general duality theory, which allows the Palm distributions with
respect to an associated random measure to be expressed in terms of conditional
densities with suitable martingale and continuity properties. The stated
approximation property then becomes equivalent to a certain asymptotic relation
involving conditional hitting probabilities. As an application, we consider the
Palm distributions of regenerative sets with respect to their local time random
measures.