We are concerned with the boundedness of the generalized fractional maximal operator \(M_{\rho ,\lambda }\) , \(\lambda \ge 1\) , on Musielak-Orlicz-Morrey spaces \(L^{\Phi ,\kappa ,\theta _1}(X)\) over unbounded metric measure spaces X, where \(\rho (x,r)\) is a positive function on \(X \times (0, \infty )\) satisfying certain conditions, as an extension of earlier results. As an important special case, we prove the boundedness of \(M_{\rho ,\lambda }\) in the framework of double phase functionals with variable exponents \(\Phi (x,t) = t^{p(x)} + a(x) t^{s(x)}, \ x \in X, \ t \ge 0\) , where \(p(x)<s(x)\) for \(x\in X\) , \(a(\cdot )\) is a non-negative, bounded and Hölder continuous function of order \(\theta \in (0,1]\) . The main novelty is that the underlying space need not be bounded, even in the case of the doubling metric measure space or the case of \(\Phi (x,t) = t^{p(x)} \left( \log (e + t)\right) ^{q(x)}\) .
