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We study existence problem for semilinear equations with Borel measure dataand operator generated by a symmetric Markov semigroup. We assume merely thatthe nonlinear part satisfies the so-called sign condition. Using the method ofsub and supersolutions we show the existence of maximal measure for which thereexists a solution to the problem (the so-called reduced measure introduced byH. Brezis, M. Marcus and A.C. Ponce).

Nonlinear elliptic equations with integro-differential divergence form operators and measure data under sign condition on the nonlinearity