  In the paper we prove the existence of probabilistic solutions to systems of
the form $-Au=F(x,u)+\mu$, where $F$ satisfies a generalized sign condition and
$\mu$ is a smooth measure. As for $A$ we assume that it is a generator of a
Markov semigroup determined by a right Markov process whose resolvent is order
compact on $L^1$. This class includes local and nonlocal operators
corresponding to Dirichlet forms as well as some operators which are not in the
variational form. To study the problem we introduce new concept of compactness
property relating the underlying Markov process to almost everywhere
convergence. We prove some useful properties of the compactness property and
provide its characterization in terms of Meyer's property (L) of Markov
processes and in terms of order compactness of the associated resolvent.
