Abstract: Let (X,d,μ) be a non-homogeneous metric measure space which satisfies the upper doubling and geometrically doubling conditions, and T_α be the generalized fractional integral operator on (X,d,μ). By establishing pointwise inequality of sharp maximum function, we obtain that T_α is bounded from the weighted Lebesgue space L^p(ω) to the weighted weak Lebesgue space WL^p,κ,η(ω), and also from the weighted Morrey space L^p,κ,η(ω) to the weighted weak Morrey space WL^p,κ,η(ω).

Key words: generalized fractional integral, weighted weak estimate, weighted Morrey space, non-homogeneous metric measure space