We study the existence of weak solutions to
(-∆)^αu+g (u) =∨ (E)
in a bounded regular domain Ω in R^N (N>1) which vanish in R^N \ Ω, where (-∆)^α denotes the fractional Laplacian with 0<α<1, ∨ is a Radon measure and g is a non-decreasing function satisfying some extra hypotheses. When g satisfies a subcritical integrability condition, we prove the existence and uniqueness of a weak solution for problem (E) for any measure. In the case where ∨ is a Dirac measure, we characterize the asymptotic behavior of the solution. When g(r)=|r|^(k-1) r with k supercritical, we show that the absolute continuity of the measure with respect to some Bessel capacity is a necessary and sufficient condition in order (E) to be solved.