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Reduced measures for semilinear elliptic equations involving Dirichlet operators

We consider elliptic equations of the form (E) $-Au=f(x,u)+μ$, where $A$ is a negative definite self-adjoint Dirichlet operator, $f$ is a function which is continuous and nonincreasing with respect to $u$ and $μ$ 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 $A=Δ$ 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 $p>1$.