  We consider elliptic equations of the form (E) $-Au=f(x,u)+\mu$, where $A$ is
a negative definite self-adjoint Dirichlet operator, $f$ is a function which is
continuous and nonincreasing with respect to $u$ and $\mu$ is a Borel measure
of finite potential. We introduce a probabilistic definition of a solution of
(E), develop the theory of good and reduced measures introduced by H. Brezis,
M. Marcus and A.C. Ponce in the case where $A=\Delta$ and show basic properties
of solutions of (E). We also prove Kato's type inequality. Finally, we
characterize the set of good measures in case $f(u)=-u^p$ for some $p>1$.
